Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

Abstract

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart ($m=1$) or Morley ($m=2$) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated $L^2$ error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

1 Introduction

Motivation. Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator from a global postprocessing of respective nonconforming finite element eigensolvers like the Crouzeix–Raviart resp. Morley finite element method (FEM) for $m=1$ resp. $m=2$ [15, 16]. The maximal mesh-size $h_{\max }$ enters as an explicit parameter and this can be non-effective for an imperative adaptive mesh-refinement. This has recently motivated the design of extra-stabilized nonconforming finite element eigensolvers for $m=1,2$ that directly compute GLB under moderate mesh-size restrictions and allow an efficacious adaptive mesh-refinement [11, 24, 27]. The striking superiority of those adaptive schemes has been displayed in numerical experiments in [11, 24] and motivates the mathematical analysis of optimal convergence rates in this paper. This appears to be the first method that combines the localization of eigenvalues as GLB with their efficient approximation.

Model problem. The continuous eigenvalue problem (EVP) seeks eigenpairs $(\lambda , u) \in {\mathbb {R}}^+\times (V{\setminus }\{0\})$ with

$$\begin{aligned} a(u,v)= \lambda \, b(u,v)\quad \text {for all } v\in V \end{aligned}$$

(1.1)

in the Hilbert space $V:=H^{m}_0(\Omega )$ with its energy scalar product $a(\bullet ,\bullet ):=(D^m\bullet , D^m\bullet )_{L^2(\Omega )}$ with the gradient $D^1:=\nabla$ or the Hessian $D^2$ and the $L^2$ scalar product $b(\bullet ,\bullet ):=(\bullet , \bullet )_{L^2(\Omega )}$ on a bounded polyhedral Lipschitz domain $\Omega \subset {\mathbb {R}}^3$. The infinite but countably many eigenvalues $0<\lambda _1\leqslant \lambda _2\leqslant \dots$ with $\lim _{j\rightarrow \infty }\lambda _j=\infty$ in (1.1) are enumerated in ascending order counting multiplicities [6, 7].

Discretization. The discrete space $\varvec{V_h}\hspace{-.2em}= \hspace{-.2em}P_{m}({\mathcal {T}})\hspace{-.1em}\times \hspace{-.1em}V({{\mathcal {T}}}) \hspace{-.2em}\subset \hspace{-.2em} P_{m}({\mathcal {T}})\hspace{-.1em}\times \hspace{-.1em} P_{m}({\mathcal {T}})$ consists of piecewise polynomials of degree at most m on the shape-regular triangulation ${\mathcal {T}}$ of $\Omega \subset {\mathbb {R}}^3$ into closed tetrahedra. Throughout this paper, $V({\mathcal {T}})$ abbreviates the Crouzeix–Raviart finite element space $\textit{CR}^1_0({\mathcal {T}})$ [25] for $m=1$ and the Morley finite element space $M({\mathcal {T}})$ [40, 41] for $m=2$. The algebraic eigenvalue problem seeks eigenpairs $(\lambda _h, \varvec{u_h}) \in {\mathbb {R}}^+\times (\varvec{V_h}{\setminus }\{0\})$ with

$$\begin{aligned} \varvec{a_h}(\varvec{u_h},\varvec{v_h}) =\lambda _h \varvec{b_h}(\varvec{u_h},\varvec{v_h})\quad \text {for all }\varvec{v_h}\in \varvec{V_h}. \end{aligned}$$

(1.2)

The discrete scalar product $\varvec{a_h}$ contains the scalar product $a_{\textrm{pw}}(\bullet , \bullet ):=(D^m_\textrm{pw}\bullet ,D^m_\textrm{pw}\bullet )_{L^2(\Omega )}$ of the piecewise derivatives of order m and some stabilization with explicit (known) constant $\kappa _m>0$ from [24], while the bilinear form $\varvec{b_h}$ is the $L^2$ scalar product $b(\bullet ,\bullet )$ of the piecewise polynomial components,

$$\begin{aligned} \varvec{a_h}(\varvec{v_h},\varvec{w_h})&=a_{\textrm{pw}}(v_{\textrm{nc}},w_{\textrm{nc}}) +\kappa _{m}^{-2}(h_{{\mathcal {T}}}^{-2{m}}(v_{\textrm{pw}}-v_{\textrm{nc}}), w_{\textrm{pw}}-w_{\textrm{nc}})_{L^2(\Omega )},\\ \varvec{b_h}(\varvec{v_h},\varvec{w_h})&=b(v_{\textrm{pw}},w_{\textrm{pw}}) \qquad \text {for all }\varvec{v_h}=(v_{\textrm{pw}},v_{\textrm{nc}}),\, \varvec{w_h}=(w_{\textrm{pw}},w_{\textrm{nc}})\in \varvec{V_h}. \end{aligned}$$

The piecewise constant mesh-size function $h_{{\mathcal {T}}}\in P_0({\mathcal {T}})$ has the value $h_{{\mathcal {T}}}|_T=h_T:=diam (T)$ in each tetrahedron $T\in {\mathcal {T}}$ and $h_{\max }:=\max _{T\in {\mathcal {T}}}h_T$ denotes the maximal mesh-size. The $M:=dim (P_m({\mathcal {T}}))$ finite discrete eigenvalues of (1.2) are enumerated in ascending order $0< \lambda _h(1)\leqslant \lambda _h(2)\leqslant \cdots \leqslant \lambda _h(M)<\infty$ counting multiplicity.

GLB. For the biharmonic operator ($m=2$) the discrete eigenvalue problem (1.2) is analysed in [24]. For the Laplace operator ($m=1$) in 2D, (1.2) describes the lowest-order skeleton method in [27]; for 3D it is different and suggested in [24]. The discrete eigenvalue problem (1.2) directly computes guaranteed lower bounds [24, Thm. 1.1] in that

$$\begin{aligned} \min \{\lambda _h({k}),\lambda _k\} \kappa ^2_m h_{\max }^{2{m}}\leqslant 1 \quad \text { implies } \quad \lambda _h({k})\leqslant \lambda _{k} \quad \text { for all }k=1,\ldots , M. \end{aligned}$$

(1.3)

AFEM. The adaptive algorithm [12, 26, 30, 39] is based on the refinement indicator $\eta (T)$ defined in (1.4) below for any triangulation ${\mathcal {T}}$ and any tetrahedron $T\in {\mathcal {T}}$. Let $\big (\lambda _h, \varvec{u_h}\big )\in {\mathbb {R}}^+\times \varvec{V_h}$ denote the k-th eigenpair of (1.2) with $\lambda _h:=\lambda _h(k)$ and $\varvec{u_h}=(u_{\textrm{pw}},u_{\textrm{nc}})\in \varvec{V_h}$. For any tetrahedron $T\in {\mathcal {T}}$ with volume |T| and set of faces ${\mathcal {F}}(T)$, the local estimator contribution $\eta ^2(T)=(\eta (T))^2$ reads

$$\begin{aligned} \eta ^2(T) = |T|^{2{m}/3}\Vert \lambda _h u_{\textrm{nc}} \Vert ^2_{L^2(T)} +|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)}\Vert [{D}^{m}_{\textrm{pw}} u_{\textrm{nc}}]_F\times \nu _F\Vert ^2_{L^2(F)} \end{aligned}$$

(1.4)

with the tangential components $[D^m_{\textrm{pw}} u_{\textrm{nc}}]_F\times \nu _F$ of the jump $[D^m_{\textrm{pw}} u_{\textrm{nc}}]_F$ along any face $F\in {\mathcal {F}}(T)$ and the (piecewise) gradient $D^1_{\textrm{pw}}\hspace{-.2em}=\hspace{-.2em}\nabla _{\textrm{pw}}$ (${m}\hspace{-.2em}=\hspace{-.2em}1$) or Hessian $D^2_{\textrm{pw}}$ (${m\hspace{-.2em}}=\hspace{-.2em}2$). Let ${\mathbb {T}}:={\mathbb {T}}({\mathcal {T}}_0)$ denote the set of all admissible regular triangulations computed by successive newest-vertex bisection (NVB) [35, 48] of a regular initial triangulation ${\mathcal {T}}_0$ of $\Omega \subset {\mathbb {R}}^3$. The AFEM algorithm with Dörfler marking and newest-vertex bisection abbreviates $\eta _\ell (T)$ for any $T\in {\mathcal {T}}:={\mathcal {T}}_\ell \in {\mathbb {T}}$ and $\eta _\ell ^2:=\eta ^2({\mathcal {T}}_\ell ):=\sum _{T\in {\mathcal {T}}_\ell }\eta _\ell ^2(T)$. The selection of the set ${\mathcal {M}}_\ell$ in the step Mark of AFEM4EVP with minimal cardinality is possible at linear cost [44].

Optimal convergence rates. The optimal convergence rates of AFEM4EVP in the error estimator means that the outputs $({\mathcal {T}}_\ell )_{\ell \in {\mathbb {N}}_0}$ and $(\eta _\ell )_{\ell \in {\mathbb {N}}_0}$ of AFEM4EVP satisfy

$$\begin{aligned} \sup _{\ell \in {\mathbb {N}}_0} (1+|{\mathcal {T}}_\ell | - |{\mathcal {T}}_0|)^s \eta _\ell \approx \sup _{N\in {\mathbb {N}}_0} (1+N)^s \min \{\eta ({\mathcal {T}}):\, {\mathcal {T}}\in {\mathbb {T}} \text { with } |{\mathcal {T}}|\leqslant |{\mathcal {T}}_0|+ N\}\nonumber \\ \end{aligned}$$

(1.5)

for any $s>0$ and the counting measure $\vert \bullet \vert =card (\bullet )$. In other words, if the estimator $\eta ({\mathcal {T}})$ converges with rate $s>0$ for some optimal selection of triangulations ${\mathcal {T}}\in {\mathbb {T}}$, then the output $\eta _\ell$ of AFEM4EVP converges with the same rate.

Theorem 1.1

(rate optimality of AFEM4EVP) Suppose that $\lambda _k=\lambda$ is a simple eigenvalue of (1.1), then there exist $\varepsilon >0$ and $0<\theta _0<1$ such that ${\mathcal {T}}_0\in {\mathbb {T}}(\varepsilon ):=\{{\mathcal {T}}\in {\mathbb {T}}:\,h_{\max }:=\max _{T\in {\mathcal {T}}}h_T\leqslant \varepsilon \}$ and $\theta$ with $0<\theta \leqslant \theta _0$ imply (1.5) for any $s>0$.

At first glance the discrete problem (1.2) involves a stabilization that is expected to generate the additional term $\kappa _m^{-2}|T|^{-2m/3}\Vert u_{\textrm{pw}}-u_{\textrm{nc}}\Vert ^2_{L^2(T)}$ in the error estimator (1.4). The negative power of the mesh-size in the latter term prevents a reduction property [12, 26, 39] and has to be circumvented. The only other known affirmative result for optimal convergence rates of an adaptive algorithm with stabilization (and negative powers of the mesh-size in the discrete problem) is [5] on discontinuous Galerkin (dG) schemes. An over-penalization therein diminishes the influence of the stabilization and eventually shows the dominance of the remaining a posteriori error terms. In the present case, the stabilization parameter $\kappa _m$ is fixed to maintain the GLB property and this requires a different argument: Since (1.2) is equivalent to a rational eigenvalue problem for a nonconforming scheme, a careful perturbation analysis eventually shows efficiency and reliability of the nonconforming error estimator (1.4) for sufficiently small mesh-sizes. The verification requires a medius analysis [37], which applies arguments from a posteriori error analysis (e.g., efficiency in (3.10) below) in an a priori error analysis.

Outline. The remaining parts of this paper are devoted to the proof of Theorem 1.1 and are organized as follows. A general interpolation operator I and a right-inverse J in Sect. 2 allow for a simultaneous analysis for $m=1$ and $m=2$ in the Crouzeix–Raviart and Morley FEM. The medius analysis in Sect. 3 provides new best-approximation results and thereby prepares the proof of Theorem 1.1 in Sect. 4–5. The proof of the optimal convergence rates requires a framework extended from [12, 26] in Appendix A. While more general boundary conditions appear feasible as in [15, 31], non-constant coefficients in a general elliptic differential operator of order 2m appear a less straightforward extension from the m-harmonic operator $(-1)^ m \Delta ^m$. An expected extension revisits [24] for the question of lower eigenvalue bounds, while the convergence analysis of an adaptive algorithm expects extra terms for the perturbations of the piecewise polynomial approximation of inhomogeneous coefficients as in [22]; this is therefore left for future research. This first paper on optimal convergence rates of an adaptive algorithm for the direct guaranteed lower eigenvalue bounds focuses on a model problem. The results hold in 2D and 3D and are presented in 3D for brevity.

2 Preliminaries

This section summarizes abstract conditions (I1)–(I4) on an interpolation operator $I:V\rightarrow V({\mathcal {T}})$ and (J1)–(J4) on a right inverse $J:V({\mathcal {T}})\rightarrow V$. The conditions hold for the Crouzeix–Raviart and the Morley finite element space in the two model examples for the Laplacian ${m}=1$ and the bi-Laplacian ${m}=2$.

2.1 Notation

Standard notation on Lebesgue and Sobolev spaces applies throughout this paper; $(\bullet ,\bullet )_{L^2(\Omega )}$ abbreviates the $L^2$ scalar product and $H^{m}(T)$ abbreviates $H^{m}(int (T ))$ for a tetrahedron $T\in {\mathcal {T}}$. The vector space $H^{m}({\mathcal {T}}):=\{v\in L^2(\Omega ):\, v|_T\in H^{m}(T)\}$ consists of piecewise $H^{m}$ functions and is equipped with the semi-norm $|||\bullet |||_{\textrm{pw}}^2:=(D^m_{\textrm{pw}}\bullet ,D^m_{\textrm{pw}}\bullet )_{L^2(\Omega )}$. The piecewise gradient $D^1_{\textrm{pw}}$ or piecewise Hessian $D^2_{\textrm{pw}}$ is understood with respect to the (non-displayed) regular triangulation ${\mathcal {T}}\in {\mathbb {T}}$ of the bounded polyhedral Lipschitz domain $\Omega \subset {\mathbb {R}}^3$ into tetrahedra. The triangulation ${\mathcal {T}}$ is computed by successive newest-vertex bisection (NVB) [35, 48] of a regular initial triangulation ${\mathcal {T}}_0$ (plus some initialization of tagged tetrahedra) of $\Omega \subset {\mathbb {R}}^3$. The set ${\mathbb {T}}:={\mathbb {T}}({\mathcal {T}}_0)$ of all admissible triangulations is (uniformly) shape-regular. For any ${\mathcal {T}}\in {\mathbb {T}}$, let ${\mathbb {T}}({\mathcal {T}})$ abbreviate the set of all admissible refinements of ${\mathcal {T}}$. For any $0<\varepsilon <1$ let ${\mathbb {T}}(\varepsilon ):=\{{\mathcal {T}}\in {\mathbb {T}}:\,h_{\max }:=\max _{T\in {\mathcal {T}}}h_T\leqslant \varepsilon \}$ denote the set of all admissible triangulations with maximal mesh-size $h_{\max }\leqslant \varepsilon$. The context-depending notation $|\bullet |$ denotes the Euclidean length of a vector, the cardinality of a finite set, as well as the non-trivial three-, two-, or one-dimensional Lebesgue measure of a subset of ${\mathbb {R}}^3$. For any positive, piecewise polynomial $\varrho \in P_k({\mathcal {T}})$ with $\varrho \geqslant 0$, $k\in {\mathbb {N}}_0$, $(\bullet , \bullet )_\varrho :=(\varrho \bullet ,\bullet )_{L^2(\Omega )}$ abbreviates the weighted $L^2$ scalar product with induced $\varrho$-weighted $L^2$ norm $\Vert \bullet \Vert _\varrho :=\Vert \varrho ^{1/2} \bullet \Vert _{L^2(\Omega )}$. The discrete space $P_{m}({\mathcal {T}}):=\{p_m \in L^2(\Omega ):\, p_m|_T\in P_m(T) \text { is a polynomial of degree at most } m\text { for any }T\in {\mathcal {T}}\}$ consists of piecewise polynomials, the spaces $\textit{CR}^1_0({\mathcal {T}})$ resp. $M({\mathcal {T}})$ will be defined in Sect. 2.4.1 resp. 2.4.2 below. Given a function $v\in L^2(\omega )$, define the integral mean . The $L^2$ projection $\Pi _0$ onto the piecewise constant functions $P_0({\mathcal {T}})$ reads for all $f\in L^2(\Omega )$ and $T\in {\mathcal {T}}$. Let $\sigma :=\min \{1,\sigma _{\textrm{reg}}\}$ denote the minimum of one and the index of elliptic regularity $\sigma _{\textrm{reg}}>0$ for the source problem of the m-Laplacian $(-1)^m\Delta ^m$ in $H^m_0(\Omega )$: Given any right-hand side $f\in L^2(\Omega )$, the weak solution $u\in V$ to $(-1)^m\Delta ^m u=f$ satisfies

$$\begin{aligned} u\in H^{{m}+\sigma }(\Omega ) \text { and } \Vert u\Vert _{H^{m+\sigma }(\Omega )}\leqslant C(\sigma )\Vert f\Vert _{L^2(\Omega )}. \end{aligned}$$

(2.1)

(This is well-established for $m=1$ [1, 28, 34, 36, 42] and $m=2$ in 2D [8] with $\sigma _{\textrm{reg}}>1/2$ and otherwise a hypothesis throughout this paper.) The Sobolev space $H^{m+s}(\Omega )$ is defined for $0<s<1$ by complex interpolation of $H^m(\Omega )$ and $H^{m+1}(\Omega )$, $m\in {\mathbb {N}}_0$. Throughout this paper, $a \lesssim b$ abbreviates $a\leqslant Cb$ with a generic constant C depending on $\sigma$ in (2.1) and the shape-regularity of ${\mathcal {T}}\in {\mathbb {T}}$ only; $a \approx b$ stands for $a\lesssim b\lesssim a$.

2.2 Interpolation

The operators I and J concern the (nonconforming) discrete space ${V({\mathcal {T}})}\subset P_{m}({\mathcal {T}})$ and $V:=H^m_0(\Omega )$ for an admissible triangulation ${\mathcal {T}}\in {\mathbb {T}}$. An advantage of separate interest is that the analysis with I and J is performed simultaneously for $m\geqslant 1$, while the examples in Sect. 2.4 below concern $m=1,2$.

Suppose that, for each admissible triangulation ${\mathcal {T}}\in {\mathbb {T}}$, there exists a linear interpolation operator I onto $V({\mathcal {T}})$ that is defined on $V+V(\widehat{{\mathcal {T}}})$ for any refinement $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$ and that satisfies the following properties with universal positive constants $\kappa _{m}$ and $\kappa _{d}$; in all examples below $\kappa _{m}$ is known and the existence of $\kappa _{d}$ is clarified.

• (I1) Any $T\in {\mathcal {T}}$ and $v\in H^{m}(T)$ satisfy $\Vert v- I v\Vert _{L^2(T)}\leqslant \kappa _{m} h_T^{m} \vert v-Iv\vert _{H^m(T)}$.

• (I2) The piecewise derivative $D^m_{\textrm{pw}}$ of any $v\in V+V(\widehat{{\mathcal {T}}})$ satisfies $D^m_{\textrm{pw}}Iv=\Pi _0D^m_{\textrm{pw}}v$.

• (I3) The operator I acts as identity in non-refined tetrahedra in that $(1-I){\widehat{v}}_{\textrm{nc}}|_T=0 \text { in }T\in {\mathcal {T}}\cap \widehat{{\mathcal {T}}} \text { for all }{\widehat{v}}_{\textrm{nc}}\in V(\widehat{{\mathcal {T}}}).$ The interpolation operator ${\widehat{I}}$ associated with $V(\widehat{{\mathcal {T}}})$ satisfies $I\circ {\widehat{I}}=I$ in $V+V(\widehat{{\mathcal {T}}})$.

• (I4) Any $T\in {\mathcal {T}}$ and ${\widehat{v}}_{\textrm{nc}}\in V(\widehat{{\mathcal {T}}})$ satisfy $\Vert {\widehat{v}}_{\textrm{nc}}- I {\widehat{v}}_{\textrm{nc}}\Vert _{L^2(T)} \leqslant {\kappa _d} h_T^{m} \vert {\widehat{v}}_{\textrm{nc}}-I{\widehat{v}}_{\textrm{nc}}\vert _{H^m(T)}.$

Corollary 2.1

(properties of I )

• (a) Given $\widehat{{\mathcal {T}}}\,{\in }\,{\mathbb {T}}({\mathcal {T}})$, any $v\,{\in }\, V+V(\widehat{{\mathcal {T}}})$ and $w_{\textrm{nc}}\,{\in }\, V({\mathcal {T}})$ satisfy $\displaystyle a_{\textrm{pw}}(v-Iv,w_{\textrm{nc}})=0$ and $\displaystyle |||v-Iv|||_{\textrm{pw}}=\min _{v_{\textrm{nc}}\in V({\mathcal {T}})}||| v-v_{\textrm{nc}}|||_{\textrm{pw}}$.

• (b) Any $v\in H^{m+s}(\Omega )$ with $1/2<s\leqslant 1$ satisfies $||| (1-I)v|||_{\textrm{pw}}\leqslant (h_{\max }/\pi )^{s} \Vert v\Vert _{H^{m+s}(\Omega )}.$

• (c) Any $v,\, w\in V$ and $v_{\textrm{nc}}\in V({\mathcal {T}})$ satisfy $a_{\textrm{pw}}(v, v_{\textrm{nc}})=a_{\textrm{pw}}(Iv,v_{\textrm{nc}})$ and

  $a_{\textrm{pw}}(v, (1-I)w)=a_{\textrm{pw}}((1-I)v, (1-I)w) \leqslant \min _{v_{\textrm{nc}}\in V({\mathcal {T}})} ||| v-v_{\textrm{nc}}|||_{\textrm{pw}} \min _{w_{\textrm{nc}}\in V({\mathcal {T}})} ||| w-w_{\textrm{nc}}|||_{\textrm{pw}}.$

• (d) Any $w\in V$ and $v\in V+V({\mathcal {T}})$ satisfy $b(v, (1-I)w)\leqslant \Vert h_{{\mathcal {T}}}^m v\Vert _{L^2(\Omega )}\Vert h_{{\mathcal {T}}}^{-m} (1-I)w\Vert _{L^2(\Omega )} \leqslant$ $\kappa _m\Vert h_{{\mathcal {T}}}^m v\Vert _{L^2(\Omega )}\min _{w_{\textrm{nc}}\in V({\mathcal {T}})} ||| w-w_{\textrm{nc}}|||_{\textrm{pw}}.$

Proof

Since $D^m_{\textrm{pw}}w_{\textrm{nc}}\in P_0({\mathcal {T}};{\mathbb {R}}^{3^m})$, (I2) implies (a). In combination with a piecewise Poincaré inequality, (I2) implies (b) (see [24, Cor. 2.2.a] for details). The first claim in (c) follows from (a). The combination of (a) with the Cauchy–Schwarz inequality proves (c). The Cauchy–Schwarz inequality, the approximation property (I1), and (c) conclude the proof of (d). $\square$

2.3 Conforming companion

Given any tetrahedron $T\in {\mathcal {T}}$ in a triangulation ${\mathcal {T}}\in {\mathbb {T}}$, let ${\mathcal {V}}(T)$ denote the set of its vertices (0-subsimplices) and let ${\mathcal {F}}(T)$ denote the set of its faces (2-subsimplices). A linear operator $J: {V({\mathcal {T}})}\rightarrow V$ is called conforming companion if (J1)–(J4) hold with universal constants $M_1,M_2,\,M_4$ (that exclusively depend on ${\mathbb {T}}$).

• (J1) J is a right inverse to the interpolation I in the sense that $I\circ J$ acts as identity in $V({\mathcal {T}})$.

• (J2) $\displaystyle \Vert h_{{\mathcal {T}}}^{-{m}} (1-J)v_{\textrm{nc}}\Vert _{L^2(\Omega )} +||| (1-J)v_{\textrm{nc}}|||_{\textrm{pw}} \leqslant \Big (M_1 \sum _{T\in {\mathcal {T}}}|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)}\Vert [D^m_{\textrm{pw}} v_{\textrm{nc}}]_F\times \nu _F\Vert ^2_{L^2(F)}\Big )^{1/2}$ $\leqslant {M_2} \min _{v\in V}||| v_{\textrm{nc}}-v|||_{\textrm{pw}}$ for any $v_{\textrm{nc}}\in V({\mathcal {T}})$.

• (J3) $\displaystyle (1-J)(V({\mathcal {T}}))\perp P_{m}({\mathcal {T}})$ holds in $L^2(\Omega )$.

• (J4) $\displaystyle \vert v_{\textrm{nc}} -Jv_{\textrm{nc}}\vert _{H^m(K)}^2 \leqslant {M_4} \sum _{T\in {\mathcal {T}}(\Omega (K))}|T|^{1/3} \sum _{F\in {\mathcal {F}}(T)}\Vert [D^{m}_{\textrm{pw}} v_{\textrm{nc}}]_F\times \nu _F\Vert _{L^2(F)}^2$ holds for any $v_{\textrm{nc}}\in V({\mathcal {T}})$ and $K\in {\mathcal {T}}$ with the set ${\mathcal {T}}(\Omega (K)):=\{T\in {\mathcal {T}}: \textrm{dist}(T,K)=0\}$ of adjacent tetrahedra.

The properties (J1)–(J4) [18, 24, 32] are stated for convenient quotation throughout this paper. The localized version (J4) applies at the very end (in Theorem 4.6) and implies parts of (J2). The second inequality in (J2) is the efficiency of a posteriori error estimators.

Remark 2.2

(on (J4)) For any refinement $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$ of a triangulation ${\mathcal {T}}\in {\mathbb {T}}$, let ${\mathcal {R}}_1:=\{K\in {\mathcal {T}}:\,\exists \, T\in {\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}} \text { with }dist (K,T)=~0\}\subset ~{\mathcal {T}}$ denote the set of coarse but not fine tetrahedra plus one layer of coarse tetrahedra around. Then (J4) and a finite overlap argument imply the existence of $M_5>0$ such that any $v_{\textrm{nc}}\in V({\mathcal {T}})$ satisfies

$$\begin{aligned} \Vert D^{m}_{\textrm{pw}}(v_{\textrm{nc}} -Jv_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 \leqslant M_5 \sum _{T\in {\mathcal {R}}_1}|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)}\Vert [D^{m}_{\textrm{pw}} v_{\textrm{nc}}]_F\times \nu _F\Vert _{L^2(F)}^2. \end{aligned}$$

The superset ${\mathcal {R}}_1$ of ${\mathcal {T}}\setminus \widehat{{\mathcal {T}}}$ serves as a simple example and could indeed be replaced by ${\mathcal {T}}\setminus \widehat{{\mathcal {T}}}$ provided J may depend on $\widehat{{\mathcal {T}}}$; cf. [23, §6] for details in the two model problems below.$\Box$

Corollary 2.3

(properties of J ) Any $w\in V$ and $v_{\textrm{nc}}\in V({\mathcal {T}})$ satisfy

• (a) $\displaystyle \Vert v_{\textrm{nc}}-Jv_{\textrm{nc}}\Vert _{L^2(\Omega )}=\Vert (1-I)Jv_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant \kappa _{m} ||| h_{{\mathcal {T}}}^{m}(v_{\textrm{nc}}-Jv_{\textrm{nc}})|||_{\textrm{pw}}$ $\displaystyle \leqslant h_{\max }^{m}\kappa _{m}{M_2}\min _{v\in V}||| v_{\textrm{nc}}-v|||_{\textrm{pw}};$

• (b) $\displaystyle b(w, v_{\textrm{nc}}-Jv_{\textrm{nc}})=b(w-Iw,v_{\textrm{nc}}-Jv_{\textrm{nc}}) \leqslant \Vert w-Iw\Vert _{L^2(\Omega )}\Vert v_{\textrm{nc}}-Jv_{\textrm{nc}}\Vert _{L^2(\Omega )}$ $\displaystyle \leqslant h_{\max }^{2{m}} \kappa _{m}^2{M_2} \min _{w_{\textrm{nc}}\in V({\mathcal {T}})}||| w-w_{\textrm{nc}}|||_{\textrm{pw}} \min _{v\in V}||| v_{\textrm{nc}}-v|||_{\textrm{pw}};$

• (c) $\displaystyle a_{\textrm{pw}}(w, v_{\textrm{nc}}-Jv_{\textrm{nc}}) =a_{\textrm{pw}}(w-Iw,v_{\textrm{nc}}-Jv_{\textrm{nc}})\leqslant ||| w-Iw|||_{\textrm{pw}} ||| v_{\textrm{nc}}-Jv_{\textrm{nc}}|||_{\textrm{pw}}$ $\displaystyle \leqslant {M_2} \min _{w_{\textrm{nc}}\in V({\mathcal {T}})} ||| w-w_{\textrm{nc}}|||_{\textrm{pw}}\min _{v\in V}||| v-v_{\textrm{nc}}|||_{\textrm{pw}}.$

Proof

The combination of (J1), (I1), and (J2) proves (a). The claim (b) follows from (J3), the Cauchy–Schwarz inequality, (I1), and (a). Corollary 2.1.c and (J1)–(J2) lead to (c). $\square$

2.4 Examples

Two examples for $V({\mathcal {T}})\subset P_m({\mathcal {T}})$ are analysed simultaneously in this paper for $m=1,2$. It is appealing to follow our methodology for $m\geqslant 3$ [52] in future research.

2.4.1 Crouzeix–Raviart finite elements for the Laplacian ($\varvec{m=1}$)

Given the shape-regular triangulation ${\mathcal {T}}\in {\mathbb {T}}$, let ${\mathcal {F}}$ (resp. ${\mathcal {F}}(\Omega )$ or ${\mathcal {F}}(\partial \Omega )$) denote the set of all (resp. interior or boundary) faces. Throughout this paper, the model problem with ${m}=1$ approximates the Dirichlet eigenvectors $u\in H^1_0(\Omega )$ of the Laplacian $-\Delta u=\lambda u$ in the Crouzeix–Raviart finite element space [25]

$$\begin{aligned} V({\mathcal {T}}):=\textit{CR}^1_0({\mathcal {T}}) :=\{v\in P_1({\mathcal {T}}):\ {}&v \text { is continuous at }mid (F)\text { for all }F\in {\mathcal {F}}(\Omega ) \text { and }\\&v(mid (F))=0\text { for all }F\in {\mathcal {F}}(\partial \Omega ) \}. \end{aligned}$$

Given the face-oriented basis functions $\psi _{F}\in \textit{CR}^1({\mathcal {T}})$ with $\psi _F(mid (E))=\delta _{EF}$ for all faces $E,F\in {\mathcal {F}}$ ($\delta _{EF}$ is Kronecker’s delta), the standard interpolation operator reads

The interpolation operator $I_{\text {CR}}$ satisfies (I1)–(I4) with $\kappa _1:=\sqrt{{1}/{\pi ^2}+{1}/{120}}$, see [23, Sec. 4.2–4.4] and the references therein. The constant $\kappa _1$ is provided in [15, 16, 27].

The design of the conforming companion $J:\textit{CR}^1_0({\mathcal {T}})\rightarrow S^{5}_0({\mathcal {T}}):=P_{5}({\mathcal {T}})\cap C_0(\Omega )$ with (J1)–(J4) is a straightforward generalization of [18, Prop. 2.3] to 3D. The arguments in [18, Prop. 2.3] can be localized [10, Thm. 5.1] and lead with [9, Thm. 3.2], [17, Thm. 4.9] to (J2) and (J4).

2.4.2 Morley finite elements for the bi-Laplacian ($\varvec{m=2}$)

Given the shape-regular triangulation ${\mathcal {T}}\in {\mathbb {T}}$, let ${\mathcal {E}}$ (resp. ${\mathcal {E}}(\Omega )$ or ${\mathcal {E}}(\partial \Omega )$) denote the set of all (resp. interior or boundary) edges. Let ${\mathcal {F}}(E):=\{F\in {\mathcal {F}}:\, E\subset {\overline{F}}\}$ denote the set of all faces containing the edge $E\in {\mathcal {E}}$. For any face $F\in {\mathcal {F}}$, let $\nu _F$ denote the unit normal with fixed orientation and $[\bullet ]_F$ the jump across F. The model problem with ${m}=2$ approximates the Dirichlet eigenvectors $u\in H^2_0(\Omega )$ of the bi-Laplacian $\Delta ^2 u=\lambda u$ in the discrete Morley finite element space [40, 41]

Given the nodal basis functions $\Phi _E,\Phi _F$ for any $E\in {\mathcal {E}}$ and $F\in {\mathcal {F}}$ (see [24, Eq. (2.1)–(2.2)] for details), the standard interpolation operator [15, 23, 24, 32] reads

The operator $I_M$ satisfies (I1)–(I4) with $\kappa _2:={\kappa _{1}}/{\pi }+\sqrt{({3\kappa _{1}^2+2\kappa _{1} })/80}$ as discussed in [23, 24]; $\kappa _2$ is provided in [15, 24]; cf. also [38] for GLB in 2m-th order eigenvalue problems in n-dimension.

There exists a conforming companion $J:M({\mathcal {T}})\rightarrow V$ based on the Hsieh–Clough–Tocher FEM [21, Chap. 6] with (J1)–(J4) in [23, 32, 50] in 2D and on the Worsey–Farin FEM [46, 51] with (J1)–(J3) in [24] in 3D. Since the arguments in the proof of (J2) in [24, Thm. 3.1.b] are local, (J4) follows in 3D as well.

3 Medius analysis

This section shows that (I1)–(I2) and (J1)–(J3) lead to best-approximation and error estimates in weaker Sobolev norms.

3.1 Main result and layout of the proof

Throughout this paper, $k\in {\mathbb {N}}$ is the number of a simple exact eigenvalue $\lambda \equiv \lambda _k$. The aim of this section is the proof of Theorem 3.1 with $\Vert \bullet \Vert _{\delta }$ defined in (3.1) below.

Theorem 3.1

(best-approximation) Let $(\lambda ,u)\in {\mathbb {R}}^+\times V$ denote the k-th continuous eigenpair of (1.1) with a simple eigenvalue $\lambda \equiv \lambda _k$ and $\Vert u\Vert _{L^2(\Omega )}=1$. There exist $\varepsilon _5>0$ and $C_0>0$ such that, for all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _5):=\{{\mathcal {T}}\in {\mathbb {T}}:\,h_{\max }\leqslant \varepsilon _5\}$, there exists a discrete eigenpair $(\lambda _h,\varvec{u_h})\in {\mathbb {R}}^+\times \varvec{V_h}$ of number k of (1.2) with $\lambda _h\equiv \lambda _h({k})$, $\varvec{u_h}=(u_{\textrm{pw}},u_{\textrm{nc}})$, $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$, and $b(u,u_{\textrm{nc}})> 0$ such that

• (a) $\lambda _h(k)$ is a simple algebraic eigenvalue of (1.2) with ${\lambda _k}/{2}\leqslant \lambda _h(k)$,

• (b) $\lambda _h(j)\leqslant \lambda _j$ for all $j=1,\ldots ,k+1$,

• (c) $\displaystyle |\lambda -\lambda _h|+||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2 +h_{\max }^{-2\sigma }\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2+\Vert u_{\textrm{nc}}\Vert _{\delta }^2 \leqslant C_0 ||| u- I u|||_{\textrm{pw}}^2$.

Some comments on related results and an outline of the proof of Theorem 3.1 are in order before Sects. 3.2–3.5 provide details.

Remark 3.2

(known convergence results) The analysis in [24] (§ 2.3.3 for $m=1$ and Thm. 1.2 for $m=2$) guarantees the convergence of the eigenvalues $\lambda _h$ to $\lambda$ and the component $u_{\textrm{pw}}\in P_m({\mathcal {T}})$ to $u\in V$. The assumption that $\lambda =\lambda _{k}$ is a simple eigenvalue of (1.1) and the convergence $\lambda _h({k})\equiv \lambda _h\rightarrow \lambda$ as $h_{\max }\rightarrow 0$ lead to the existence of $\varepsilon _0>0$ such that the number $M:=dim (P_m({\mathcal {T}}))$ of discrete eigenvalues of (1.2) is larger than $k+1$ and $\lambda _h({k}-1)<\lambda _h({k})\equiv \lambda _h<\lambda _h({k}+1)$ as well as $\lambda _k/{2}\leqslant \lambda _h(k)$ for all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _0)$. Then the eigenfunction $\varvec{u_h}=(u_{\textrm{pw}},u_{\textrm{nc}})\in \varvec{V_h}{\setminus }\{0\}$ is unique.

The convergence analysis in [24] displays convergence of the eigenvector $u_{\textrm{pw}}\in P_m({\mathcal {T}})$ but not for the nonconforming component $u_{\textrm{nc}}\in V({\mathcal {T}})$. This section focusses on the convergence analysis for $u_{\textrm{nc}}\in V({\mathcal {T}})$. Recall that $k\in {\mathbb {N}}$ is fixed and $(\lambda , u)$ denotes the k-th eigenpair of (1.1) with a simple eigenvalue $\lambda \equiv \lambda _k>0$ and $\Vert u\Vert _{L^2(\Omega )}=1$. Set $\varepsilon _1:=\min \{\varepsilon _0, (2\lambda _{k+1}\kappa _m^2)^{-1/(2\,m)}\}$ and suppose ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _1)$. Let $(\lambda _h,\varvec{u_h})$ denote the k-th discrete eigenpair in (1.2) with $\lambda _h\equiv \lambda _h(k)>0$, $\varvec{u_h}=(u_{\textrm{pw}},u_{\textrm{nc}})\in \varvec{V_h}$, $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$, and $b(u,u_{\textrm{nc}})\geqslant 0$.

Proof of Theorem 3.1.a

This follows from Remark 3.2 for $\varepsilon _1:=\min \{\varepsilon _0, (2\lambda _{k+1}\kappa _m^2)^{-1/(2\,m)}\}$. $\square$

Proof of Theorem 3.1.b

The choice $\varepsilon _1:=\min \{\varepsilon _0, (2\lambda _{k+1}\kappa _m^2)^{-1/(2\,m)}\}$ implies for all $j=1,\ldots ,k$ that $\lambda _j\kappa _{m}^2\,h^{2{m}}_{\max }\leqslant \lambda _{k+1}\kappa _{m}^2\varepsilon _1^{2{m}}{\leqslant } 1/2$. Hence (1.3) proves Theorem 3.1.b. $\square$

Remark 3.3

(weight $\delta$) The piecewise constant weight $\delta \in P_0({\mathcal {T}})$ in the weighted $L^2$ norm $\Vert \bullet \Vert _\delta :=\Vert \sqrt{\delta }\bullet \Vert _{L^2(\Omega )}$ on the left-hand side of Theorem 3.1.c reads

$$\begin{aligned} \delta :=\frac{1}{1-\lambda _h\kappa _{m}^2h_{{\mathcal {T}}}^{2{m}}}-1 =\frac{\lambda _h\kappa _{m}^2h_{{\mathcal {T}}}^{2{m}}}{1-\lambda _h\kappa _{m}^2h_{{\mathcal {T}}}^{2{m}}} =\lambda _h\kappa _{m}^2h_{{\mathcal {T}}}^{2{m}}(1+\delta )\in P_0({\mathcal {T}}). \end{aligned}$$

(3.1)

Notice that $h_{\max }\leqslant \varepsilon _1$ implies $\delta \leqslant \delta _{\max }:=(1-\lambda _h \kappa _{m}^2h_{\max }^{2{m}})^{-1}-1\leqslant 1$. The constant $C_\delta :=2\lambda \kappa _m^2$ satisfies $\delta \leqslant C_{\delta }h_{{\mathcal {T}}}^{2{m}}\leqslant C_{\delta }h_{\max }^{2{m}}$ (because $\lambda _h\leqslant \lambda$ from Theorem 3.1.b) and $\delta$ converges to zero as the maximal mesh-size $h_{\max }\rightarrow 0$ approaches zero.

Remark 3.4

(related work) This section extends the analysis in [18, Section 2–3] to a simultaneous analysis of the Crouzeix–Raviart and Morley FEM and to the extra-stabilized discrete eigenvalue problem (EVP) (1.2) and to 3D.

Remark 3.5

(equivalent problem) Since $\lambda _h\kappa _{m}^2\,h^{2{m}}_{\max }\leqslant \lambda _{k+1}\kappa _{m}^2\varepsilon _1^{2{m}}{\leqslant }1/2$, (1.2) is equivalent to a reduced rational eigenvalue problem that seeks $(\lambda _h,u_{\textrm{nc}})\in {\mathbb {R}}^+\times (V({\mathcal {T}}){\setminus }\{0\})$ with

$$\begin{aligned} a_{\textrm{pw}}(u_{\textrm{nc}},v_{\textrm{nc}}) =\lambda _h \Big (\frac{u_{\textrm{nc}}}{1-\lambda _h\kappa _{m}^2 h_{{\mathcal {T}}}^{2{m}}},v_{\textrm{nc}}\Big )_{L^2(\Omega )} \qquad \text {for all }v_{\textrm{nc}}\in V({\mathcal {T}}) \end{aligned}$$

(3.2)

and $u_{\textrm{pw}}=(1-\lambda _h\kappa _{m}^2h_{{\mathcal {T}}}^{{2m}})^{-1}u_{\textrm{nc}}$ [24, Prop. 2.5, § 2.3.3].

Outline of the proof of Theorem3.1.c. The outline of the proof of Theorem 3.1.c provides an overview and clarifies the various steps for a reduction of $\varepsilon _1$ to $\varepsilon _5$, before the technical details follow in the subsequent subsections. The coefficient $(1-\lambda _h\kappa _m^2h_{{\mathcal {T}}}^{2m})^{-1}=1+\delta \in P_0({\mathcal {T}})$ with $\lambda _h\equiv \lambda _h({k})$ on the right-hand side of (3.2) is frozen in the intermediate EVP.

Definition 3.6

(intermediate EVP) Recall $(\bullet ,\bullet )_{1+\delta }:=((1+\delta )\bullet ,\bullet )_{L^2(\Omega )}$. Let $(\mu ,\phi )\in {\mathbb {R}}^+\times (V({\mathcal {T}})\setminus \{0\})$ solve the (algebraic) eigenvalue problem

$$\begin{aligned} a_{\textrm{pw}}(\phi ,v_{\textrm{nc}}) =\mu (\phi ,v_{\textrm{nc}})_{1+\delta } \quad \text {for all }v_{\textrm{nc}}\in V({{\mathcal {T}}}). \end{aligned}$$

(3.3)

The two coefficient matrices in (3.3) are SPD and there exist $N:=dim \,V({\mathcal {T}})$ (algebraic) eigenpairs $(\mu _1,\phi _1),\ldots ,(\mu _N,\phi _N)$ of (3.3). The eigenvectors $\phi _1,\ldots ,\phi _N$ are $(\bullet ,\bullet )_{1+\delta }$-orthonormal and the eigenvalues $\mu _1\leqslant \cdots \leqslant \mu _N$ are enumerated in ascending order counting multiplicities.

Since $\lambda _h$ is an eigenvalue of the rational problem (3.2), $\lambda _h\in \{\mu _1,\ldots ,\mu _N\}$ belongs to the eigenvalues of (3.3). Lemma 3.9 below guarantees the convergence $|\mu _j- \lambda _h(j)|\rightarrow 0$ as $h_{\max }\rightarrow 0$ for $j =1,\ldots ,k+1$. Hence there exist positive $\varepsilon _2\leqslant \min \{1/2,\varepsilon _1\}$ and $M_6$ such that ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _2)$ implies

• (H1) $\ \ \mu _k=\lambda _h(k)$ is a simple algebraic eigenvalue of (3.3),

• (H2) $\displaystyle \max _{\begin{array}{c} j=1,\ldots , N\\ j\not =k \end{array}}\frac{\lambda _k}{|\lambda _k-\mu _j|}\leqslant M_6$.

The intermediate EVP and the following associated source problem allow for the control of the extra-stabilization.

Definition 3.7

(auxiliary source problem) Let $z_{\textrm{nc}}\in V({{\mathcal {T}}})$ denote the solution to

$$\begin{aligned} a_{\textrm{pw}}(z_{\textrm{nc}},v_{\textrm{nc}})=(\lambda u,v_{\textrm{nc}})_{1+\delta } \quad \text {for all }v_{\textrm{nc}}\in V({{\mathcal {T}}}). \end{aligned}$$

(3.4)

For any ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _2)$, Sect. 3.3 below provides $C_1,C_2>0$ that satisfy

$$\begin{aligned} \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}&\leqslant C_1 \Vert u-z_{\textrm{nc}}\Vert _{L^2(\Omega )}, \end{aligned}$$

(3.5)

$$\begin{aligned} C_2^{-1}\Vert u-z_{\textrm{nc}}\Vert _{L^2(\Omega )}&\leqslant h_{\max }^{\sigma }||| u-z_{\textrm{nc}}|||_{\textrm{pw}}+\Vert \delta \lambda u\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.6)

The proof of (3.5) in Sect. 3.3 extends [18, Lem. 2.4]. The proof of (3.6) utilizes another continuous source problem with the right-hand side $u-Jz_{\textrm{nc}}$. For all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _2)$, Sect. 3.4 below provides a constant $C_3>0$ such that

$$\begin{aligned} C_3^{-1}||| u-z_{\textrm{nc}}|||_{\textrm{pw}}\leqslant ||| u-Iu|||_{\textrm{pw}} +\Vert \delta \lambda u\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.7)

The proof of (3.7) below rests upon a decomposition of $||| u-z_{\textrm{nc}}|||_{\textrm{pw}}^2$ into terms controlled by the conditions (I1)–(I2) and (J1)–(J3). Since $h_{\max }\leqslant 1$, the combination of (3.5)–(3.7) reads

$$\begin{aligned} \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant C_1C_2\big (C_3h_{\max }^\sigma ||| u-Iu|||_{\textrm{pw}}+(1+C_3) \Vert \delta \lambda u\Vert _{L^2(\Omega )}\big ). \end{aligned}$$

(3.8)

The control of $\Vert \delta \lambda u\Vert _{L^2(\Omega )}$ on the right-hand side of (3.8) consists of two steps and leads to $c_1:=2\lambda ^2\kappa _m^2 C_1C_2(1+C_3)$ and $\varepsilon _3:=\min \{\varepsilon _2,(2c_1)^{-1/2m}\}$. A triangle inequality $\Vert \delta \lambda u\Vert _{L^2(\Omega )} \leqslant \Vert \delta \lambda (u-u_{\textrm{nc}})\Vert _{L^2(\Omega )}+\Vert \delta \lambda u_{\textrm{nc}}\Vert _{L^2(\Omega )}$, the estimate $\delta \leqslant 2\lambda \kappa _m^2 h_{\max }^{2m}$ in Remark 3.3, and (3.8) imply

$$\begin{aligned} \Vert \delta \lambda u\Vert _{L^2(\Omega )} \leqslant \frac{c_1C_3h_{\max }^{2m}}{1+C_3}h_{\max }^{\sigma } ||| u-Iu|||_{\textrm{pw}} +c_1h_{\max }^{2m}\Vert \delta \lambda u\Vert _{L^2(\Omega )} +\Vert \delta \lambda u_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

The choice of $\varepsilon _3$ shows $c_1h_{\max }^{2m}\Vert \delta \lambda u\Vert _{L^2(\Omega )}\leqslant \Vert \delta \lambda u\Vert _{L^2(\Omega )}/2$ for any ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _3)$. Therefore

$$\begin{aligned} \Vert \delta \lambda u\Vert _{L^2(\Omega )}\leqslant C_3/(1+C_3) h_{\max }^\sigma ||| u-Iu|||_{\textrm{pw}} +2\Vert \delta \lambda u_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.9)

Notice that $\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant 2\lambda \kappa _m^2 h_{\max }^m \Vert h_{{\mathcal {T}}}^{m} u_{\textrm{nc}}\Vert _{L^2(\Omega )}$ (from Remark 3.3) allows for the application of an efficiency estimate

$$\begin{aligned} C_4^{-1} \Vert h_{{\mathcal {T}}}^{m} u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant h_{\max }^{m}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}+\lambda ^{-1} ||| u-Iu|||_{\textrm{pw}} \end{aligned}$$

(3.10)

based on Verfürth’s bubble-function methodology [49]; see Sect. 3.4 for the proof of (3.10). Abbreviate $c_2:=4\lambda ^2\kappa _m^2 C_1C_2(1+C_3)C_4$ and $C_5:=2{C_1C_2}\big (2C_3+4\lambda \kappa _m^2 (1+C_3)C_4\big )$. The combination of (3.9)–(3.10) controls $\Vert \delta \lambda u\Vert _{L^2(\Omega )}$ in (3.8) and shows

$$\begin{aligned} \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant \frac{C_5}{2} h_{\max }^\sigma ||| u-Iu|||_{\textrm{pw}} +c_2h_{\max }^{2m} \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.11)

The choice $\varepsilon _4:=\min \{\varepsilon _3, (2c_2)^{-1/2m}\}<1$ shows $c_2h_{\max }^{2m} \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}/2$ for ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _4)$. This and (3.11) show the central estimate in Theorem 3.1.c

$$\begin{aligned} \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant C_5 h_{\max }^{\sigma } ||| u-Iu|||_{\textrm{pw}}. \end{aligned}$$

(3.12)

Note that (3.12) and Corollary 2.1.b from (I2) imply the convergence $\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}\rightarrow 0$ as $h_{\max }\rightarrow 0$. This and some $\varepsilon _5\leqslant \varepsilon _4$ ensures $b(u,u_{\textrm{nc}})>0$ for all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _5)$. Based on this outline, it remains to prove (3.5)–(3.7), (3.10), and hence (3.12) and to identify $C_0,\ldots , C_4$ below. The remaining estimates in Theorem 3.1.c follow in Sect. 3.5.

3.2 Intermediate EVP

Recall $\varepsilon _1:=\min \{\varepsilon _0, (2\lambda _{k+1}\kappa _m^2)^{-1/(2\,m)}\}$ and that $(\lambda _h,\varvec{u_h})$ denotes the k-th eigenpair of (1.2) with $\lambda _h\equiv \lambda _h(k)>0$, $\varvec{u_h}=(u_{\textrm{pw}},u_{\textrm{nc}})\in \varvec{V_h}$, $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$, and $b(u,u_{\textrm{nc}})\geqslant 0$. Recall the intermediate EVP (3.3) and that $(\lambda _h,u_{\textrm{nc}})\in {\mathbb {R}}^+\times V({\mathcal {T}})$ solves the rational EVP (3.2).

Remark 3.8

($\Vert \bullet \Vert _{1+\delta }\approx \Vert \bullet \Vert _{L^2(\Omega )}$) The weighted norm $\Vert \bullet \Vert _{1+\delta }$ is equivalent to the $L^2$-norm. Since $\lambda _h \kappa _{m}^2 \varepsilon _1^{2{m}}< \lambda _{k+1} \kappa _{m}^2 \varepsilon _1^{2{m}}\leqslant 1/2$ and $1\leqslant (1+\delta )|_T\leqslant 2$ for all $T\in {\mathcal {T}}\in {\mathbb {T}}(\varepsilon _1)$, $\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant \Vert v_{\textrm{nc}}\Vert _{1+\delta } \leqslant \sqrt{2}\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}$ holds for any $v_{\textrm{nc}}\in V({{\mathcal {T}}})$.

$\square$

Lemma 3.9

(comparison of (1.2) with (3.3)) Given ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _1)$, let $\lambda _h({j})$ denote the j-th eigenvalue of (1.2), and $\mu _{j}$ the j-th eigenvalue of (3.3) for any ${j} =1,\ldots ,k+1$. Then

$$\begin{aligned} (1-\lambda _{k+1}\kappa _{m}^2 h_{\max }^{2{m}})\mu _{j} \leqslant (1-\lambda _h(j)\kappa _{m}^2 h_{\max }^{2{m}})\mu _{j} \leqslant \lambda _h({j})&\leqslant \mu _{j}+2\lambda _h^2\kappa _m^2 h_{\max }^{2m}. \end{aligned}$$

(3.13)

The upper bound $\lambda _h({j})\leqslant \mu _{j}+2\lambda _h^2\kappa _m^2 h_{\max }^{2m}$ holds for all ${j}=1,\ldots , N$; $N:=dim \,V({\mathcal {T}})$.

Proof of the upper bound

Since the eigenfunctions $\phi _1,\ldots ,\phi _N$ of (3.3) are $(\bullet , \bullet )_{1+\delta }$-orthonormal, $a_{\textrm{pw}}(\phi _{j},\phi _\ell )=\mu _{j} \delta _{{j}\ell }$ and $(\phi _{j},\phi _\ell )_{1+\delta }=\delta _{{j}\ell }$ for all $j,\ell =1,\ldots , N$. Set $\psi _{j}:=(1+\delta )\phi _{j}$ and $\varvec{U_{j}}:=span \{(\psi _1,\phi _1), \ldots , (\psi _{j},\phi _{j})\}\subset \varvec{V_h}$. Since $b(\psi _j,\phi _\ell )=(\phi _j,\phi _\ell )_{1+\delta }=\delta _{j\ell }$, the functions $\phi _1,\ldots ,\phi _N$ are linear independent and so $dim (\varvec{U_{j}})={j}$ for any ${j} =1,\ldots ,N$. The discrete min-max principle [7, 45] for the algebraic eigenvalue problem (1.2) shows

$$\begin{aligned} \lambda _h({j})\leqslant \max _{\varvec{v_h}\in \varvec{U_{j}}\setminus \{0\}} {\varvec{a_h}(\varvec{v_h},\varvec{v_h})}/{\varvec{b_h}(\varvec{v_h},\varvec{v_h})}. \end{aligned}$$

(3.14)

The maximum in (3.14) is attained for some $\varvec{v_h}=(\psi ,\phi )\in \varvec{U_{j}}\setminus \{0\}$ with $\phi =\sum _{\ell =1}^{{j}}\alpha _\ell \phi _\ell \in V({{\mathcal {T}}})$, $\psi =\sum _{\ell =1}^{{j}}\alpha _\ell \psi _\ell =(1+\delta )\phi \in P_{m}({\mathcal {T}})$, and $1=\Vert \phi \Vert _{1+\delta }^2=\sum _{\ell =1}^{{j}}\alpha _\ell ^2$. Then $\varvec{b_h}(\varvec{v_h},\varvec{v_h})=\Vert (1+\delta )\phi \Vert _{L^2(\Omega )}^2\geqslant 1$ and $\varvec{a_h}(\varvec{v_h},\varvec{v_h}) =||| \phi |||_{\textrm{pw}}^2+ \Vert \kappa _{m}^{-1}h_{{\mathcal {T}}}^{-{m}}(\psi -\phi )\Vert _{L^2(\Omega )}^2.$ Since $a_{\textrm{pw}}(\phi _{j},\phi _\ell )=\mu _{j} \delta _{{j}\ell }$ for $\ell ,{j}=1,\ldots , N$, $\sum _{\ell =1}^{{j}}\alpha _\ell ^2=1$ implies $||| \phi |||_{\textrm{pw}}^2=\sum _{\ell =1}^{{j}}\alpha _\ell ^2\mu _\ell \leqslant \mu _{j}$. Since $\delta =\lambda _h\kappa _{m}^2h_{{\mathcal {T}}}^{2{m}}(1+\delta )$ a.e. in $\Omega$, the stabilization term in $\varvec{a_h}$ reads

$$\begin{aligned}&\Vert \kappa _{m}^{-1}h_{{\mathcal {T}}}^{-{m}}(\psi -\phi )\Vert _{L^2(\Omega )}^2 =\Vert \kappa _{m}^{-1} h_{{\mathcal {T}}}^{-{m}} \delta \phi \Vert ^2_{L^2(\Omega )} = \lambda _h^2\kappa _{m}^2\Vert h_{{\mathcal {T}}}^{m}(1+ \delta )\phi \Vert ^2_{L^2(\Omega )}. \end{aligned}$$

The bound $1+\delta \leqslant 2$ from Remark 3.3 and $\Vert \phi \Vert _{1+\delta }=1$ imply $\Vert h_{{\mathcal {T}}}^{m}(1+ \delta )\phi \Vert ^2_{L^2(\Omega )}\leqslant 2h_{\max }^{2{m}}$. Consequently, $\Vert \kappa _{m}^{-1}h_{{\mathcal {T}}}^{-{m}}(\psi -\phi )\Vert _{L^2(\Omega )}^2\leqslant 2\lambda _h^2\kappa _m^2h_{\max }^{2{m}}$. The substitution of the resulting estimates $\varvec{b_h}(\varvec{v_h},\varvec{v_h})\geqslant 1$ and $\varvec{a_h}(\varvec{v_h},\varvec{v_h})\leqslant \mu _j+ 2\lambda _h^2\kappa _m^2h_{\max }^{2{m}}$ in (3.14) concludes the proof of $\lambda _h(j)\leqslant \mu _{j}+2\lambda _h^2\kappa _m^2h_{\max }^{2{m}}$ in (3.13) for $j=1, \ldots , N$. $\square$

Proof of the lower bound

This situation is similar to [27, Thm. 6.4] and adapted below for completeness. For $j=1,\ldots , k+1$, let $(\lambda _h({j}),\varvec{\phi _h}({j}))\in {\mathbb {R}}^+\times \varvec{V_h}$ denote the first $\varvec{b_h}$-orthonormal eigenpairs of (1.2) with $\varvec{\phi _h}({j})=(\phi _{\textrm{pw}}({j}),\phi _{\textrm{nc}}({j}))$. The test functions $(v_{\textrm{nc}},v_{\textrm{nc}})\in V({\mathcal {T}})\times V({\mathcal {T}})\subset \varvec{V_h}$ and $(v_{\textrm{pw}},0)\in \varvec{V_h}$ in (1.2) show

$$\begin{aligned} a_{\textrm{pw}}(\phi _{\textrm{nc}}({j}),v_{\textrm{nc}})&=\lambda _h(j) b(\phi _{\textrm{pw}}({j}),v_{\textrm{nc}})\quad \text {and}\nonumber \\ \phi _{\textrm{pw}}({j})-\phi _{\textrm{nc}}({j})&=\lambda _h({j})\kappa _{m}^2h_{{\mathcal {T}}}^{2m}\phi _{\textrm{pw}}({j}). \end{aligned}$$

(3.15)

For $\xi =(\xi _1,\ldots ,\xi _{j})\in {\mathbb {R}}^{j}$ with $\sum _{{\ell }=1}^{j}\xi _{\ell }^2=1$, set

$$\begin{aligned} v_{\textrm{nc}}:=\sum _{{\ell }=1}^{j} \xi _{\ell } \phi _{\textrm{nc}}({\ell }),\quad v_{\textrm{pw}}:=\sum _{{\ell }=1}^{j} \xi _{\ell } \phi _{\textrm{pw}}({\ell }),\quad \text { and } \quad w_{\textrm{pw}}:=\sum _{{\ell }=1}^{j} \xi _{\ell } \lambda _h(\ell )\phi _{\textrm{pw}}({\ell }). \end{aligned}$$

Since $(\phi _{\textrm{pw}}(\alpha ),\phi _{\textrm{pw}}(\beta ))_{L^2(\Omega )}=\delta _{\alpha \beta }$ for $\alpha ,\beta =1,\ldots , k+1$, $\Vert v_{\textrm{pw}}\Vert _{L^2(\Omega )}=1$ and $\Vert w_{\textrm{pw}}\Vert _{L^2(\Omega )}=\sqrt{\sum _{{\ell }=1}^{j} \xi _{\ell }^2 \lambda _h({\ell })^2}\leqslant \lambda _h({j})$. The combination of this with (3.15) and a Cauchy–Schwarz inequality leads to $||| v_{\textrm{nc}}|||_{\textrm{pw}}^2=b(w_{\textrm{pw}},v_{\textrm{nc}}) \leqslant \lambda _h(j)\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}$ and $v_{\textrm{pw}}-v_{\textrm{nc}}=\kappa _{m}^2h_{{\mathcal {T}}}^{2m}w_{\textrm{pw}}.$ This and a reverse triangle inequality result in

$$\begin{aligned} 0&<1-\lambda _h(j)\kappa _{m}^2 h_{\max }^{2{m}} \leqslant 1-\kappa _{m}^2 h_{\max }^{2m} \Vert w_{\textrm{pw}}\Vert _{L^2(\Omega )} \nonumber \\&\leqslant \Vert v_{\textrm{pw}}- \kappa _{m}^2h_{{\mathcal {T}}}^{2m}w_{\textrm{pw}}\Vert _{L^2(\Omega )}= \Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.16)

This holds for all $v_{\textrm{nc}}\in U_{{j}}:=span \{\phi _{\textrm{nc}}(1),\ldots ,\phi _{\textrm{nc}}({j})\} \subset V({{\mathcal {T}}})$ with coefficients $(\xi _1,\ldots ,\xi _j)\in {\mathbb {R}}^j$ of Euclidean norm one. Hence $dim ({U_{j}})={j}$ and the discrete min-max principle [7, 45] for (3.3) show

$$\begin{aligned} \mu _{j} \leqslant \max _{v_{\textrm{nc}}\in U_{j}\setminus \{0\}} {||| v_{\textrm{nc}}|||^2_{\textrm{pw}}}/{\Vert v_{\textrm{nc}}\Vert _{1+\delta }^2}. \end{aligned}$$

(3.17)

Let $v_{\textrm{nc}}=\sum _{\ell =1}^{{j}}\alpha _\ell \phi _{\textrm{nc}}(\ell )\in U_{{j}}$ denote a maximizer in (3.17) with $\sum _{\ell =1}^{{j}}\alpha _\ell ^2=1$. The combination of $||| v_{\textrm{nc}}|||^2_{\textrm{pw}} \leqslant \lambda _h(j)\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}$, (3.16)–(3.17), and $\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant \Vert v_{\textrm{nc}}\Vert _{1+\delta }$ from Remark 3.8 provides

$$\begin{aligned} \mu _{j} \leqslant \frac{||| v_{\textrm{nc}}|||^2_{\textrm{pw}}}{\Vert v_{\textrm{nc}}\Vert _{1+\delta }^2} \leqslant \frac{||| v_{\textrm{nc}}|||^2_{\textrm{pw}}}{\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}^2} \leqslant \frac{\lambda _h({j})}{1-\lambda _h(j)\kappa _{m}^2 h_{\max }^{2{m}}}. \end{aligned}$$

Recall $\lambda _h(j)\leqslant \lambda _h(k+1)\leqslant \lambda _{k+1}$ from the lower bound property in Theorem 3.1.b to conclude the proof of the associated lower bound for all $j=1,\ldots ,k$. $\square$

The subsequent corollaries adapt the notation $\mu _j,\, \lambda _h(j), \, \lambda _j$ from Lemma 3.9.

Corollary 3.10

For any $j=1,\ldots ,k+1$, it holds $|\mu _{{j}}-\lambda _h({j})|+|\mu _{j} -\lambda _{j}|\rightarrow 0$ as $h_{\max }\rightarrow 0$.

Proof

The a priori convergence analysis [24, Thm. 1.2] implies $\lim _{h_{\max }\rightarrow 0}\lambda _h({j})\rightarrow \lambda _{j}$. Lemma 3.9 shows $|\lambda _h(j)-\mu _j|\leqslant h_{\max }^{2m}\kappa _{m}^2\max \{ 2\lambda _h^2, \lambda _h(j)\mu _{j}\}\rightarrow 0$ as $h_{\max }\rightarrow 0$. $\square$

Corollary 3.11

There exists $0<\varepsilon _2\leqslant \min \{1/2,\varepsilon _1\}$ such that (H1)–(H2) hold for ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _2)$.

Proof

Corollary 3.10 and $\lambda _h=\lambda _h({k})\in \{\mu _1,\ldots , \mu _N\}$ lead to $\varepsilon _a>0$ such that $\lambda _h=\lambda _h({k})=\mu _{k}$ has the correct index k for all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _a)$. It also leads to some $\varepsilon _b>0$ such that $\mu _{{k}-1}<\mu _{k}<\mu _{{k}+1}$ for all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _b)$. Then $\varepsilon _2:= \min \{1/2,\varepsilon _1,\varepsilon _a,\varepsilon _b\}$ and ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _2)$ imply (H1)–(H2). $\square$

3.3 Proof of (3.5)–(3.6) for the $L^2$ error control

Recall $M_6$ from (H2), $\delta$ from Remark 3.3, the norm equivalence from Remark 3.8, and the auxiliary source problem (3.4).

Proof of (3.5)

Recall the following straightforward result from [18, Eq. (2.8)]: Any $u,v\in L^2(\Omega )$ with $\Vert u\Vert _{L^2(\Omega )}=\Vert v\Vert _{L^2(\Omega )}=1$ satisfy

$$\begin{aligned} \big (1+b(u,v)\big )\Vert u-v\Vert ^2_{L^2(\Omega )}=2\min _{t\in {\mathbb {R}}}{\Vert u-tv\Vert _{L^2(\Omega )}^2}. \end{aligned}$$

This, a triangle inequality, $t:=(z_{\textrm{nc}},u_{\textrm{nc}})_{1+\delta } \Vert \phi _{k}\Vert _{L^2(\Omega )}^2$, and $v_{\textrm{nc}}:=z_{\textrm{nc}}-t u_{\textrm{nc}}$ lead to

$$\begin{aligned} 2^{-1/2}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant \Vert u-t u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant \Vert u-z_{\textrm{nc}}\Vert _{L^2(\Omega )}+\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.18)

Since the eigenvectors $\phi _1,\ldots ,\phi _N$ of (3.3) are $(\bullet ,\bullet )_{1+\delta }$-orthonormal and form a basis of $V({\mathcal {T}})$, there exist Fourier coefficients $\alpha _1,\ldots , \alpha _N\in {\mathbb {R}}$ with $v_{\textrm{nc}}=\sum _{j=1}^N \alpha _j\phi _j$ and $\Vert v_{\textrm{nc}}\Vert _{1+\delta }^2=\sum _{{j}=1}^N\alpha _{j}^2$. Since $(\lambda _h, u_{\textrm{nc}})$ solves (3.2), (H1) implies $u_{\textrm{nc}}\in span \{\phi _k\}$ with $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$. Hence $u_{\textrm{nc}}=\pm \phi _{k}/\Vert \phi _{k}\Vert _{L^2(\Omega )}$, $t=\pm (z_{\textrm{nc}},\phi _{k})_{1+\delta }\Vert \phi _{k}\Vert _{L^2(\Omega )}$, and $(u_{\textrm{nc}},\phi _{k})_{1+\delta }=\pm \Vert \phi _{k}\Vert _{L^2(\Omega )}^{-1}$. Consequently,

$$\begin{aligned} \alpha _{k}=(v_{\textrm{nc}},\phi _{k})_{1+\delta } =(z_{\textrm{nc}},\phi _{k})_{1+\delta }-t (u_{\textrm{nc}},\phi _{k})_{1+\delta } =0. \end{aligned}$$

Since $(u_{\textrm{nc}},\phi _j)_{1+\delta }=0$ for all $j=1,\ldots , N$ with $j\not =k$, $\alpha _{j} =(v_{\textrm{nc}},\phi _{j})_{1+\delta }=(z_{\textrm{nc}},\phi _{j})_{1+\delta }$. Since $\phi _j$ is an eigenvector in (3.3) and $z_{\textrm{nc}}$ solves (3.4), it follows

$$\begin{aligned} \alpha _{j} =(z_{\textrm{nc}},\phi _{j})_{1+\delta } =\frac{1}{\mu _{j}}a_{\textrm{pw}}(z_{\textrm{nc}},\phi _{{j}}) =\frac{\lambda }{\mu _{j}}(u,\phi _{{j}})_{1+\delta }. \end{aligned}$$

Hence $(u-z_{\textrm{nc}},\phi _{j})_{1+\delta }=(\mu _{j}/\lambda -1)\alpha _{j}$. These values for the coefficients $\alpha _j$ and the separation condition (H2) imply

$$\begin{aligned} \Vert v_{\textrm{nc}}\Vert _{1+\delta }^2&=\sum _{ {j} \ne {k}}\alpha _{j}^2 =\sum _{ {j} \ne {k}} \Big \vert \frac{\lambda }{\mu _{j}-\lambda }\Big \vert \vert \alpha _{{j}}\vert \vert (u-z_{\textrm{nc}},\phi _{j})_{1+\delta } \vert \\&\leqslant M_6 \sum _{{j} \ne {k}} (u-z_{\textrm{nc}},\alpha ^\prime _{{j}}\phi _{j})_{1+\delta } \end{aligned}$$

for a sign in $\alpha _{j}^\prime \in \{\pm \alpha _{j}\}$ such that $\vert (u-z_{\textrm{nc}},\alpha _{{j}}\phi _{j})_{1+\delta }\vert =(u-z_{\textrm{nc}},\alpha _{j}^\prime \phi _{j})_{1+\delta }$ and with the abbreviation $\sum _{j\ne k}=\sum _{{{j}=1, {j} \ne {k}}}^N$. This and a Cauchy–Schwarz inequality show

$$\begin{aligned} M_6^{-1}\Vert v_{\textrm{nc}}\Vert _{1+\delta }^2&\leqslant \Big (u-z_{\textrm{nc}},\sum _{ {j} \ne {k}}\alpha _{j}^\prime \phi _{j}\Big )_{1+\delta } \leqslant \Vert u-z_{\textrm{nc}}\Vert _{1+\delta } \Vert v_{\textrm{nc}}\Vert _{1+\delta }. \end{aligned}$$

The norm equivalence in Remark 3.8 proves $\Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant \Vert v_{\textrm{nc}}\Vert _{1+\delta } \leqslant \sqrt{2}M_6\Vert u-z_{\textrm{nc}}\Vert _{L^2(\Omega )}$. This and (3.18) conclude the proof of (3.5) with $C_1:=\sqrt{2}(1+\sqrt{2} M_6)$. $\square$

Proof of (3.6)

Given the solution $z_{\textrm{nc}}\in V({\mathcal {T}})$ to (3.4), let ${w}\in V:=H^m_0(\Omega )$ solve

$$\begin{aligned} a({w},\varphi )=b(u-Jz_{\textrm{nc}},\varphi )\quad \text {for all }\varphi \in V. \end{aligned}$$

(3.19)

Since $u-Jz_{\textrm{nc}}\in V\subset L^2(\Omega )$, the elliptic regularity (2.1) guarantees ${w}\in H^{{m}+\sigma }(\Omega )$ and

$$\begin{aligned} \Vert {w}\Vert _{H^{{m}+\sigma }(\Omega )}\leqslant C(\sigma ) \Vert u-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.20)

The combination of (3.20) with Corollary 2.1.b shows

$$\begin{aligned} ||| {w}-I{w}|||_{\textrm{pw}}\leqslant \left( \frac{h_{\max }}{\pi }\right) ^{\sigma }\Vert {w}\Vert _{H^{{m}+\sigma }(\Omega )} \leqslant C(\sigma )\left( \frac{h_{\max }}{\pi }\right) ^{\sigma }\Vert u-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

(3.21)

The test function $\varphi =u-Jz_{\textrm{nc}}$ in the auxiliary problem (3.19) leads to

$$\begin{aligned} \Vert u-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )}^2&= a(u, {w}-JI{w})+a_{\textrm{pw}}({w}, z_{\textrm{nc}}-Jz_{\textrm{nc}})\nonumber \\&\quad +a(u, JI{w})-a_{\textrm{pw}}({w},z_{\textrm{nc}}). \end{aligned}$$

(3.22)

Since (J1) asserts $I({w}-JI{w})=0$, Corollary 2.1.c and a triangle inequality show

$$\begin{aligned} a(u,{w}-JI{w})&= a_{\textrm{pw}}(u, (1-I)({w}-JI{w})) \\ {}&\leqslant ||| u-z_{\textrm{nc}}|||_{\textrm{pw}} (||| {w}-I{w}|||_{\textrm{pw}} +||| I{w}-JI{w}|||_{\textrm{pw}}). \end{aligned}$$

Then (J2) implies that $a(u,{w}-JI{w}) \leqslant (1+{{M_2}}) ||| {w}-I{w}|||_{\textrm{pw}}||| u-z_{\textrm{nc}}|||_{\textrm{pw}}.$ Corollary 2.3.c proves for the second term in the right-hand side of (3.22) that

$$\begin{aligned} a_{\textrm{pw}}({w}, z_{\textrm{nc}}-Jz_{\textrm{nc}})&\leqslant M_2||| {w}-I{w}|||_{\textrm{pw}}||| u-z_{\textrm{nc}}|||_{\textrm{pw}}. \end{aligned}$$

Corollary 2.1.c ensures $a_{\textrm{pw}}({w},z_{\textrm{nc}})=a_{\textrm{pw}}(I{w},z_{\textrm{nc}})$. Since $(\lambda , u)$ is an eigenpair of (1.1) and $z_{\textrm{nc}}$ satisfies (3.4), this implies

$$\begin{aligned} a(u, JI{w})-a_{\textrm{pw}}({w},z_{\textrm{nc}})&=b(\lambda u, JI{w})-a_{\textrm{pw}}(I{w},z_{\textrm{nc}}) =\lambda b(u, JI{w} -I{w}- \delta I{w}). \end{aligned}$$

Corollary 2.3.b shows $b( u, JI{w} -I{w}) \leqslant {{M_2}}\kappa _{m}^{2} h_{\max }^{2{m}} ||| u-z_{\textrm{nc}}|||_{\textrm{pw}} ||| {w}-I{w}|||_{\textrm{pw}}.$ The discrete Friedrichs inequality

$$\begin{aligned} \Vert v_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant C_{\textrm{dF}}||| v_{\textrm{nc}}|||_{\textrm{pw}} \text { for all } v_{\textrm{nc}}\in V({\mathcal {T}})\text { with } C_{\textrm{dF}}:=C_{F}(1+{M_2})+{M_2} h^m_{\max } \end{aligned}$$

(3.23)

is a direct consequence of the Friedrichs inequality $\Vert v\Vert _{L^2(\Omega )}\leqslant C_F||| v|||$ for any $v\in V$ and (J2); cf. [19, Cor. 4.11] for details in case $m=1$; the proof for $m=2$ is analogous. This, (I2), and the boundedness of $\Pi _0$ imply $C_{\textrm{dF}}^{-1}\Vert I {w}\Vert _{L^2(\Omega )}\leqslant ||| Iw|||_{\textrm{pw}}=\Vert \Pi _0D^m w\Vert _{L^2(\Omega )} \leqslant \Vert {w}\Vert _{H^{m}(\Omega )}.$ The Cauchy–Schwarz inequality leads to

$$\begin{aligned} - b(\lambda u, \delta I{w})\leqslant \Vert \delta \lambda u\Vert _{L^2(\Omega )} \Vert I{w}\Vert _{L^2(\Omega )} \leqslant C_{\textrm{dF}}\Vert \delta \lambda u\Vert _{L^2(\Omega )} \Vert {w}\Vert _{H^{{m}+\sigma }(\Omega )}. \end{aligned}$$

This bounds the last term on the right-hand side of (3.22). The substitution in (3.22) and $\lambda \kappa _m^2 h_{\max }^{2m}\leqslant 1/2$ result in

$$\begin{aligned} \Vert u-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )}^2&\leqslant (1+5M_2/2) ||| {w}-I{w}|||_{\textrm{pw}}||| u-z_{\textrm{nc}}|||_{\textrm{pw}} \\&\quad + C_{\textrm{dF}}\Vert \delta \lambda u\Vert _{L^2(\Omega )} \Vert {w}\Vert _{H^{{m}+\sigma }(\Omega )}. \end{aligned}$$

This and (3.20)–(3.21) imply

$$\begin{aligned} C(\sigma )^{-1}\Vert u-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant (h_{\max }/\pi )^{\sigma }(1+5M_2/2) ||| u-z_{\textrm{nc}}|||_{\textrm{pw}} +C_{\textrm{dF}}\Vert \delta \lambda u\Vert _{L^2(\Omega )}. \end{aligned}$$

Corollary 2.3.a implies $\Vert z_{\textrm{nc}}-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant {{M_2}}\kappa _{m} h_{\max }^{m} ||| u-z_{\textrm{nc}}|||_{\textrm{pw}}.$ This, $0<\sigma \leqslant 1\leqslant m$, $h_{\max }<1$, and a triangle inequality show

$$\begin{aligned} \Vert u-z_{\textrm{nc}}\Vert _{L^2(\Omega )}&\leqslant \Vert Jz_{\textrm{nc}}-z_{\textrm{nc}}\Vert _{L^2(\Omega )}+\Vert u-Jz_{\textrm{nc}}\Vert _{L^2(\Omega )}\\&\leqslant C_2 \big (h_{\max }^{\sigma }||| u-z_{\textrm{nc}}|||_{\textrm{pw}} +\Vert \delta \lambda u\Vert _{L^2(\Omega )}\big ) \end{aligned}$$

with the constant $C_2:=\max \big \{C(\sigma )(1+5M_2/2)/\pi ^{\sigma }+{{M_2}} \kappa _{m},C(\sigma )C_{\textrm{dF}}\big \}.$ $\square$

3.4 Proof of (3.7) and (3.10) for the energy error control

Recall $\delta$ from Remark 3.3 and that $z_{\textrm{nc}}\in V({{\mathcal {T}}})$ solves (3.4).

Proof of (3.7)

Elementary algebra with $a_{\textrm{pw}}(z_{\textrm{nc}},u)=a_{\textrm{pw}}(z_{\textrm{nc}},Iu)$ from Corollary 2.1.c shows

$$\begin{aligned} ||| u-z_{\textrm{nc}}|||^2_{\textrm{pw}}&= a (u, u-JIu)+a_{\textrm{pw}}(u, Jz_{\textrm{nc}}-z_{\textrm{nc}})\nonumber \\&\quad + a(u,JIu-Jz_{\textrm{nc}}) +a_{\textrm{pw}}(z_{\textrm{nc}},z_{\textrm{nc}}- Iu). \end{aligned}$$

(3.24)

Corollary 2.1.c and Corollary 2.3.c control the first two terms in the decomposition

$$\begin{aligned}&a (u, u-JIu)+ a_{\textrm{pw}}(u, Jz_{\textrm{nc}}-z_{\textrm{nc}})\\&\quad =a_{\textrm{pw}}(u,u-Iu)+a_{\textrm{pw}}(u,Iu-JIu)+ a_{\textrm{pw}}(u, Jz_{\textrm{nc}}-z_{\textrm{nc}})\\&\quad \leqslant (1+{{M_2}}) ||| u-Iu|||_{\textrm{pw}}^2+{{M_2}} ||| u-Iu|||_{\textrm{pw}} ||| u-z_{\textrm{nc}}|||_{\textrm{pw}}. \end{aligned}$$

Recall that $(\lambda ,u)$ is an eigenpair of (1.1) and $z_{\textrm{nc}}$ satisfies (3.4). Consequently,

$$\begin{aligned} a(u,JIu-Jz_{\textrm{nc}})+a_{\textrm{pw}}(z_{\textrm{nc}}, z_{\textrm{nc}}-Iu)&= b(\lambda u, JIu-Jz_{\textrm{nc}}+ (1+\delta )(z_{\textrm{nc}}-Iu))\\&=\lambda b( u, (J-1)(Iu-z_{\textrm{nc}}))\\&\quad +\lambda b(\delta u, z_{\textrm{nc}}-Iu). \end{aligned}$$

Corollary 2.3.b, $\kappa _m^2\lambda h_{\max }^{2m}\leqslant 1/2$, and a triangle inequality show

$$\begin{aligned} \lambda b(u,(J-1)(Iu-z_{\textrm{nc}})) \leqslant M_2/2\,||| u-Iu|||_{\textrm{pw}}(||| u-Iu|||_{\textrm{pw}} +||| u-z_{\textrm{nc}}|||_{\textrm{pw}}) . \end{aligned}$$

Since Cauchy–Schwarz and triangle inequalities show $b(\delta \lambda u, z_{\textrm{nc}}-Iu) \leqslant \Vert \delta \lambda u \Vert _{L^2(\Omega )} (\Vert u-z_{\textrm{nc}} \Vert _{L^2(\Omega )}+\Vert u-Iu\Vert _{L^2(\Omega )}),$ (I1) provides the first and (3.6) the second estimate in

$$\begin{aligned} b(\delta \lambda u, z_{\textrm{nc}}-Iu)&\leqslant \Vert \delta \lambda u \Vert _{L^2(\Omega )} (\Vert u-z_{\textrm{nc}} \Vert _{L^2(\Omega )} +\kappa _m h_{\max }^{m}||| u-Iu|||_{\textrm{pw}})\\&\leqslant \Vert \delta \lambda u \Vert _{L^2(\Omega )} (C_2 h_{\max }^{\sigma }||| u-z_{\textrm{nc}}|||_{\textrm{pw}}+C_2\Vert \delta \lambda u \Vert _{L^2(\Omega )} \\&\quad +\kappa _m h_{\max }^{m}||| u-Iu|||_{\textrm{pw}}). \end{aligned}$$

Since $h_{\max }^{m}||| u-Iu|||_{\textrm{pw}}\leqslant h_{\max }^{\sigma }||| u-z_{\textrm{nc}}|||_{\textrm{pw}}$ from Corollary 2.1.a, a weighted Young inequality shows $b(\delta \lambda u, z_{\textrm{nc}}-Iu) \leqslant ((C_2+\kappa _m)^2h_{\max }^{2\sigma }+C_2)\Vert \delta \lambda u\Vert _{L^2(\Omega )}^2 +||| u-z_{\textrm{nc}}|||_{\textrm{pw}}^2/4.$ The substitution of the displayed estimates in (3.24) shows

$$\begin{aligned} ||| u-z_{\textrm{nc}}|||^2_{\textrm{pw}} \leqslant&(1+3M_2/2) ||| u-Iu|||_{\textrm{pw}}^2+ 3 M_2/2\, ||| u-Iu|||_{\textrm{pw}}||| u-z_{\textrm{nc}}|||_{\textrm{pw}} \\&+ ((C_2+\kappa _m)^2h_{\max }^{2\sigma }+C_2)\Vert \delta \lambda u\Vert _{L^2(\Omega )}^2 +||| u-z_{\textrm{nc}}|||_{\textrm{pw}}^2/4. \end{aligned}$$

This and $3M_2/2||| u-Iu|||_{\textrm{pw}}||| u-z_{\textrm{nc}}|||_{\textrm{pw}} \leqslant 9M_2^2/4 ||| u-Iu|||_{\textrm{pw}}^2+ ||| u-z_{\textrm{nc}}|||_{\textrm{pw}} ^2/4$ conclude the proof of (3.7) with $C_3^2:=2 \max \{1+3{{M_2}}/2+9{{M_2}}^2/4, (C_2+\kappa _m)^2h_{\max }^{2\sigma }+C_2\}$. $\square$

Proof of (3.10)

The proof of the efficiency estimate of the volume residual is based on Verfürth’s bubble-function methodology [49], comparable to [3, Thm. 2], [33, Prop. 3.1], and given here for completeness. Let $\varphi _z\in S^1({\mathcal {T}}):=P_1({\mathcal {T}})\cap C(\Omega )$ denote the nodal basis function associated with the vertex $z\in {\mathcal {V}}$. For any $T\in {\mathcal {T}}$, let $b_T:=4^{4\,m}\prod _{z\in {\mathcal {V}}(T)}\varphi _z^m\in P_{4\,m}(T)\cap W^{m,\infty }_0(T)\subset V$ denote the volume-bubble-function with $supp (b_T)=T$ and $\Vert b_T\Vert _{\infty }=1$. An inverse estimate $\Vert p\Vert _{L^2(T)}\leqslant c_b \Vert p\Vert _{b_T}$ for any polynomial $p\in P_m(T)$ leads to

$$\begin{aligned} c_b^{-2}\Vert u_{\textrm{nc}}\Vert _{L^2(T)}^2 \leqslant \Vert u_{\textrm{nc}}\Vert _{b_T}^2 = (u_{\textrm{nc}},u)_{b_T}-( u_{\textrm{nc}},u-u_{\textrm{nc}})_{b_T}. \end{aligned}$$

(3.25)

The Cauchy–Schwarz inequality and $\Vert b_T\Vert _{\infty }=1$ show $( u_{\textrm{nc}},u-u_{\textrm{nc}})_{b_T} \leqslant \Vert u_{\textrm{nc}}\Vert _{L^2(T)}\Vert u-u_{\textrm{nc}}\Vert _{L^2(T)}.$ An integration by parts proves $\int _T D^{m}(b_T u_{\textrm{nc}})\,d x=0$ since $b_T u_{\textrm{nc}}\in H^{m}_0(T)$, i.e., $D^{m}{(}b_Tu_{\textrm{nc}}{)}$ is $L^2$-orthogonal to $P_0(T)$. Recall that $(\lambda ,u)$ is an eigenpair of (1.1) and the support of $b_Tu_{\textrm{nc}}$ is T. This, (I2), and the Cauchy–Schwarz inequality result in

$$\begin{aligned} \lambda b(u,b_{T}u_{\textrm{nc}})&=a_{\textrm{pw}}(u,b_{T}u_{\textrm{nc}})=(D^{m} u, D^{m} (b_T u_{\textrm{nc}}))_{L^2(T)} \\&\leqslant \vert u-Iu\vert _{H^m(T)}\vert b_T u_{\textrm{nc}}\vert _{H^m(T)}. \end{aligned}$$

An inverse estimate for polynomials in $P_{5m}(T)$ with the constant $c_{\textrm{inv}}$ and the boundedness of $b_T$ show $\lambda b(u,b_{T}u_{\textrm{nc}}) \leqslant c_{\textrm{inv}} h_T^{-m} \vert u-Iu\vert _{H^m(T)}\Vert u_{\textrm{nc}}\Vert _{L^2(T)}$. This provides $c_b^{-2}h_T^{m}\Vert u_{\textrm{nc}}\Vert _{L^2(T)} \leqslant h_T^{m} \Vert u-u_{\textrm{nc}}\Vert _{L^2(T)}+c_{\textrm{inv}}\lambda ^{-1}\vert u-Iu\vert _{H^m(T)}$ for all $T\in {\mathcal {T}}$ in (3.25). The sum over all $T\in {\mathcal {T}}$ concludes the proof of (3.10) with $C_4=c_b^2\max \{1,c_{\textrm{inv}}\}$. $\square$

3.5 Proof of Theorem 3.1.c

Proof of (3.12) for $\varvec{\varepsilon _4}\varvec{>0}$ Recall $c_1:=2\lambda ^2\kappa _m^2 C_1C_2(1+C_3)$ and (3.8) as a result of (3.5)–(3.7). A triangle inequality, Remark 3.3, and (3.8) show

$$\begin{aligned} \Vert \delta \lambda u\Vert _{L^2(\Omega )}&\leqslant 2\lambda ^2\kappa _m^2 h_{\max }^{2m}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}+ \Vert \delta \lambda u_{\textrm{nc}}\Vert _{L^2(\Omega )} \\&\leqslant \frac{c_1C_3h_{\max }^{2m}}{1+C_3}h_{\max }^{\sigma } ||| u-Iu|||_{\textrm{pw}} +c_1h_{\max }^{2m}\Vert \delta \lambda u\Vert _{L^2(\Omega )} +\Vert \delta \lambda u_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

Since $0<\varepsilon _3:=\min \{\varepsilon _2,(2c_1)^{-1/2m}\}$ ensures $c_1h_{\max }^{2m}\leqslant 1/2$ for all ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _3)$, the previously displayed estimate reads $\Vert \delta \lambda u\Vert _{L^2(\Omega )} \leqslant \frac{c_1C_3h_{\max }^{2m}}{1+C_3}h_{\max }^{\sigma } ||| u-Iu|||_{\textrm{pw}} +\Vert \delta \lambda u\Vert _{L^2(\Omega )}/2 +\Vert \delta \lambda u_{\textrm{nc}}\Vert _{L^2(\Omega )}.$ This implies (3.9). The bound (3.9) for $\Vert \delta \lambda u\Vert _{L^2(\Omega )}$ recasts (3.8) as

$$\begin{aligned} C_1^{-1}C_2^{-1}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant 2C_3 h_{\max }^\sigma ||| u-Iu|||_{\textrm{pw}} +2(1+C_3)\lambda \Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}. \end{aligned}$$

Remark 3.3 and (3.10) control the last term in

$$\begin{aligned} (2\kappa _m^2 C_4)^{-1}\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}&\leqslant C_4^{-1}\lambda h_{\max }^m \Vert h_{{\mathcal {T}}}^{m} u_{\textrm{nc}}\Vert _{L^2(\Omega )} \\ {}&\leqslant \lambda h_{\max }^{2m}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} +h_{\max }^m||| u-Iu|||_{\textrm{pw}}. \end{aligned}$$

Recall that $c_2:=4\lambda ^2\kappa _m^2 C_1C_2(1+C_3)C_4$ and $\varepsilon _4:=\min \{\varepsilon _3, (2c_2)^{-1/2m}\}<1$ ensure $c_2 h_{\max }^{2m}\leqslant 1/2$. Hence the last term in (3.11) is $\leqslant \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} /2$ and can be absorbed. This concludes the proof of (3.12) with $C_5:=2{C_1C_2} \big (2C_3+4\kappa _m^2 \lambda (1+C_3)C_4\big ).$ $\square$

Recall $0<\varepsilon _5\leqslant \varepsilon _4$ such that $b(u,u_{\textrm{nc}})>0$ for any ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _5)$.

Proof of Theorem 3.1.c for $\varvec{\varepsilon _5}$ Recall $\lambda _h\leqslant \lambda$ and $\Vert u\Vert _{L^2(\Omega )}=\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$. The continuous eigenpair $(\lambda ,u)$ in (1.1) satisfies $\lambda =||| u|||^2$. The discrete eigenpair $(\lambda _h,u_{\textrm{nc}})$ solves (3.2) and so $\lambda _h=||| u_{\textrm{nc}}|||_{\textrm{pw}}^2/\Vert u_{\textrm{nc}}\Vert _{1+\delta }^2$ with $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$. Then

$$\begin{aligned} ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2&=\lambda -2a_{\textrm{pw}}(u,u_{\textrm{nc}})+\lambda _h \Vert u_{\textrm{nc}}\Vert _{1+\delta }^2 \quad \text {and} \\ \Vert u_{\textrm{nc}}\Vert ^2_{1+\delta }-1&=b(\delta u_{\textrm{nc}},u_{\textrm{nc}}) =\Vert u_{\textrm{nc}}\Vert _{\delta }^2. \end{aligned}$$

This and elementary algebra show for the left-hand side of Theorem 3.1.c that

$$\begin{aligned} LHS :=\lambda -\lambda _h+||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2 +\Vert u_{\textrm{nc}}\Vert _{\delta }^2&= 2\lambda -2 a_{\textrm{pw}}(u,u_{\textrm{nc}}) +(1+\lambda _h) \Vert u_{\textrm{nc}}\Vert _{\delta }^2. \end{aligned}$$

Since u is the eigenfunction in (1.1) and $2b(u,u-u_{\textrm{nc}})=\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2$ from $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1=\Vert u\Vert _{L^2(\Omega )}$, it follows

$$\begin{aligned} \lambda&=\lambda b(u,u_{\textrm{nc}})+\lambda b(u,u-u_{\textrm{nc}}) \\&= \lambda b(u,u_{\textrm{nc}}-Ju_{\textrm{nc}})+a_{\textrm{pw}}(u, Ju_{\textrm{nc}}) +\lambda /2\ \Vert u-u_{\textrm{nc}}\Vert ^2_{L^2(\Omega )}. \end{aligned}$$

The combination of the last two displayed identities eventually leads to

$$\begin{aligned} LHS&=(1+\lambda _h)\Vert u_{\textrm{nc}}\Vert _{\delta }^2+ \lambda \Vert u-u_{\textrm{nc}}\Vert ^2_{L^2(\Omega )} + 2\lambda b(u,u_{\textrm{nc}}-Ju_{\textrm{nc}})\nonumber \\&\quad +2a_{\textrm{pw}}(u, Ju_{\textrm{nc}}-u_{\textrm{nc}}). \end{aligned}$$

(3.26)

Recall $2\lambda \kappa _m^2h_{\max }^{2m}\leqslant 1$. The combination of Remark 3.3 and (3.10) implies that

$$\begin{aligned} \Vert u_{\textrm{nc}}\Vert _{\delta }&\leqslant \sqrt{2}\kappa _m \lambda ^{1/2}\Vert h_{{\mathcal {T}}}^m u_{{\textrm{nc}}}\Vert _{L^2(\Omega )} \leqslant C_4\Vert u -u_{\textrm{nc}}\Vert _{L^2(\Omega )}+\sqrt{2/\lambda }\kappa _m C_4||| u-Iu|||_{\textrm{pw}} \end{aligned}$$

and (3.12) controls $\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant C_5h_{\max }^{\sigma }||| u-Iu|||_{\textrm{pw}}.$ Corollary 2.3.b asserts $2\lambda b(u,u_{\textrm{nc}}-Ju_{\textrm{nc}}) \leqslant M_2 ||| u-Iu |||_{\textrm{pw}}||| u-u_{\textrm{nc}} |||_{\textrm{pw}}.$ Corollary 2.3.c shows $a_{\textrm{pw}}(u, Ju_{\textrm{nc}}-u_{\textrm{nc}}) \leqslant {{M_2}} ||| u-Iu|||_{\textrm{pw}} ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}.$ Since $\lambda _h\leqslant \lambda$, these estimates lead in (3.26) to

$$\begin{aligned} LHS \hspace{-0.2em}&\leqslant \big (\hspace{-0.1em}(1\hspace{-0.1em} +\hspace{-0.1em} \lambda ) C_4^2 (C_5h_{\max }^{\sigma }+\sqrt{2/\lambda }\kappa _m)^2 \hspace{-0.1em} +\hspace{-0.1em} \lambda C_5^2h_{\max }^{2\sigma } \big ) ||| u-Iu|||^2 \hspace{-0.1em}\\&\quad +\hspace{-0.1em} 3M_2 ||| u-Iu |||_{\textrm{pw}}||| u_{\textrm{nc}}-u |||_{\textrm{pw}}. \end{aligned}$$

A weighted Young inequality and the absorption of $||| u_{\textrm{nc}}-u |||_{\textrm{pw}}^2/2$ conclude the proof of Theorem 3.1.c with $C_0:=\max \{C_5^2, 2((1+\lambda ) C_4^2 (C_5h_{\max }^{\sigma }+\sqrt{2/\lambda }\kappa _m)^2+ \lambda C_5^2h_{\max }^{2\sigma }) +9M_2^2\}$. $\square$

4 Optimal convergence rates

This section verifies some general axioms of adaptivity [12, 26] sufficient for optimal rates for AFEM4EVP and prepares the conclusion of the proof of Theorem 1.1 in Sect. 5.

4.1 Stability and reduction

The 2-level notation of Table 1 concerns one coarse triangulation ${\mathcal {T}}\in {\mathbb {T}}$ and one fine triangulation $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$. Let $(\lambda ,u)\in {\mathbb {R}}^+\times V$ denote the k-th continuous eigenpair of (1.1) with a simple eigenvalue $\lambda \equiv \lambda _k$ and the normalization $\Vert u\Vert _{L^2(\Omega )}=1$. Choose $\varepsilon _5>0$ as in Theorem 3.1, suppose ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _5)$, and let $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$ be any admissible refinement of ${\mathcal {T}}$.

Definition 4.1

(2-level notation) Let $(\lambda _h,\varvec{u_h})\in {\mathbb {R}}^+\times \varvec{V_h}$ (resp. $({\widehat{\lambda }}_h,\varvec{{\widehat{u}}_h})\in {\mathbb {R}}^+\times \varvec{{\widehat{V}}_h}$) with $\varvec{u_h}=(u_{\textrm{pw}},u_{\textrm{nc}})\in \varvec{V_h}:=P_m({\mathcal {T}})\times V({{\mathcal {T}}})$ (resp. $\varvec{{\widehat{u}}_h}=({\widehat{u}}_{\textrm{pw}},{\widehat{u}}_{\textrm{nc}}) \in \varvec{{\widehat{V}}_h}:=P_{m}(\widehat{{\mathcal {T}}})\times V(\widehat{{\mathcal {T}}})$) denote the k-th discrete eigenpair of (1.2) with the simple algebraic eigenvalue $\lambda _h\equiv \lambda _h({k})$ (resp. ${\widehat{\lambda }}_h\equiv {\widehat{\lambda }}_h(k)$), the normalization $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$ (resp. $\Vert {\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$), and the sign convention $b(u,u_{\textrm{nc}})> 0$ (resp. $b(u,{\widehat{u}}_{\textrm{nc}})> 0$). Recall ${\widehat{h}}_{\max }:=\max _{T\in \widehat{{\mathcal {T}}}}h_T\leqslant h_{\max }:=\max _{T\in {\mathcal {T}}}h_T\leqslant \varepsilon _5$, $\lambda _h,\,{\widehat{\lambda }}_h\leqslant \lambda$ from Theorem 3.1.b, and ${\delta }$ from Remark 3.3 with its analogue ${\widehat{\delta }}:=(1-{\widehat{\lambda }}_h\kappa _{m}^2h_{\widehat{{\mathcal {T}}}}^{2{m}})^{-1}-1 \in P_0(\widehat{{\mathcal {T}}})$ on the fine level. The constant $C_{\delta }:=2\lambda \kappa _m^2$ satisfies $\delta \leqslant C_\delta h_{{\mathcal {T}}}^{2{m}}$ and ${\widehat{\delta }}\leqslant C_\delta h_{\widehat{{\mathcal {T}}}}^{2{m}}$. Recall the estimator $\eta ^2(T)$ for any $T\in {\mathcal {T}}$ from (1.4) and define ${\widehat{\eta }}^2(T)$, for any $T\in \widehat{{\mathcal {T}}}$ with volume |T| and the set of faces $\widehat{{\mathcal {F}}}(T)$, by

$$\begin{aligned} {\widehat{\eta }}^2(T) := |T|^{2{m}/3}\Vert {\widehat{\lambda }}_h {\widehat{u}}_{\textrm{nc}} \Vert ^2_{L^2(T)} +|T|^{1/3}\sum _{F\in \widehat{{\mathcal {F}}}(T)}\Vert [{D}^{m}_{\textrm{pw}} {\widehat{u}}_{\textrm{nc}}]_F\times \nu _F\Vert ^2_{L^2(F)}. \end{aligned}$$

(4.1)

Table 1 2-level notation with respect to ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon )$ (left) and an admissible refinement $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$ (right)

The sum conventions $\eta ^2({\mathcal {M}}):= \sum _{T\in {\mathcal {M}}}\eta ^2(T)$ for ${\mathcal {M}}\subset {\mathcal {T}}$ and ${\widehat{\eta }}^2(\widehat{{\mathcal {M}}}):= \sum _{T\in \widehat{{\mathcal {M}}}}{\widehat{\eta }}^2(T)$ for $\widehat{{\mathcal {M}}}\subset \widehat{{\mathcal {T}}}$ from Table 1 apply throughout this section. Abbreviate the distance function

$$\begin{aligned} \delta ^2({\mathcal {T}},\widehat{{\mathcal {T}}}) :=\Vert \lambda _h u_{\textrm{nc}}-{\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 + ||| u_{\textrm{nc}}-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}}^2. \end{aligned}$$

(4.2)

Theorem 4.2

(stability and reduction) There exist $\Lambda _1,\Lambda _2~>~0$, such that, for any ${\mathcal {T}}$ and $\widehat{{\mathcal {T}}}$ from Definition 4.1, the following holds

• (A1) Stability. $\displaystyle \big |{\eta ({\mathcal {T}}\cap \widehat{{\mathcal {T}}}) -{{\widehat{\eta }}}( {\mathcal {T}}\cap \widehat{{\mathcal {T}}})}\big |\leqslant \Lambda _1 \delta ({\mathcal {T}},\widehat{{\mathcal {T}}}),$

• (A2) Reduction. $\displaystyle {\widehat{\eta }}(\widehat{\mathcal T}{\setminus }{\mathcal {T}}) \leqslant 2^{-1/12} \eta ({\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}}) + \Lambda _2 \delta ({\mathcal {T}},\widehat{{\mathcal {T}}}).$

Proof

A reverse triangle inequality in ${\mathbb {R}}^L$ for the number $L:=|{\mathcal {T}}\cap \widehat{{\mathcal {T}}}|$ of tetrahedra in ${\mathcal {T}}\cap \widehat{{\mathcal {T}}}$ and one for each common tetrahedra $T\in {\mathcal {T}}\cap \widehat{{\mathcal {T}}}$ and each of its faces $F\in {\mathcal {F}}(T)$ lead to

$$\begin{aligned} \big \vert \eta ({\mathcal {T}}\cap \widehat{{\mathcal {T}}}) -{\widehat{\eta }}({\mathcal {T}}\cap \widehat{{\mathcal {T}}})\big \vert ^2 \leqslant&\sum _{T\in {\mathcal {T}}\cap \widehat{{\mathcal {T}}}}\Big (|T|^{2{m}/3}\Vert \lambda _hu_{\textrm{nc}} -{\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(T)}^2 \\&+|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)} \Vert [D^{m}_{\textrm{pw}}(u_{\textrm{nc}}-{\widehat{u}}_{\textrm{nc}})]_F\times \nu _F\Vert _{L^2(F)}^2\Big ). \end{aligned}$$

The discrete jump control from [26, Lem. 5.2] with constant $C_{\textrm{jc}}(\ell )$ (that only depends on the shape-regularity of ${\mathbb {T}}$ and the polynomial degree $\ell \in {\mathbb {N}}_0$) reads

$$\begin{aligned} \sum _{T\in {\mathcal {T}}}|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)}\Vert [g]_F\Vert ^2_{L^2(F)} \leqslant C_{\textrm{jc}}(\ell )^2\Vert g\Vert _{L^2(\Omega )}^2 \quad \text { for any }g\in P_\ell ({\mathcal {T}}). \end{aligned}$$

The combination of the two displayed estimates concludes the proof of (A1) with $\Lambda _1^2=\max \big \{\max _{T\in {\mathcal {T}}_0}|T|^{2m/3}, C_{\textrm{jc}}(0)^2\big \}$. For any tetrahedron $K\in {\mathcal {T}}\setminus \widehat{{\mathcal {T}}}$, let $\widehat{{\mathcal {T}}}(K):=\{T\in \widehat{{\mathcal {T}}}:\, T\subset K\}$ denote its fine triangulation. The newest-vertex bisection guarantees $|{T}|\leqslant |K|/2$ for the volume |T| of any ${T}\in \widehat{{\mathcal {T}}}(K)$. This, a triangle inequality, and $(a+b)^2\leqslant (1+\beta )a^2+(1+1/\beta )b^2$ for $a,\,b\geqslant 0,\,\beta =2^{1/6}-1>0$ show

$$\begin{aligned} {{\widehat{\eta }}^2(\widehat{{\mathcal {T}}}(K))}&\leqslant 2^{-1/6} {\eta ^2(K)} +(1+1/\beta )\sum _{T\in \widehat{{\mathcal {T}}}(K)}\Big (|T|^{2{m}/3} \Vert \lambda _h u_{\textrm{nc}}-{\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(K)}^2\\&\quad +|T|^{1/3}\sum _{F\in \widehat{{\mathcal {F}}}(T)} \Vert [D^{m}_{\textrm{pw}}({\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}})]_F\times \nu _F\Vert _{L^2(F)}^2\Big ). \end{aligned}$$

The summation over all $K\in {\mathcal {T}}\setminus \widehat{{\mathcal {T}}}$ and the above jump control conclude the proof of (A2) with $\Lambda _2^2=2^{1/6}/(2^{1/6}-1)\,\max \big \{\max _{T\in {\mathcal {T}}_0}|T|^{2m/3}, C_{\textrm{jc}}(0)^2\big \}$. The arguments for (A1)–(A2) are similar for other problems; cf., e.g., [12, 20, 22, 26] for more details. $\square$

4.2 Towards discrete reliability

Given the 2-level notation of Definition 4.1 with respect to ${\mathcal {T}}$ and $\widehat{{\mathcal {T}}}$, let ${\mathcal {R}}_1:=\{K\in {\mathcal {T}}:\,\exists \, T\in {\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}} \text { with } dist (K,T)=0\} \subset {\mathcal {T}}$ denote the set of coarse but not fine tetrahedra plus one layer of coarse tetrahedra around. Lemma 4.3–4.5 prepare the proof of the discrete reliability in Theorem 4.6 below. Let ${\widehat{I}}:V+V(\widehat{{\mathcal {T}}})\rightarrow V(\widehat{{\mathcal {T}}})$ denote the interpolation operator on the fine level of $\widehat{{\mathcal {T}}}$ so that (I3) and a Cauchy–Schwarz inequality show, for any $v\in V+V(\widehat{{\mathcal {T}}})$ and any $w\in V+V({{\mathcal {T}}})+V(\widehat{{\mathcal {T}}})$, that

$$\begin{aligned} \begin{aligned} \vert b( (I-{\widehat{I}})v,w)\vert&\leqslant \Vert (I-{\widehat{I}})v\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \Vert w\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})},\\ \vert a_{\textrm{pw}}((I-{\widehat{I}})v,w)\vert&\leqslant \Vert D^{m}_{\textrm{pw}} (I-{\widehat{I}})v\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \Vert D^{m}_{\textrm{pw}}w\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}. \end{aligned} \end{aligned}$$

(4.3)

Lemma 4.3

(distance control I) There exists $C_6>0$ such that any ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _5)$ and the difference $e:= {\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}}$ satisfy

$$\begin{aligned} C_6^{-1}||| e|||_{\textrm{pw}}^2\leqslant & {} \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 +\Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 + \Vert e\Vert _{L^2(\Omega )}^2\\{} & {} +\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 +\Vert {\widehat{\delta }}{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )}^2. \end{aligned}$$

Proof

Corollary 2.1.c shows $a_{\textrm{pw}}(e, {\widehat{u}}_{\textrm{nc}}- Ju_{\textrm{nc}}) = a_{\textrm{pw}}\big ({\widehat{u}}_{\textrm{nc}}, {\widehat{u}}_{\textrm{nc}}- {\widehat{I}}Ju_{\textrm{nc}}\big ) -a_{\textrm{pw}}\big (u_{\textrm{nc}}, I({\widehat{u}}_{\textrm{nc}}- Ju_{\textrm{nc}})\big ).$ Since $(\lambda _h,u_{\textrm{nc}})$ and $({\widehat{\lambda }}_h,{\widehat{u}}_{\textrm{nc}})$ solve (3.2), this and (J1) lead to

$$\begin{aligned} a_{\textrm{pw}}(e, {\widehat{u}}_{\textrm{nc}}- Ju_{\textrm{nc}})&= b\big ({\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}, (1+{\widehat{\delta }})({\widehat{u}}_{\textrm{nc}}- {\widehat{I}}Ju_{\textrm{nc}})\big )\nonumber \\&\quad -b\big (\lambda _h u_{\textrm{nc}},(1+\delta )(I{\widehat{u}}_{\textrm{nc}}- u_{\textrm{nc}})\big )\nonumber \\&=b({\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}-\lambda _h u_{\textrm{nc}},e) +b({\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}},{\widehat{\delta }}e)-b(\lambda _h u_{\textrm{nc}},\delta e)\nonumber \\&\quad +b\big ({\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}, (1\hspace{-0.1em} +\hspace{-0.1em} {\widehat{\delta }})(u_{\textrm{nc}}-{\widehat{I}}Ju_{\textrm{nc}})\big )\nonumber \\&\quad + b\big (\lambda _h u_{\textrm{nc}},(1\hspace{-0.1em} +\hspace{-0.1em} \delta )({\widehat{u}}_{\textrm{nc}}-I{\widehat{u}}_{\textrm{nc}})\big ). \end{aligned}$$

(4.4)

Elementary algebra with $\Vert u_{\textrm{nc}}\Vert _{L^2(\Omega )}=\Vert {\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )}=1$ shows (as, e.g., in [13, Lem. 3.1])

$$\begin{aligned} b({\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}-\lambda _h u_{\textrm{nc}},e)&=\frac{{\widehat{\lambda }}_h+\lambda _h}{2}\Vert e\Vert _{L^2(\Omega )}^2 +\frac{{\widehat{\lambda }}_h-\lambda _h}{2}b({\widehat{u}}_{\textrm{nc}} +u_{\textrm{nc}},{\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}})\\&=\frac{{\widehat{\lambda }}_h+\lambda _h}{2}\Vert e\Vert _{L^2(\Omega )}^2. \end{aligned}$$

Cauchy–Schwarz inequalities verify

$$\begin{aligned} b({\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}},{\widehat{\delta }}e)-b(\lambda _h u_{\textrm{nc}},\delta e) \leqslant \Vert e\Vert _{L^2(\Omega )}\big ({\widehat{\lambda }}_h \Vert {\widehat{\delta }}{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )} +\lambda _h\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}\big ). \end{aligned}$$

Since $1+{\widehat{\delta }}\leqslant 2$ and ${\widehat{\lambda }}_h\leqslant \lambda$ from Table 1, the right inverse property (J1) and (4.3) result in

$$\begin{aligned} b\big ((1+{\widehat{\delta }}){\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}, u_{\textrm{nc}}-{\widehat{I}}Ju_{\textrm{nc}}\big )&=b\big ((1+{\widehat{\delta }}){\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}, (I-{\widehat{I}})Ju_{\textrm{nc}}\big ) \\&\leqslant 2 \Vert h_{{\mathcal {T}}}^{m}{\lambda }{\widehat{u}}_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})} \Vert h_{{\mathcal {T}}}^{-{m}} (I-{\widehat{I}})Ju_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}. \end{aligned}$$

The triangle inequality $\Vert h_{{\mathcal {T}}}^{m}{\lambda }{\widehat{u}}_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}{\setminus } \widehat{{\mathcal {T}}})} \leqslant h_{\max }^{m}\lambda \Vert e\Vert _{L^2(\Omega )} +\Vert h_{{\mathcal {T}}}^{m}\lambda u_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}{\setminus } \widehat{{\mathcal {T}}})}$ and $\lambda /\lambda _h\leqslant 2$ from Theorem 3.1.a imply $\Vert h_{{\mathcal {T}}}^{m}\lambda u_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}{\setminus } \widehat{{\mathcal {T}}})} \leqslant 2\Vert h_{{\mathcal {T}}}^{m}\lambda _h u_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}{\setminus } \widehat{{\mathcal {T}}})}$. Since the interpolation operators I and ${\widehat{I}}$ satisfy (I3)–(I4), it follows that

$$\begin{aligned} \Vert h_{{\mathcal {T}}}^{-{m}} (I-{\widehat{I}})Ju_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}&=\Vert h_{{\mathcal {T}}}^{-{m}} (1-I){\widehat{I}}Ju_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}\\&\leqslant \kappa _d \Vert D^{m}_{\textrm{pw}}(1-I){\widehat{I}}Ju_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}. \end{aligned}$$

Recall $D^m_{\textrm{pw}}u_{\textrm{nc}}\in P_0({\mathcal {T}};{\mathbb {R}}^{3^m})$. The condition (I2) and the $L^2$-orthogonal projections $\Pi _0$ (resp. ${\widehat{\Pi }}_0$) onto $P_0({\mathcal {T}})$ (resp. $P_0(\widehat{{\mathcal {T}}})$) lead to the estimate

$$\begin{aligned}&\kappa _d^{-1} \Vert h_{{\mathcal {T}}}^{-{m}} (I-{\widehat{I}})Ju_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}\leqslant \Vert (\Pi _0-{\widehat{\Pi }}_0) D^{m}Ju_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}\\&\quad = \Vert (\Pi _0-{\widehat{\Pi }}_0) D^{m}_{\textrm{pw}} (Ju_{\textrm{nc}}-u_{\textrm{nc}})\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})} \leqslant \Vert D^{m}_{\textrm{pw}}(Ju_{\textrm{nc}}-u_{\textrm{nc}})\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})}. \end{aligned}$$

The estimate (4.3) and $\delta \leqslant 1$ from Table 1 imply the first inequality and (I4) and Corollary 2.1.a the second in

$$\begin{aligned} b\big (\lambda _h u_{\textrm{nc}},(1+\delta )({\widehat{u}}_{\textrm{nc}}-I{\widehat{u}}_{\textrm{nc}})\big )&=b\big (\lambda _hu_{\textrm{nc}},(1+\delta )({\widehat{I}}-I){\widehat{u}}_{\textrm{nc}}\big ) \\&\leqslant 2 \Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})} \Vert h_{{\mathcal {T}}}^{-{m}}({\widehat{u}}_{\textrm{nc}} -I{\widehat{u}}_{\textrm{nc}})\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{{\mathcal {T}}})} \\&\leqslant 2 \kappa _d \Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{ {\mathcal {T}}})} ||| e||| _{\textrm{pw}}. \end{aligned}$$

The combination of the six previously displayed estimates and $\lambda _h,{\widehat{\lambda }}_h\leqslant \lambda$ lead in (4.4) to

$$\begin{aligned} a_{\textrm{pw}}(e, {\widehat{u}}_{\textrm{nc}}- Ju_{\textrm{nc}})&\leqslant 2\kappa _d \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}}) \Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \big (\lambda h_{\max }^{m}\Vert e\Vert _{L^2(\Omega )} \\&\quad +2\Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}\big )+\lambda \Vert e\Vert _{L^2(\Omega )}\big ( \Vert e\Vert _{L^2(\Omega )}+\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )} \\&\quad +\Vert {\widehat{\delta }}\widehat{ u}_{\textrm{nc}}\Vert _{L^2(\Omega )}\big ) +2\kappa _d||| e||| _{\textrm{pw}}\Vert h_{{\mathcal {T}}}^{m} \lambda _h u_{\textrm{nc}} \Vert _{L^2({{\mathcal {T}}}\setminus \widehat{ {\mathcal {T}}})}.&\end{aligned}$$

Additionally, Corollary 2.3.c and (4.3) show

$$\begin{aligned} a_{\textrm{pw}}(e, Ju_{\textrm{nc}}-u_{\textrm{nc}})&=a_{\textrm{pw}}((1-I)e, Ju_{\textrm{nc}}-u_{\textrm{nc}}) =a_{\textrm{pw}}\big (({\widehat{I}}-I){\widehat{u}}_{\textrm{nc}}, Ju_{\textrm{nc}}-u_{\textrm{nc}}\big )\\&\leqslant \Vert D^{m}_{\textrm{pw}}(1-I)e\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}. \end{aligned}$$

Condition (I2) and the boundedness of $\Pi _0$ show $\Vert D^{m}_{\textrm{pw}}(1-I)e\Vert _{L^2({\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}})} \leqslant ||| e|||_{\textrm{pw}}$. This and the combination of the two previously displayed estimates with a triangle inequality prove

$$\begin{aligned} ||| e|||_{\textrm{pw}}^2&=a_{\textrm{pw}}(e, Ju_{\textrm{nc}}-u_{\textrm{nc}})+a_{\textrm{pw}}(e, {\widehat{u}}_{\textrm{nc}}- Ju_{\textrm{nc}})\\&\leqslant \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}}) \Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \big (||| e|||_{\textrm{pw}}+2\kappa _d \lambda h_{\max }^{m}\Vert e\Vert _{L^2(\Omega )}\\&\quad +4\kappa _d \Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}} \Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}\big )+\lambda \Vert e\Vert _{L^2(\Omega )}\big ( \Vert e\Vert _{L^2(\Omega )}\\&\quad +\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}+\Vert {\widehat{\delta }}\widehat{ u}_{\textrm{nc}}\Vert _{L^2(\Omega )}\big ) +2\kappa _d||| e||| _{\textrm{pw}}\Vert h_{{\mathcal {T}}}^{m}\lambda _h u_{\textrm{nc}} \Vert _{L^2({{\mathcal {T}}}\setminus \widehat{ {\mathcal {T}}})}\\&\leqslant (1+4\kappa _d^2+\kappa _d^2\lambda ^2h_{\max }^{2m}) \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 +\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 \\&\quad +\Vert {\widehat{\delta }}\widehat{ u}_{\textrm{nc}}\Vert _{L^2(\Omega )}^2+(1+\lambda +\lambda ^2/2) \Vert e\Vert _{L^2(\Omega )}^2 \\&\quad +(1+4\kappa _d^2) \Vert h_{{\mathcal {T}}}^{m}\lambda _h u_{\textrm{nc}}\Vert _{L^2({{\mathcal {T}}}\setminus \widehat{ {\mathcal {T}}})}^2 +{||| e||| _{\textrm{pw}}^2}/{2} \end{aligned}$$

with weighted Young inequalities in the last step. This concludes the proof with $C_6:=2\max \{1+4\kappa _d^2+\kappa _d^2\lambda ^2h_{\max }^{2m}, 1+\lambda +\lambda ^2/2\}$. $\square$

4.2.1 Reliability and efficiency

A first consequence of Lemma 4.3 is the reliability of the error estimator $\eta ({\mathcal {T}})$ from (1.4).

Theorem 4.4

(reliability and efficiency) There exist $C_{\textrm{rel}},\,C_{\textrm{eff}},$ and $\varepsilon _6>0$ such that $C_{\textrm{eff}}^{-1}\eta ({\mathcal {T}}) \leqslant ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}\leqslant C_{\textrm{rel}} \,\eta ({\mathcal {T}})$ holds for ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _6)$.

Proof of reliability

Lemma 4.3 holds for any refinement $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$ of ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _5)$ and we may consider a sequence $\widehat{{\mathcal {T}}}=\widehat{{\mathcal {T}}}_\ell$ of uniform mesh-refinements of ${\mathcal {T}}$. The reliability follows in the limit as ${\widehat{h}}_{\max }\rightarrow 0$ for $\ell \rightarrow \infty$ and $||| u-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}}\rightarrow 0$ from Theorem 3.1.c. The left-hand side of Lemma 4.3 converges to $C_6^{-1}||| u-u_{\textrm{nc}}|||_{\textrm{pw}}$. On the right-hand side, $\Vert {\widehat{\delta }}\widehat{ u}_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant C_{\delta } {\widehat{h}}_{\max }^{2m}$ converges to zero and $\Vert e\Vert _{L^2(\Omega )}\rightarrow \Vert u- u_{\textrm{nc}}\Vert _{L^2(\Omega )}$ as ${\widehat{h}}_{\max }\rightarrow 0$. Moreover the shape-regularity $h_T\leqslant C_{\textrm{sr}}|T|^{1/3}$ for $T\in {\mathcal {T}}\in {\mathbb {T}}$, (J2), and $\Vert {\delta }{ u}_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant 2\kappa _m^2 {h}_{\max }^{m}\Vert h_{{\mathcal {T}}}^m\lambda _h{ u}_{\textrm{nc}}\Vert _{L^2(\Omega )}$ show

$$\begin{aligned}{} & {} \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2(\Omega )}^2 +\Vert h_{{\mathcal {T}}}^{m}\lambda _h u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 +\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2\\{} & {} \quad \leqslant \max \{M_1,C_{\textrm{sr}}^{2m}(1+4\kappa _m^4 {h}_{\max }^{2m})\} \eta ^2({\mathcal {T}}). \end{aligned}$$

For the remaining term on the right-hand side, (3.12) and Corollary 2.1.a show

$$\begin{aligned} C_5^{-1}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant h_{\max }^{\sigma } ||| u-Iu|||_{\textrm{pw}} \leqslant h_{\max }^{\sigma } ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}. \end{aligned}$$

A reduction to $\varepsilon _6:=\min \{\varepsilon _5,(2C_5^2{C_6})^{-1/(2\sigma )}\}$ such that $C_5^{2}C_6h_{\max }^{2\sigma }\leqslant 1/2$ allows for the absorption of $C_5^2{C_6}h_{\max }^{2\sigma }||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2\leqslant ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2/2$ and concludes the proof with $C_{\textrm{rel}}^2:=2C_6\max \{M_1,C_{\textrm{sr}}^{2m}(1+4\kappa _m^4 {h}_{\max }^{2m})\}$. $\square$

Proof of efficiency

The condition (J2) guarantees

$$\begin{aligned} M_1/M_2^2\sum _{T\in {\mathcal {T}}}|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)}\Vert [D^{m}_{\textrm{pw}}u_{\textrm{nc}}]_F\times \nu _F\Vert _{L^2(F)}^2&\leqslant \min _{v\in V} ||| v-u_{\textrm{nc}}|||_{\textrm{pw}}^2\\&\leqslant ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2. \end{aligned}$$

The combination of $|T|^{1/3}\leqslant h_T$, $\lambda _h\leqslant \lambda$, and the efficiency (3.10) with $||| u-Iu|||_{\textrm{pw}}\leqslant ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}$ from Corollary 2.1.a implies that

$$\begin{aligned} \sum _{T\in {\mathcal {T}}}|T|^{2m/3}\Vert \lambda _h u_{\textrm{nc}}\Vert _{L^2(T)}^2&\leqslant \Vert h_{{\mathcal {T}}}^m \lambda _h u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 \\&\leqslant 2C_4^2 \big (\lambda ^2h_{\max }^{2m}\Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 + ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}^2\big ). \end{aligned}$$

Theorem 3.1.c concludes the proof with $C_{\textrm{eff}}^2:=M_2^2/M_1+2C_4^2+2C_4^2 C_0\lambda ^2h_{\max }^{2m+2\sigma }$. $\square$

4.2.2 Discrete reliability

Lemma 4.5

(distance control II) There exists a constant $C_7>0$ such that $\displaystyle \Vert {\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}-\lambda _hu_{\textrm{nc}}\Vert _{L^2(\Omega )} \hspace{-0.1em}+\hspace{-0.1em}\Vert {\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}}\Vert _{L^2(\Omega )} \hspace{-0.1em}+\hspace{-0.1em}\Vert {\widehat{\delta }} {\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )} \hspace{-0.1em}+\hspace{-0.1em}\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )} \hspace{-0.1em}\leqslant \hspace{-0.1em} C_7 h_{\max }^{\sigma }||| u-u_{\textrm{nc}}|||_{\textrm{pw}} \hspace{-0.1em}\leqslant \hspace{-0.1em} C_7C_{\textrm{rel}} h_{\max }^{\sigma }\eta ({\mathcal {T}})$ holds for any ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _6)$.

Proof

Triangle inequalities and the normalization $\Vert u\Vert _{L^2(\Omega )}=1$ show

$$\begin{aligned} \Vert {\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}-\lambda _h u_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant&\lambda _h \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} +{\widehat{\lambda }}_h \Vert u-{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )} +\vert {\widehat{\lambda }}_h-\lambda _h\vert . \end{aligned}$$

Theorem 3.1.c and Corollary 2.1.b imply $\vert \lambda -\lambda _h\vert \leqslant C_0||| u-Iu |||_{\textrm{pw}}^2\leqslant C_0(h_{\max }/\pi )^{2\sigma }\Vert u\Vert _{H^{m+\sigma }(\Omega )}^2$. Since the eigenfunction $u\in V$ in (1.1) solves the source problem with right-hand side $\lambda u\in L^2(\Omega )$, (2.1) implies $\Vert u\Vert _{H^{m+\sigma }(\Omega )}\leqslant C(\sigma ) \Vert \lambda u\Vert _{L^2(\Omega )}=C(\sigma )\lambda$. The same arguments apply to $\vert \lambda -{\widehat{\lambda }}_h\vert$. This and ${\widehat{h}}_{\max }^{\sigma } ||| u-{\widehat{I}}u |||_{\textrm{pw}}\leqslant h_{\max }^{\sigma }||| u-Iu |||_{\textrm{pw}}$ result in

$$\begin{aligned} \vert {\widehat{\lambda }}_h-\lambda _h\vert \leqslant \vert \lambda -\lambda _h\vert +\vert \lambda -{\widehat{\lambda }}_h\vert \leqslant 2C_0C(\sigma )\lambda /\pi ^{\sigma } h_{\max }^{\sigma }||| u-Iu |||_{\textrm{pw}}. \end{aligned}$$

Recall $\lambda _h,{\widehat{\lambda }}_h\leqslant \lambda$, $\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant C_{\delta } ^{{1/2}} h_{\max }^m\Vert u_{\textrm{nc}}\Vert _{\delta }$, and $\Vert {\widehat{\delta }} {\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )} \leqslant C_{\delta }^{{1/2}} {\widehat{h}}_{\max }^m\Vert {\widehat{u}}_{\textrm{nc}} \Vert _{{\widehat{\delta }}}$ from Table 1. The last two displayed estimates, a triangle inequality, and Theorem 3.1.c show

$$\begin{aligned}&\Vert {\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}-\lambda _hu_{\textrm{nc}}\Vert _{L^2(\Omega )} + \Vert {\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}}\Vert _{L^2(\Omega )} +\Vert {\widehat{\delta }} {\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )} +\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )} \\&\quad \leqslant 2\big ( (C_0C(\sigma )\lambda /\pi ^{\sigma }+C_0^{1/2}(1+\lambda )) h_{\max }^{\sigma }+{(}C_{\delta }C_0{)}^{1/2} h_{\max }^{m}\big ) ||| u-Iu|||_{\textrm{pw}} \end{aligned}$$

with $||| u-{\widehat{I}}u |||_{\textrm{pw}}\leqslant ||| u-Iu |||_{\textrm{pw}}$ and ${\widehat{h}}_{\max }\leqslant h_{\max }$. Since $h_{\max }\leqslant \varepsilon _6<1$ and $1/2<\sigma \leqslant 1\leqslant m$, Corollary 2.1.a concludes the proof of the first bound in Lemma 4.5 with $C_7:=2C_0C(\sigma )\lambda /\pi ^{\sigma }+2C_0^{1/2}(1+\lambda +C_{\delta }^{{1/2}})$. The second claim follows from Theorem 4.4. $\square$

Theorem 4.6

(discrete reliability) There exist constants $\Lambda _3, \, M_3>0$ such that ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _6)$ with maximal mesh-size $h_{\max }\leqslant \varepsilon _6$ ($\varepsilon _6$ from Theorem 4.4) and $\epsilon _3:=M_3{h}_{\max }^{2\sigma }$ imply

(A${3_{\varepsilon }}$) Discrete reliability. $\displaystyle \delta ^2({\mathcal {T}},\widehat{{\mathcal {T}}}) \leqslant \Lambda _3 \eta ^2({\mathcal {R}}_1) +\epsilon _3\eta ^2({\mathcal {T}}).$

Proof

Recall that Lemma 4.3 shows

$$\begin{aligned} C_6^{-1}||| {\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}}|||_{\textrm{pw}}^2&\leqslant \Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 +\Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 \\&\quad + \Vert {\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2+\Vert {\widehat{\delta }}{\widehat{u}}_h\Vert _{L^2(\Omega )}^2 +\Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2. \end{aligned}$$

This and Lemma 4.5 lead with $M_3:=C_7^2 C_{\textrm{rel}}^2\max \{1,C_6\}$ to

$$\begin{aligned} \delta ^2({\mathcal {T}},\widehat{{\mathcal {T}}})&=\Vert {\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}-\lambda _h u_{\textrm{nc}}\Vert _{L^2(\Omega )}^2 +||| {\widehat{u}}_{\textrm{nc}}-u_{\textrm{nc}}|||_{\textrm{pw}}^2 \\&\leqslant C_6\Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 +C_6\Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2 + M_3h_{\max }^{2\sigma }\eta ^2({\mathcal {T}}). \end{aligned}$$

The shape regularity $h_T\leqslant C_{\textrm{sr}}|T|^{1/3}$ for any $T\in {\mathcal {T}}\in {\mathbb {T}}$ guarantees

$$\begin{aligned} \Vert h_{{\mathcal {T}}}^{m}\lambda _hu_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \leqslant C_{\textrm{sr}} ^{m} |T|^{m/3}\Vert \lambda _hu_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \leqslant C_{\textrm{sr}} ^{m} \eta ({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})\leqslant C_{\textrm{sr}} ^{m} \eta ({\mathcal {R}}_1). \end{aligned}$$

with ${\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}}\subset {\mathcal {R}}_1$ in the last step. Remark 2.2 asserts

$$\begin{aligned} M_5^{-1}\Vert D^{m}_{\textrm{pw}} (u_{\textrm{nc}}-Ju_{\textrm{nc}})\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}^2&\leqslant \sum _{T\in {\mathcal {R}}_1}|T|^{1/3}\sum _{F\in {\mathcal {F}}(T)} \Vert [D^{m}_{\textrm{pw}} u_{\textrm{nc}}]_F\times \nu _F\Vert _{L^2(F)}^2\\&\quad \leqslant \eta ^2({\mathcal {R}}_1). \end{aligned}$$

The combination of the last three displayed inequalities concludes the proof of (A${3_{\varepsilon }}$) with $\Lambda _3:=C_6(C_{\textrm{sr}}^{2m}+ M_5)$. $\square$

4.3 Quasiorthogonality

The quasiorthogonality in Theorem 4.7 below concerns the outcome $({\mathcal {T}}_j)_{j\in {\mathbb {N}}_0}$ of AFEM4EVP. Let $u_j\in V({\mathcal {T}}_j)$ abbreviate the nonconforming component of the discrete solution ${\varvec{u}}_j=(u_{\textrm{pw}}, u_\textrm{nc})=:(u_{\textrm{pw}}, u_j)\in P_m({\mathcal {T}}_j)\times V({\mathcal {T}}_j)$ with $b(u,u_j)> 0$, $\Vert u_j\Vert _{L^2{(\Omega })}=1$, and $\lambda _{j}(k)\leqslant \lambda$ the associated eigenvalue from AFEM4EVP on the level $j\in {\mathbb {N}}_0$. Recall the distance

$$\begin{aligned} \delta ^2({\mathcal {T}}_j,{\mathcal {T}}_{j+1})=\Vert \lambda _j(k)u_j-\lambda _{j+1}(k)u_{j+1}\Vert _{L^2(\Omega )}^2+||| u_j-u_{j+1}|||_{\textrm{pw}}^2 \end{aligned}$$

for the triangulations ${\mathcal {T}}_j$ and ${\mathcal {T}}_{j+1}$. Set $h_0:=\max _{T\in {\mathcal {T}}_0}h_T$ and recall $\varepsilon _6>0$ from Theorem 4.4.

Theorem 4.7

(quasiorthogonality) For any $0<\beta \leqslant C_{\textrm{eff}}^2/C_{\textrm{rel}}^2$, there exist $\Lambda _4$, ${\widetilde{\Lambda }}_4$, and $\epsilon _4:={\widetilde{\Lambda }}_4(\beta +h_0^{2\sigma }(1+\beta ^{-1}))>0$, such that ${\mathcal {T}}_0\in {\mathbb {T}}(\varepsilon _6)$ implies that the output $(\eta _j)_{j\in {\mathbb {N}}_0}$ and $({\mathcal {T}}_j)_{j\in {\mathbb {N}}_0}$ of AFEM4EVP satisfies

(A${4_{\varepsilon }}$) Quasiorthogonality. $\displaystyle \sum _{j=\ell }^{\ell +L}\delta ^2({\mathcal {T}}_j,{\mathcal {T}}_{j+1}) \leqslant \Lambda _{4}(1+\beta ^{-1}) \eta ^2_\ell +\epsilon _4 \sum _{j=\ell }^{\ell +L}\eta _j^2\quad \text { for any }\ell ,L\in {\mathbb {N}}_0.$

The following Lemma 4.8 in the 2-level notation of Definition 4.1 prepares the proof of Theorem 4.7 below.

Lemma 4.8

(2-level quasiorthogonality) There exists $C_{\textrm{qo}}>0$ such that, for ${\mathcal {T}}\in {\mathbb {T}}(\varepsilon _6)$, $\displaystyle { a_{\textrm{pw}}(u-{\widehat{u}}_{\textrm{nc}}, u_{\textrm{nc}}-{\widehat{u}}_{\textrm{nc}})} \leqslant C_{\textrm{qo}}\big (h_{\max }^{\sigma } ||| u-{u}_{\textrm{nc}}|||_{\textrm{pw}} +\Vert h_{{\mathcal {T}}}^{m}\lambda u\Vert _{L^2({\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}})}\big )$${||| u-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}}}$ holds.

Proof

Since $(\lambda _h,u_{\textrm{nc}})$ (resp. $({\widehat{\lambda }}_h,{\widehat{u}}_{\textrm{nc}})$) solves (3.2) with respect to ${\mathcal {T}}$ and ${\delta }\in P_0({{\mathcal {T}}})$ (resp. $\widehat{{\mathcal {T}}}$ and ${\widehat{\delta }}\in P_0(\widehat{{\mathcal {T}}})$ from Table 1), Corollary 2.1.c and elementary algebra show that

$$\begin{aligned}&a_{\textrm{pw}}(u_{\textrm{nc}}-{\widehat{u}}_{\textrm{nc}},u-{\widehat{u}}_{\textrm{nc}}) =a_{\textrm{pw}}\big (u_{\textrm{nc}}, I(u-{\widehat{u}}_{\textrm{nc}})\big ) -a_{\textrm{pw}}\big ({\widehat{u}}_{\textrm{nc}},{\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}\big )\nonumber \\&\quad = b\big (\lambda _hu_{\textrm{nc}}(1+\delta ), I(u-{\widehat{u}}_{\textrm{nc}})\big ) - b\big ( {\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}(1+{\widehat{\delta }}), {\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}\big )\nonumber \\&\quad = b\big (\lambda _hu_{\textrm{nc}}(1+\delta ) -{\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}(1+{\widehat{\delta }}),{\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}\big ) +\big (\lambda _hu_{\textrm{nc}}, (I-{\widehat{I}})(u-{\widehat{u}}_{\textrm{nc}})\big )_{1+\delta }. \end{aligned}$$

(4.5)

The Cauchy–Schwarz inequality, $\lambda _h,{\widehat{\lambda }}_h\leqslant \lambda$, and Lemma 4.5 in the last step prove

$$\begin{aligned} t_1&:=b\big (\lambda _hu_{\textrm{nc}}(1+\delta ) -{\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}(1+{\widehat{\delta }}),{\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}\big )\\&\leqslant \Big (\Vert \lambda _hu_{\textrm{nc}}-{\widehat{\lambda }}_h{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )} +\lambda _h \Vert \delta u_{\textrm{nc}}\Vert _{L^2(\Omega )} +{\widehat{\lambda }}_h\Vert {\widehat{\delta }} {\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )}\Big ) \Vert {\widehat{I}}u-{\widehat{u}}_{\textrm{nc}} \Vert _{L^2(\Omega )}\\&\leqslant \max \{1,\lambda \} C_7 h_{\max }^{\sigma } ||| u-{u}_{\textrm{nc}}|||_{\textrm{pw}} \Vert {\widehat{I}}u-{\widehat{u}}_{\textrm{nc}} \Vert _{L^2(\Omega )}. \end{aligned}$$

The discrete Friedrichs inequality (3.23) with respect to $V(\widehat{{\mathcal {T}}})$, (I2), and the $L^2$-projection ${\widehat{\Pi }}_0$ onto $P_0(\widehat{{\mathcal {T}}})$ lead to

$$\begin{aligned} C_{\textrm{dF}}^{-1}\Vert {\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}\Vert _{L^2(\Omega )}\leqslant ||| {\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}} =\Vert {\widehat{\Pi }}_0 D^{m}_{\textrm{pw}} (u-{\widehat{u}}_{\textrm{nc}})\Vert _{L^2(\Omega )} \leqslant ||| u-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}}. \end{aligned}$$

Consequently, $t_1\leqslant \max \{1,\lambda \} C_7C_{\textrm{dF}} h_{\max }^{\sigma } ||| u-{u}_{\textrm{nc}}|||_{\textrm{pw}} ||| u-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}}.$ Since $1+\delta \leqslant 2$ from Table 1, the arguments behind (4.3) also show

$$\begin{aligned} t_2:=&\big (\lambda _hu_{\textrm{nc}}, (I-{\widehat{I}})(u-{\widehat{u}}_{\textrm{nc}})\big )_{1+\delta } \leqslant 2 \Vert h_{{\mathcal {T}}}^{m}\lambda _h u_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} \Vert h_{{\mathcal {T}}} ^{-{m}}(I-{\widehat{I}})(u-{\widehat{u}}_{\textrm{nc}}) \Vert _{L^2(\Omega )}. \end{aligned}$$

Since (I3) implies $I({\widehat{I}}u)=Iu$, (I2) and (I4) for I and (I2) for ${\widehat{I}}$ show $\Vert h_{{\mathcal {T}}} ^{-{m}}(I-{\widehat{I}})(u-{\widehat{u}}_{\textrm{nc}}) \Vert _{L^2(\Omega )} =\Vert h_{{\mathcal {T}}} ^{-{m}}(1-I)({\widehat{I}}u-{\widehat{u}}_{\textrm{nc}}) \Vert _{L^2(\Omega )} \leqslant \kappa _d ||| (1-I){\widehat{I}}(u-{\widehat{u}}_{\textrm{nc}})|||_{\textrm{pw}} \leqslant \kappa _d ||| u-{\widehat{u}}_{\textrm{nc}}|||_{\textrm{pw}}.$ On the other hand, $\lambda _h\leqslant \lambda$, a triangle inequality, (3.12), and Corollary 2.1.a imply

$$\begin{aligned} \Vert h_{{\mathcal {T}}}^{m}\lambda _h u_{\textrm{nc}}\Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})}&\leqslant \Vert h_{{\mathcal {T}}}^{m}\lambda u \Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} + \lambda h_{\max }^m \Vert u-u_{\textrm{nc}}\Vert _{L^2(\Omega )} \\&\leqslant \Vert h_{{\mathcal {T}}}^{m}\lambda u \Vert _{L^2({\mathcal {T}}\setminus \widehat{{\mathcal {T}}})} + C_5\lambda h_{\max }^{m+\sigma } ||| u-u_{\textrm{nc}}|||_{\textrm{pw}}. \end{aligned}$$

Hence the upper bound $t_1+t_2$ in (4.5) is controlled and the above estimates lead to the assertion with $C_{\textrm{qo}}:=\max \{2\kappa _d, \max \{1,\lambda \}C_7C_{\textrm{dF}}+2C_5\lambda h_{\max }^m\kappa _d\}$. $\square$

Proof of Theorem 4.7

Recall that $u_j\in V({\mathcal {T}}_j)$ is the nonconforming component of the discrete solution ${\varvec{u}}_j=(u_{\textrm{pw}}, u_\textrm{nc})=:(u_{\textrm{pw}}, u_j)\in P_m({\mathcal {T}}_j)\times V({\mathcal {T}}_j)$ and that $\lambda _{j}(k)\leqslant \lambda$ is the associated eigenvalue from AFEM4EVP on the j-th level for $\ell \leqslant j\leqslant \ell +L$. Since ${\mathcal {T}}_j,\, {\mathcal {T}}_{j+1}\in {\mathbb {T}}({\mathcal {T}}_0)$ for $\ell \leqslant j\leqslant \ell +L$, Lemma 4.5 shows

$$\begin{aligned} \delta ^2({\mathcal {T}}_j, {\mathcal {T}}_{j+1})&\leqslant ||| u_j-u_{j+1}|||_{\textrm{pw}}^2+C_7^2C_{\textrm{rel}}^2h_0^{2\sigma }\eta ^2_j. \end{aligned}$$

Elementary algebra, Lemma 4.8, and two weighted Young inequalities show

$$\begin{aligned} ||| u_j-u_{j+1}|||_{\textrm{pw}}^2 -&||| u-u_j|||_{\textrm{pw}}^2+||| u-u_{j+1}|||_{\textrm{pw}}^2 =2 a_{\textrm{pw}}(u-u_{j+1}, u_j-u_{j+1})\\ \leqslant&2 C_{\textrm{qo}} \Big (h_0^{\sigma } ||| u-{u}_j|||_{\textrm{pw}} +\Vert h_{{\mathcal {T}}_j}^{m}\lambda u\Vert _{L^2({\mathcal {T}}_j\setminus {{\mathcal {T}}_{j+1}})}\Big ) ||| u-{u}_{j+1}|||_{\textrm{pw}} \\ \leqslant&\frac{2 C_{\textrm{qo}}^2}{\beta } h_0^{2\sigma } C_{\textrm{rel}}^2\eta _j^2 +\beta C_{\textrm{rel}}^2 \eta _{j+1}^2 +\frac{2 C_{\textrm{qo}}^2}{\beta } \Vert h_{{\mathcal {T}}_j}^{m} \lambda u\Vert ^2_{L^2({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1})} \end{aligned}$$

with Theorem 4.4 in the last step. Theorem 4.4 controls the telescoping sum

$$\begin{aligned} \sum _{j=\ell }^{\ell +L}\big (||| u-u_{j}|||_{\textrm{pw}}^2-||| u-u_{j+1}|||_{\textrm{pw}}^2\big )= & {} ||| u-u_{\ell }|||_{\textrm{pw}}^2-||| u-u_{\ell +L+1}|||_{\textrm{pw}}^2\\\leqslant & {} C_{\textrm{rel}}^2\eta _{\ell }^2-C_{\textrm{eff}}^2\eta _{\ell +L+1}^2. \end{aligned}$$

Since $\beta \leqslant C_{\textrm{eff}}^2/C_{\textrm{rel}}^2$ implies $(\beta C_{\textrm{rel}}^2-C_{\textrm{eff}}^2)\,\eta _{\ell +L+1}^2\leqslant 0$, the last three displayed estimates show

$$\begin{aligned} \sum _{j=\ell }^{\ell +L} \delta ^2({\mathcal {T}}_j, {\mathcal {T}}_{j+1})&\leqslant \sum _{j=\ell }^{\ell +L}\Bigg (||| u-u_{j}|||_{\textrm{pw}}^2- ||| u-u_{j+1}|||_{\textrm{pw}}^2\Bigg )\nonumber \\&\quad +\left( \left( \frac{2C_{\textrm{qo}}^2}{\beta }+C_7^2\right) h_0^{2\sigma }+\beta \right) C_{\textrm{rel}}^2\sum _{k=\ell }^{\ell +L}\eta ^2_j \nonumber \\&\quad +\beta C_{\textrm{rel}}^2\eta _{\ell +L+1}^2 +\frac{2C_{\textrm{qo}}^2}{\beta }\sum _{j=\ell }^{\ell +L} \Vert h_{{\mathcal {T}}_j}^{m}\lambda u\Vert ^2_{L^2({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1})} \nonumber \\&\leqslant C_{\textrm{rel}}^2\eta _{\ell }^2 +\left( \left( \frac{2C_{\textrm{qo}}^2}{\beta }+C_7^2\right) h_0^{2\sigma }+\beta \right) C_{\textrm{rel}}^2\sum _{k=\ell }^{\ell +L}\eta ^2_j\nonumber \\&\quad +\frac{2C_{\textrm{qo}}^2}{\beta } \sum _{j=\ell }^{\ell +L} \Vert h_{{\mathcal {T}}_j}^{m}\lambda u\Vert ^2_{L^2({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1})}. \end{aligned}$$

(4.6)

Recall that $h_{{\mathcal {T}}_j}|_T:=diam (T)$ for any $T\in {\mathcal {T}}_j$ and compare it with the piecewise constant function ${\tilde{h}}_j\in P_0({\mathcal {T}}_j)$ defined by ${\tilde{h}}_j|_T:=|T|^{1/3}\leqslant h_T\leqslant C_{\textrm{sr}}|T|^{1/3}$ (from shape-regularity) for any $T\in {\mathcal {T}}_j$ and $j\in {\mathbb {N}}_0$. Then ${\tilde{h}}_j\approx h_{{\mathcal {T}}_j}\in P_0({\mathcal {T}}_j)$ and ${\tilde{h}}_j\in P_0({\mathcal {T}}_j)$ satisfies the reduction ${\tilde{h}}_{j+1}\leqslant {\tilde{h}}_{j}/2^{1/3}$ a.e. in the set of refined tetrahedra $\bigcup \big ({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1}\big )$. Hence ${\tilde{h}}_j^{m}\leqslant \frac{2^{m/3}}{\sqrt{4^{m/3}-1}}\,\sqrt{{\tilde{h}} _j^{2{m}}-{\tilde{h}}_{j+1}^{2{m}}}$ a.e. in $\bigcup \big ({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1}\big )$ and

$$\begin{aligned} C_{\textrm{sr}}^{-2m}&\frac{{4^{m/3}-1}}{4^{m/3}} \sum _{j=\ell }^{\ell +L}\Vert h_{{\mathcal {T}}_j}^{m}\lambda u\Vert ^2_{L^2({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1})} \leqslant \frac{{4^{m/3}-1}}{4^{m/3}}\sum _{j=\ell }^{\ell +L}\Vert {\tilde{h}}_j^{m} \lambda u\Vert ^2_{L^2({\mathcal {T}}_j\setminus {\mathcal {T}}_{j+1})}\\&\leqslant \sum _{j=\ell }^{\ell +L}\Big \Vert \sqrt{{\tilde{h}}_j^{2{m}}-{\tilde{h}}_{j+1}^{2{m}}}\lambda u\Big \Vert ^2_{L^2(\Omega )} = \int _{\Omega }({\tilde{h}}_\ell ^{2{m}}-{\tilde{h}}_{\ell +L+1}^{2{m}})(\lambda u)^2d x \leqslant \Vert {\tilde{h}}_\ell ^{m} \lambda u\Vert _{L^2(\Omega )}^2. \end{aligned}$$

Since ${\tilde{h}}_\ell \leqslant h_{{\mathcal {T}}_\ell }\leqslant h_0:=\max _{T\in {\mathcal {T}}_0}h_T\leqslant \varepsilon _6$, a triangle inequality implies

$$\begin{aligned} \Vert {\tilde{h}} _\ell ^{m}\lambda u\Vert _{L^2(\Omega )}^2 \leqslant 2(\lambda /\lambda _\ell (k))^{2}\Vert {\tilde{h}} _\ell ^{m}\lambda _\ell (k) u_\ell \Vert _{L^2(\Omega )}^2 + 2\lambda ^2 h_0^{2m}\Vert u-u_\ell \Vert _{L^2(\Omega )}^2. \end{aligned}$$

Theorem 3.1.a and (1.4) show $(\lambda /\lambda _\ell (k))^{2}\Vert {\tilde{h}} _\ell ^{m}\lambda _\ell (k) u_\ell \Vert _{L^2(\Omega )}^2\leqslant 4\eta _\ell ^2$. Corollary 2.1.a, Theorem 4.4, and (3.12) imply $\Vert u-u_\ell \Vert _{L^2(\Omega )}^2 \leqslant h_{0}^{2\sigma } C_5^2C_\textrm{rel}^2\eta _\ell ^2.$ The substitution in (4.6) concludes the proof with $\Lambda _4:=\max \{C_{\textrm{rel}}^2,C_{\textrm{qo}}^2C_{\textrm{sr}}^{2m}\frac{4^{m/3+1}}{{4^{m/3}-1}}(4+h_{0}^{2m+2\sigma }C_5^2C_{\textrm{rel}}^2\lambda ^2)\}$ and ${\widetilde{\Lambda }}_4:=C_{\textrm{rel}}^2\max \{1,2C_{\textrm{qo}}^2,C_7^2\}$. $\square$

5 Conclusion and comments

5.1 Proof of Theorem 1.1

The proven properties (A1)–(A${4_{\varepsilon }}$) are the axioms of adaptivity in [12, 26] and known to imply (1.5). Compared to [12, 26] the discrete reliability in Theorem 4.6 is extended in that (A${3_{\varepsilon }}$) includes the additional term $M_3 h_{\max }^{2\sigma }\eta ^2({\mathcal {T}})$. Minor modifications of the arguments in [12, 26] prove that (A1)–(A${4_{\varepsilon }}$) imply (1.5). This is stated and proven as Theorem A.1 in Appendix A for some $\varepsilon :=\varepsilon _{{8}}\leqslant \varepsilon _6$. $\Box$

5.2 Optimal convergence rates of the error

The reliability and efficiency in Theorem 4.4 provide the equivalence $||| u-u_{\ell }|||_{\textrm{pw}}\approx \eta _\ell ({\mathcal {T}}_\ell )$. This and Theorem 1.1 lead to optimal convergence rates for the error as well.

5.3 Global convergence

This paper on the asymptotic convergence rates justifies that a small initial mesh-size guarantees the asymptotic convergence from the beginning. Although the reasons are presented in several steps for $\varepsilon _0, \ldots ,\varepsilon _{{8}}$, the computation of $\varepsilon _{{8}}$ may be cumbersome and a huge overestimation in practice. To guarantee global convergence without a priori knowledge of $\varepsilon _{{8}}$, we may modify the marking step in AFEM4EVP as follows: Enlarge the set ${\mathcal {M}}_\ell$ in AFEM4EVP by one tetrahedron of maximal mesh-size in ${\mathcal {T}}_\ell$. This guarantees that the maximal mesh-size tends to zero as the level $\ell \rightarrow \infty$. Consequently there exists some $L\in {\mathbb {N}}$ such that ${\mathcal {T}}_\ell \in {\mathbb {T}}(\varepsilon _{{8}})$ for all $\ell =L,L+1,L+2,\ldots$ Relabel ${\mathcal {T}}_L$ by ${\mathcal {T}}_0$ so that Theorem 1.1 leads to optimal convergence rates for $\eta _L,\eta _{L+1},\eta _{L+2},\ldots$, whence for the entire outcome of the adaptive algorithm. However, the constant in the overhead control [48, Thm. 6.1] depends on ${\mathcal {T}}_L$ and this possibly enlarges the equivalence constants in (1.5).

5.4 Numerical experiments

Numerical experiments in [11, 24] show an asymptotic convergences of AFEM4EVP with $\theta =0.5$ even for coarse initial triangulation and confirm the optimal convergence rates of Theorem 1.1 even for one example with a multiple eigenvalue. The extension to eigenvalue clusters requires an algorithm from [4, 29, 33]. This paper assumes exact solve of the algebraic eigenvalue problem (1.2), but perturbation results in numerical linear algebra [43] can be included as in [14].

A. Appendix – A review and extension of the axioms of adaptivity

The framework (A1)–(A${4_{\varepsilon }}$) in Sect. 4 is a modification of [12, 26] with a more general discrete reliability (A${3_{\varepsilon }}$). Theorem A.1 below proves that the modified axioms are sufficient for optimal convergence rates of the AFEM algorithm with Dörfler marking and newest-vertex bisection [12, Algorithm 2.2]. On level $\ell \in {\mathbb {N}}_0$ of the general purpose adaptive algorithm AFEM there is given a regular triangulation ${\mathcal {T}}_\ell$ of $\Omega \subset {\mathbb {R}}^n$ into closed simplices and an undisplayed discrete problem with a discrete solution $u_\ell$. These allow for the computation of $\eta _\ell (T)$ for all $T\in {\mathcal {T}}_\ell$ in the step compute. The step mark uses the sum convention $\eta _\ell ^2({\mathcal {M}}):=\sum _{T\in {\mathcal {M}}}\eta _\ell ^2(T)$ for any ${\mathcal {M}}\subseteq {\mathcal {T}}_\ell$ and $\eta _\ell ^2:=\eta ^2_\ell ({\mathcal {T}}_\ell )$. The selection of a set ${\mathcal {M}}_\ell$ with almost minimal cardinality in this step means that there exists a constant $\Lambda _{\textrm{opt}}\geqslant 1$ such that the cardinality satisfies $|{\mathcal {M}}_\ell |\leqslant \Lambda _{\textrm{opt}}|{\mathcal {M}}_\ell ^\star |$, where ${{\mathcal {M}}}^\star _\ell \subset {\mathcal {T}}_\ell$ denotes some set of minimal cardinality $|{{\mathcal {M}}}^\star _\ell |$ with $\theta \eta _\ell ^2 \leqslant \sum _{T\in {\mathcal {M}}_\ell ^\star }\eta _\ell ^2(T)$; cf. [12, 26, 47] for details; this is more general than in AFEM4EVP, which utilizes a minimal set $M_\ell$ with $\Lambda _{\textrm{opt}}=1$ constructed at linear cost in [44].

This appendix is written in a self-contained way based on the set ${\mathbb {T}}:={\mathbb {T}}({\mathcal {T}}_0)$ of all admissible triangulation computed by successive newest-vertex bisection [35, 48] of a regular initial triangulation ${\mathcal {T}}_0$ (plus some initialization of tagged n-simplices) of the bounded polyhedral Lipschitz domain $\Omega \subset {\mathbb {R}}^n$ into closed simplices and the subset ${\mathbb {T}}({\mathcal {T}})$ of admissible refinements of ${\mathcal {T}}\in {\mathbb {T}}$. For $N\in {\mathbb {N}}_0$, set ${\mathbb {T}}(N):=\{{\mathcal {T}}\in {\mathbb {T}}:\, |{\mathcal {T}}|\leqslant |{\mathcal {T}}_0|+ N\}$. To analyse the error estimates $\eta _\ell ({\mathcal {T}}_\ell )$ and their rates and in particular to compare with error estimators $\eta ({\mathcal {T}},\bullet )$ for any admissible triangulation ${\mathcal {T}}\in {\mathbb {T}}$, we need to assume that the error estimators are computable for any ${\mathcal {T}}\in {\mathbb {T}}$. This leads to a family $\eta ({\mathcal {T}},\bullet )\in {\mathbb {R}}^{{\mathcal {T}}}$ of error estimators parametrized by ${\mathcal {T}}\in {\mathbb {T}}$ with $\eta ({\mathcal {T}},K)\geqslant 0$ for all $K\in {\mathcal {T}}$. For any subset ${\mathcal {M}}\subseteq {\mathcal {T}}\in {\mathbb {T}}$, the sum convention reads

$$\begin{aligned} \eta ^2({\mathcal {T}},{\mathcal {M}}):= \big (\eta ({\mathcal {T}},{\mathcal {M}})\big )^2:=\sum _{T\in {\mathcal {M}}}\eta ^2({\mathcal {T}},T) \quad \text { and }\quad \eta ^2({\mathcal {T}}):=\eta ({\mathcal {T}},{\mathcal {T}}). \end{aligned}$$

(1)

For any triangulation ${\mathcal {T}}_\ell$ in the AFEM algorithm, we abbreviate $\eta _\ell (\bullet ):=\eta ({\mathcal {T}}_\ell ,\bullet )$ and $\eta _\ell :=\eta _\ell ({\mathcal {T}}_\ell )\equiv \eta ({\mathcal {T}}_\ell ,{\mathcal {T}}_\ell )$. Recall the Axioms (A1)–(A${4_{\varepsilon }}$) with constants $\Lambda _1$, $\Lambda _2$, $\Lambda _3$, $\Lambda _4$, $\Lambda _{\textrm{ref}}>0$, ${\widehat{\Lambda }}_3$, $\epsilon _3$, $\epsilon _4{ \geqslant 0}$, and $0<\rho _2<1$ for convenient reading. For any ${\mathcal {T}}\in {\mathbb {T}}$ and admissible refinement $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$, there exists a set ${\mathcal {R}}({\mathcal {T}},\widehat{{\mathcal {T}}})\subseteq {\mathcal {T}}$ with ${\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}}\subset {\mathcal {R}}({\mathcal {T}},\widehat{{\mathcal {T}}})$ and $|{\mathcal {R}}({\mathcal {T}},\widehat{{\mathcal {T}}})|\leqslant \Lambda _{\textrm{ref}}|{\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}}|$, such that ${\mathcal {T}}\in {\mathbb {T}}$, $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$, ${\mathcal {R}}({\mathcal {T}},\widehat{{\mathcal {T}}})$, and the output $({\mathcal {T}}_k)_{k\in {\mathbb {N}}_0}$ and $(\eta _k)_{k\in {\mathbb {N}}_0}$ of AFEM satisfy (A1)–(A${4_{\varepsilon }}$).

• (A1) Stability. $\displaystyle \big |{\eta ({\mathcal {T}},{\mathcal {T}}\cap \widehat{{\mathcal {T}}}) -{\eta }( \widehat{{\mathcal {T}}},{\mathcal {T}}\cap \widehat{\mathcal T})}\big |\leqslant \Lambda _1 \delta ({\mathcal {T}},\widehat{\mathcal T}).$

• (A2) Reduction. $\displaystyle \eta ( \widehat{{\mathcal {T}}},\widehat{{\mathcal {T}}}{\setminus }{\mathcal {T}}) \leqslant \rho _2 \eta ({\mathcal {T}},{\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}}) + \Lambda _2 \delta ({\mathcal {T}},\widehat{{\mathcal {T}}}).$

(A${3_{\varepsilon }}$) Discrete reliability. $\displaystyle \delta ^2({\mathcal {T}},\widehat{{\mathcal {T}}}) \leqslant \Lambda _3 \eta ^2({\mathcal {T}},{\mathcal {R}}({\mathcal {T}},\widehat{{\mathcal {T}}})) + {{\widehat{\Lambda }}}_3 \eta ^2(\widehat{{\mathcal {T}}})+\epsilon _3 \eta ^2({\mathcal {T}}).$

(A${4_{\varepsilon }}$) Quasiorthogonality. $\displaystyle \sum _{j=\ell }^{\ell +m}\delta ^2({\mathcal {T}}_j,{\mathcal {T}}_{j+1}) \leqslant \Lambda _{4} \eta ^2_\ell +\epsilon _4\sum _{j=\ell }^{\ell +m}\eta _j^2 \text { for any }\ell ,m\in {\mathbb {N}}_0.$

Theorem A.1 below contains smallness assumptions for the constants ${\widehat{\Lambda }}_3,\,\epsilon _3,$ and $\epsilon _4$. In a typical application such as Theorem 1.1 the quantities ${\widehat{\Lambda }}_3,\,\epsilon _3,\,\epsilon _4$ contain a power of the initial mesh-size $h_0:=\max _{T\in {\mathcal {T}}_0}h_T$ such that the assumptions are satisfied for a sufficiently fine initial triangulation ${\mathcal {T}}_0$. Given $\epsilon _3<\Lambda _1^{-2}$, set $\Theta :={(1-\Lambda _1^2\epsilon _3)}/{(1+\Lambda _1^2 \Lambda _3)}$. Any choice of $\mu$ and $\xi$ with $0<\mu <\rho _2^{-2}-1$ and $0<\xi <(1-(1+\mu )\rho _2^2)\Theta /(1-\Theta )$ implies

$$\begin{aligned} \rho _{12}:= & {} \Theta \rho _2^2(1+\mu )+(1-\Theta )(1+\xi )<1\;\; \text {and}\\ \Lambda _{12}:= & {} (1+1/\xi )\Lambda _1^2+(1+1/\mu )\Lambda _2^2<\infty . \end{aligned}$$

Theorem A.1

(rate optimality of the adaptive algorithm) Suppose (A1)–(A${4_{\varepsilon }}$) with

$$\begin{aligned} \Lambda _1^2\epsilon _3<1,\quad {\widehat{\Lambda }}_3(\Lambda _1^2+\Lambda _2^2)<1,\quad \epsilon _4<(1-\rho _{12})/\Lambda _{12},\quad \text {and}\quad 0<\theta < \Theta . \end{aligned}$$

The output $({\mathcal {T}}_{\ell })_{\ell \in {\mathbb {N}}_0}$ and $(\eta _\ell )_{\ell \in {\mathbb {N}}_0}$ of AFEM satisfy, for any $s>0$, the equivalence

$$\begin{aligned} \sup _{\ell \in {\mathbb {N}}_0} (1+|{\mathcal {T}}_\ell | - |{\mathcal {T}}_0|)^s \eta _\ell \approx \sup _{N\in {\mathbb {N}}_0} (1+N)^s \min _{{\mathcal {T}}\in {\mathbb {T}}(N)}\eta ({\mathcal {T}}). \end{aligned}$$

The proof of Theorem A.1 reviews parts of the analysis in [12, 26] and focusses on the relevant extensions in Theorem A.2 and Theorem A.3 below. The following results (A12), (A4), and (2) follow verbatim as in [12, 26]: (A1)–(A2) and the Dörfler marking strategy with bulk parameter $\theta<\Theta <1$ provide the estimator reduction [26, Thm. 4.1]

$$\begin{aligned} \eta ^2(\widehat{{\mathcal {T}}})\leqslant \varrho _{12}\eta ^2({\mathcal {T}})+\Lambda _{12}\delta ^2({\mathcal {T}},\widehat{{\mathcal {T}}}) \end{aligned}$$

(A12)

for any ${\mathcal {T}}\in {\mathbb {T}}$ and any admissible refinement $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$. The estimator reduction (A12), (A${4_{\varepsilon }}$), and $\Lambda _{\textrm{qo}}:=\Lambda _4+\epsilon _4(1+\Lambda _{12}\Lambda _4)/(1-\rho _{12}-\epsilon _4\Lambda _{12})>0$ guarantee the stricter quasi-orthogonality [26, Thm. 3.1]

$$\begin{aligned} \sum _{k=\ell }^{\ell +m}\delta ^2({\mathcal {T}}_k,{\mathcal {T}}_{k+1}) \leqslant \Lambda _{\textrm{qo}} \eta ^2_\ell \quad \text { for any }\ell ,m\in {\mathbb {N}}_0. \end{aligned}$$

(A4)

This and (A12) imply plain and R-linear convergence on each level for the output $(\eta _\ell )_{\ell \in {\mathbb {N}}_0}$ of AFEM in [26, Thm. 4.2]: The constants $\Lambda _c:=(1+\Lambda _{12}\Lambda _{\textrm{qo}})/(1-\rho _{12})>0$ and $q_c:=\Lambda _c/(1+\Lambda _c)<1$ satisfy

$$\begin{aligned} \sum _{k=\ell }^{\ell +m}\eta _k^2&\leqslant \Lambda _c\eta _\ell ^2 \quad \text {and}\quad \eta _{\ell +m}^2\leqslant \frac{q_c^m}{1-q_c}\eta _{\ell }^2\quad \text {for any }\ell , m\in {\mathbb {N}}_0. \end{aligned}$$

(2)

On the other hand, (A1)–(A3) are sufficient for the quasimonotonicity (QM) and the comparison lemma. But the discrete reliability is relaxed in (A${3_{\varepsilon }}$) in this paper, so the proofs of (QM) and the comparison lemma are revisited below.

Theorem A.2

(QM) The axioms (A1), (A2), (A${3_{\varepsilon }}$), and ${\widehat{\Lambda }}_3(\Lambda _1^2+\Lambda _2^2)<1$ imply the existence of $\Lambda _{\textrm{mon}}>0$ such that $\eta (\widehat{{\mathcal {T}}})\leqslant \Lambda _{\textrm{mon}}\eta ({\mathcal {T}})$ holds for any ${\mathcal {T}}\in {\mathbb {T}}$ and $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$.

Proof

This proof extends [12, Lem. 3.5] and [26, Thm. 3.2]. The axioms (A1)–(A2) apply to the decomposition $\eta ^2(\widehat{{\mathcal {T}}})=\eta ^2(\widehat{{\mathcal {T}}},{\mathcal {T}}\cap \widehat{{\mathcal {T}}}) +\eta ^2(\widehat{{\mathcal {T}}},\widehat{{\mathcal {T}}}{\setminus }{\mathcal {T}})$ of the estimator of the fine triangulation $\widehat{{\mathcal {T}}}\in {\mathbb {T}}({\mathcal {T}})$ and show

$$\begin{aligned} \eta ^2(\widehat{{\mathcal {T}}})&\leqslant \big (\eta ({\mathcal {T}},{\mathcal {T}}\cap \widehat{{\mathcal {T}}})+ \Lambda _1 \delta ({\mathcal {T}},\widehat{{\mathcal {T}}})\big )^2 +\big (\rho _2 \eta ({\mathcal {T}},{\mathcal {T}}\setminus \widehat{{\mathcal {T}}}) + \Lambda _2 \delta ({\mathcal {T}},\widehat{{\mathcal {T}}})\big )^2\\&\leqslant (1+1/\alpha )\eta ^2({\mathcal {T}})+(1+\alpha )(\Lambda _1^2+\Lambda _2^2)\delta ^2({\mathcal {T}},\widehat{\mathcal T}) \end{aligned}$$

with $(a+b)^2\leqslant (1+\alpha )a^2+(1+1/\alpha )b^2$ for any positive $a,\,b$ and $0<\alpha <\big ((\Lambda _1^2+\Lambda _2^2){\widehat{\Lambda }}_3\big )^{-1}-1$ in the second step. (For ${\widehat{\Lambda }}_3=0$, the upper bound for $0<\alpha <\infty$ is understood as infinity.) The Axiom (A${3_{\varepsilon }}$) controls the distance $\delta ^2({\mathcal {T}},\widehat{{\mathcal {T}}})$ and leads to

$$\begin{aligned} \eta ^2(\widehat{{\mathcal {T}}}) \leqslant \big (1+1/\alpha +(1+\alpha )(\Lambda _1^2+\Lambda _2^2)(\Lambda _3+\epsilon _3)\big )\eta ^2({\mathcal {T}}) +(1+\alpha )(\Lambda _1^2+\Lambda _2^2) {{\widehat{\Lambda }}}_3 \eta ^2(\widehat{{\mathcal {T}}}). \end{aligned}$$

Since $(1+\alpha )(\Lambda _1^2+\Lambda _2^2) {{\widehat{\Lambda }}}_3<1$, this proves $\eta ^2(\widehat{{\mathcal {T}}})\leqslant \Lambda _{\textrm{mon}}^2\eta ^2({\mathcal {T}})$ for

$$\begin{aligned} \Lambda _{\textrm{mon}}^2 :=\frac{ 1+1/\alpha +(1+\alpha )(\Lambda _1^2+\Lambda _2^2)(\Lambda _3+\epsilon _3)}{1-(1+\alpha )(\Lambda _1^2+\Lambda _2^2) {{\widehat{\Lambda }}}_3}. \end{aligned}$$

$\square$

The convergence is guaranteed with (2) and the optimality requires the sufficient smallness of the bulk parameter $\theta <\Theta$ in the adaptive algorithm. This enters with the help of the comparison lemma, where some $\theta _0(\varkappa ,\alpha )$ depends on parameter $\varkappa ,\alpha$ that allow for $\theta \leqslant \theta _0(\varkappa ,\alpha )<\Theta$. The lemma dates back to the seminal contribution [47].

Lemma A.3

(comparison) Suppose (QM), i.e., the axioms (A1), (A2), (A${3_{\varepsilon }}$), and ${\widehat{\Lambda }}_3(\Lambda _1^2+\Lambda _2^2)<1$. Let $0<\varkappa <1$, $0<\alpha <\infty$, and let $s>0$ satisfy

$$\begin{aligned} M:=\sup _{N\in {\mathbb {N}}_0}(N+1)^s\min _{{\mathcal {T}}\in {\mathbb {T}}(N)}\eta ({\mathcal {T}})<\infty . \end{aligned}$$

Then for any level $\ell \in {\mathbb {N}}_0$, there exist $\widehat{{\mathcal {T}}}_\ell \in {\mathbb {T}}({\mathcal {T}}_\ell )$ and

$$\begin{aligned} \theta _0(\alpha ,\varkappa ):=\frac{1- \varkappa ^2\big ( (1+\alpha )+(1+1/\alpha ) \Lambda _1^2{{\widehat{\Lambda }}}_3 \big ) - (1+1/\alpha ) \Lambda _1^2\epsilon _3 }{1+(1+1/\alpha ) \Lambda _1^2 \Lambda _3} < 1 \end{aligned}$$

such that

• (a) $\eta (\widehat{{\mathcal {T}}}_\ell )\leqslant \varkappa \eta ({\mathcal {T}}_\ell )\leqslant \Lambda _{\textrm{mon}}M|{\mathcal {T}}_\ell {\setminus } \widehat{{\mathcal {T}}}_\ell |^{-s}$ and

• (b) $\theta _0(\alpha ,\varkappa )\eta ^2({\mathcal {T}}_\ell )\leqslant \eta ^2({\mathcal {T}}_\ell ,{\mathcal {R}}_\ell )$ with ${\mathcal {T}}_\ell {\setminus } \widehat{{\mathcal {T}}}_\ell \subset {\mathcal {R}}_\ell :={\mathcal {R}}({\mathcal {T}}_\ell , \widehat{{\mathcal {T}}}_\ell )$ and $|{\mathcal {R}}_\ell |\leqslant \Lambda _{\textrm{ref}} |{\mathcal {T}}_\ell {\setminus } \widehat{{\mathcal {T}}}_\ell |$.

Proof

The proof of (a) is verbatim that of [12, Prop. 4.12] or that of [26, Lem. 4.3] based on the overlay control (i.e., (6) below) and Theorem A.2. It remains to modify the proofs in [12, Prop. 4.12] or [26, Lem. 4.3] for the verification of (b). Axiom (A1) and (a) imply that

$$\begin{aligned} \eta ({\mathcal {T}}_{\ell },{\mathcal {T}}_\ell \cap \widehat{{\mathcal {T}}}_\ell )&\leqslant \eta (\widehat{{\mathcal {T}}}_{\ell },{\mathcal {T}}_\ell \cap \widehat{{\mathcal {T}}}_\ell ) +\Lambda _1\delta ({\mathcal {T}}_{\ell },\widehat{{\mathcal {T}}}_{\ell }) \leqslant \varkappa \eta ({{\mathcal {T}}}_{\ell })+\Lambda _1\delta ({\mathcal {T}}_{\ell },\widehat{\mathcal T}_{\ell }). \end{aligned}$$

(3)

Recall $\eta ^2_\ell ({\mathcal {M}}_\ell ):=\eta ^2({\mathcal {T}}_\ell ,{\mathcal {M}}_\ell ):= \sum _{T\in {\mathcal {M}}_\ell }\eta ^2({\mathcal {T}}_\ell ,T)$ for any ${\mathcal {M}}_\ell \subset {\mathcal {T}}_\ell$ and $\eta _\ell :=\eta ({\mathcal {T}}_\ell )\equiv \eta ({\mathcal {T}}_\ell ,{\mathcal {T}}_\ell )$ and abbreviate ${\widehat{\eta }}_\ell :=\eta (\widehat{{\mathcal {T}}}_\ell )\equiv \eta (\widehat{{\mathcal {T}}}_\ell ,\widehat{{\mathcal {T}}}_\ell )$. A weighted Young inequality with $\alpha >0$, the Axiom (A${3_{\varepsilon }}$) with ${\mathcal {R}}({\mathcal {T}}_\ell ,\widehat{{\mathcal {T}}}_\ell )$ replaced by ${\mathcal {R}}_\ell$ defined in (b), and (a) show that

$$\begin{aligned} \big (\varkappa \eta _{\ell }+\Lambda _1\delta ({\mathcal {T}}_{\ell },\widehat{\mathcal T}_{\ell })\big )^2&\leqslant (1+\alpha )\varkappa ^2 \eta ^2_{\ell } +(1+1/\alpha ) \Lambda _1^2 \big ( \Lambda _3 \eta ^2_\ell ({\mathcal {R}}_\ell ) + {{\widehat{\Lambda }}}_3 {\widehat{\eta }}_\ell ^2+\epsilon _3 \eta ^2_{\ell }\big ) \nonumber \\&\leqslant (1+\alpha )\varkappa ^2 \eta ^2_{\ell } +(1+1/\alpha ) \Lambda _1^2 \big ( \Lambda _3 \eta ^2_\ell ({\mathcal {R}}_\ell ) + {{\widehat{\Lambda }}}_3 \varkappa ^2\eta ^2_{\ell }+\epsilon _3 \eta ^2_{\ell }\big ). \end{aligned}$$

(4)

Recall $\varkappa <1$, $\alpha >0$, and set

$$\begin{aligned} C_a:=(1+\alpha )\varkappa ^2+(1+1/\alpha ) \Lambda _1^2(\epsilon _3 + {{\widehat{\Lambda }}}_3 \varkappa ^2) \quad \text {and}\quad C_b:=(1+1/\alpha ) \Lambda _1^2 \Lambda _3. \end{aligned}$$

Then the combination of (3)–(4) reads

$$\begin{aligned} \eta ^2_\ell ({\mathcal {T}}_\ell \cap \widehat{{\mathcal {T}}}_\ell )&\leqslant C_a \eta ^2_{\ell }+ C_b\eta ^2_\ell ({\mathcal {R}}_\ell ). \end{aligned}$$

(5)

Since ${\mathcal {T}}_\ell \setminus \widehat{{\mathcal {T}}}_\ell \subseteq {\mathcal {R}}_\ell$, the estimate (5) implies

$$\begin{aligned} \eta ^2_{\ell }&\leqslant \eta ^2_\ell ({\mathcal {R}}_\ell )+\eta ^2_\ell ({\mathcal {T}}_\ell \cap \widehat{{\mathcal {T}}}_\ell ) \leqslant C_a \eta ^2_{\ell }+(1+C_b)\eta ^2_\ell ({\mathcal {R}}_\ell ). \end{aligned}$$

This proves (b) with

$$\begin{aligned} \frac{1-C_a}{1+C_b} =\frac{1- \big ( (1+\alpha )\varkappa ^2+(1+1/\alpha ) \Lambda _1^2(\epsilon _3 + {{\widehat{\Lambda }}}_3 \varkappa ^2)\big ) }{1+(1+1/\alpha ) \Lambda _1^2 \Lambda _3 }=\theta _0(\varkappa ,\alpha )<1. \end{aligned}$$

$\square$

The proof of Theorem A.1 can be concluded as in [12, Proof of Theorem 4.1 (ii)] or [26, Section 4.3]. The function $\theta _0(\alpha ,\varkappa )$ in Theorem A.3.b is bounded from above by $\lim _{\alpha \rightarrow \infty } \theta _0(0,\alpha )=({1-\Lambda _1^2\epsilon _3})/({1+\Lambda _1^2 \Lambda _3})$ and there exist a choice of $0<\varkappa <1$ and $0<\alpha <\infty$ such that $0<\theta<\theta _0(\alpha ,\varkappa )< \Theta$. This is the first formula on page 2655 in [26] and the remaining parts of the proof are summarized below for convenient reading and almost verbatim to Case A in [26]. The choice of $\theta$ and Theorem A.3.b show

$$\begin{aligned} \theta \eta ^2({\mathcal {T}}_\ell )\leqslant \theta _0(\alpha ,\varkappa )\eta ^2({\mathcal {T}}_\ell )\leqslant \eta ^2({\mathcal {T}}_\ell ,{\mathcal {R}}_\ell ), \end{aligned}$$

i.e., ${\mathcal {R}}_\ell$ satisfies the Dörfler marking condition. Recall that ${\mathcal {M}}_\ell$ denotes the set of marked elements on level $\ell$ in AFEM, while ${{\mathcal {M}}}^\star _\ell$ with $|{{\mathcal {M}}}^\star _\ell |=M_\ell$ is a minimal set of marked elements. Then there exists $\Lambda _{\textrm{opt}}\geqslant 1$ with $|{\mathcal {M}}_\ell |\leqslant \Lambda _{\textrm{opt}}M_\ell \leqslant \Lambda _{\textrm{opt}}|{\mathcal {R}}_\ell |$. The control over ${\mathcal {R}}_\ell :={\mathcal {R}}({\mathcal {T}}_\ell , \widehat{{\mathcal {T}}}_\ell )$ and Theorem A.3.a ensure

$$\begin{aligned} |{\mathcal {R}}_\ell |\leqslant \Lambda _{\textrm{ref}}|{\mathcal {T}}_\ell \setminus \widehat{{\mathcal {T}}}_\ell |\leqslant \Lambda _{\textrm{ref}} \big (\Lambda _{\textrm{mon}}M/(\varkappa \eta _\ell )\big )^{1/s}. \end{aligned}$$

Hence $|{\mathcal {M}}_\ell |\leqslant C_c M^{1/s}\eta _\ell ^{-1/s}$ with $C_c:=\Lambda _{\textrm{opt}}\Lambda _{\textrm{ref}}\Lambda _{\textrm{mon}}^{1/s}\varkappa ^{-1/s}$. One important ingredient of NVB is the overhead control [2, 48]

$$\begin{aligned} |{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|\leqslant \Lambda _{\textrm{BDdV}}\sum _{k=0}^{\ell -1}|{\mathcal {M}}_k| \end{aligned}$$

(6)

with a universal constant $\Lambda _{\textrm{BDdV}}$ that exclusively depends on ${\mathcal {T}}_0$. The combination of the above with the overhead control leads to

$$\begin{aligned} |{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|\leqslant \Lambda _{\textrm{BDdV}} C_c M^{1/s}\sum _{k=0}^{\ell -1}\eta _k^{-1/s}. \end{aligned}$$

(7)

The R-linear convergence (2) bounds the sum $\sum _{k=0}^{\ell -1}\eta _k^{-1/s}$ as in [26, Thm. 4.2.c]. For all $0\leqslant k<\ell$, the second identity in (2) implies $\eta _{k}^{-1/s}\leqslant \eta _{\ell }^{-1/s}{q_c^{(\ell -k)/(2\,s)}}{(1-q_c)^{-1/(2\,s)}}$. Hence the formula for the partial sum of the geometric series shows

$$\begin{aligned} \sum _{k=0}^{\ell -1}\eta _k^{-1/s}\leqslant C_d \eta _\ell ^{-1/s}\quad \text {with}\quad C_d:=\frac{q_c^{1/(2s)}}{\big (1-q_c^{1/(2s)}\big )(1-q_c)^{1/(2s)}}. \end{aligned}$$

(8)

The combination of (7)–(8) reads $|{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|\leqslant \Lambda _{\textrm{BDdV}} C_cC_d M^{1/s} \eta _\ell ^{-1/s}$. Hence $1\leqslant |{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|$ implies $(1+ |{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|)\leqslant 2(|{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|)\leqslant 2\Lambda _{\textrm{BDdV}} C_cC_d M^{1/s} \eta _\ell ^{-1/s}$, while $|{\mathcal {T}}_\ell |=|{\mathcal {T}}_0|$ implies $1\leqslant M^{1/s}\eta _\ell ^{-1/s}$. This concludes the proof of

$$\begin{aligned} \eta _\ell (1+ |{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|)^s\leqslant & {} {\max \{1,(2\Lambda _{\textrm{BDdV}} C_c C_d)^s\}} M\\{} & {} \text { with} \ M:=\sup _{N\in {\mathbb {N}}_0}(N+1)^s\min _{{\mathcal {T}}\in {\mathbb {T}}(N)}\eta ({\mathcal {T}}) \end{aligned}$$

and so of “$\lesssim$” in Theorem A.1.

For the proof of the converse implication, assume, without loss of generality, that $0<\min _{{\mathcal {T}}\in {\mathbb {T}}(N)}\eta ({\mathcal {T}})$ and so $0<\eta _\ell$ for any $\ell \in {\mathbb {N}}_0$ with $N_\ell :=|{\mathcal {T}}_\ell |-|{\mathcal {T}}_0|\leqslant N$. AFEM leads to $N_\ell <N_{\ell +1}$ (since no refinement only occurs for $\eta _\ell =0$). Hence there exists a level $\ell$ with $N_\ell <N\leqslant N_{\ell +1}$ and $(N+1)^s \min _{{\mathcal {T}}\in {\mathbb {T}}(N)}\eta ({\mathcal {T}})\leqslant (N_{\ell +1}+1)^s\eta _\ell$. On each refinement level $\ell$ each simplex creates at most a finite number K(n) (depending only on the spatial dimension n) of children in the next level $\ell +1$ [35]. In other words $|{\mathcal {T}}_{\ell +1}|\leqslant K(n)|{\mathcal {T}}_\ell |$ and $(N_{\ell +1}+1)/(N_\ell +1)\leqslant K(n) +(K(n)-1)(|{\mathcal {T}}_0|-1)\lesssim 1$.

This concludes the proof of rate optimality for AFEM in Theorem A.1. $\Box$

Proof of Theorem 1.1. The AFEM4EVP in Theorem 1.1 is a particular case with ${\mathcal {R}}({\mathcal {T}},\widehat{{\mathcal {T}}})\hspace{-1mm}:={\mathcal {R}}_1:=\{K\in {\mathcal {T}}:\,\exists \, T\in {\mathcal {T}}{\setminus }\widehat{{\mathcal {T}}} \text { with }dist (K,T)=~0\}$. Theorem 4.2, 4.6, and 4.7 guarantee (A1)–(A${4_{\varepsilon }}$) with ${\widehat{\Lambda }}_3:=0$, $\epsilon _3:= {M}_3{h}_{\max }^{2\sigma }$, and $\epsilon _4:={\widetilde{\Lambda }}_4(\beta +h_0^{2\sigma }(1+1/\beta ))>0$. Let ${\varepsilon _7}:=\min \big \{\varepsilon _6, (2\Lambda _1^2 M_3)^{-1/(2\sigma )}\big \}$ such that $\epsilon _3 <\Lambda _1^{-2}$ and select $\rho _{12}$ and $\Lambda _{12}$, then abbreviate $c_3:=(1-\rho _{12})/(2\Lambda _{12}{\widetilde{\Lambda }}_4)$, $\beta := \min \{C_{\textrm{eff}}^2/C_{\textrm{rel}}^2,c_3/2\}$, and define

$$\begin{aligned} \varepsilon :=\varepsilon _{{8}}:=\min \big \{{\varepsilon _7}, ((c_3-\beta )/(1+1/\beta ))^{1/(2\sigma )}\big \}. \end{aligned}$$

(9)

Then ${\widehat{\Lambda }}_3(\Lambda _1^2+\Lambda _2^2)=0$, $\epsilon _3 \Lambda _1^2\leqslant 1/2$, and $\epsilon _4\leqslant (1-\rho _{12})/(2\Lambda _{12})$ in Theorem A.1.

Remark A.4

(smallness assumptions on $\varepsilon _5,\varepsilon _6,\varepsilon _7,{\varepsilon _8}$) The reduction to $\varepsilon _5$ guarantees the best approximation result in Theorem 3.1, while $\varepsilon _6:=\min \{\varepsilon _5,(2C_5^2)^{-1/(2\sigma )}\}$ is sufficient for reliability in Theorem 4.4. Optimal rates follow with $\varepsilon :=\varepsilon _{{8}}$ from (9). Since $C_5$ from (3.12), $c_3:=(1-\rho _{12})/(2\Lambda _{12}{\widetilde{\Lambda }}_4)$, and $M_3$ are bounded ${\mathcal {O}}(1)$, independent of the mesh-size, $\varepsilon _6=\min \{\varepsilon _5,{\mathcal {O}}(1)\}$, $\varepsilon _7=\min \{\varepsilon _6,{\mathcal {O}}(1)\}$, and ${\varepsilon _8=\min \{\varepsilon _7,{\mathcal {O}}(1)\}}$ are not expected to be dramatically smaller than $\varepsilon _5$.

Remark A.5

(modification with global convergence) The modified algorithm of Sect. 5.3, with ${\mathcal {T}}_L,{\mathcal {T}}_{L+1},\ldots$ has no influence on the constants $1/2\leqslant \Theta (1+{\Lambda _1^2\Lambda _3})\leqslant 1$, $\Lambda _4\leqslant \Lambda _{\textrm{qo}}\leqslant 2\Lambda _4+1/\Lambda _{12}$, $1+(\Lambda _1^2+\Lambda _2^2)\Lambda _3\leqslant \Lambda _\textrm{mon}^2 \leqslant \big (1+\sqrt{(\Lambda _1^2+\Lambda _2^2)(\Lambda _3+\Lambda _1^{-2}/2)}\big )^2$. But $\Lambda _{\textrm{BDdV}}$ in the overhead control (6) (e.g. [48, Thm. 6.1]) depends on ${\mathcal {T}}_L$ and could become larger (when replacing ${\mathcal {T}}_0$ by ${\mathcal {T}}_L$) and leads to larger equivalence constants in Theorem A.1. Fortunately, the asymptotic convergence rate remains optimal and the choice of $\theta$ is not affected.

Remark A.6

(parameter choice in practice) In a practical computation, we suggest uniform mesh-refinement until the eigenvalue $\lambda _k$ of interest is resolved in that $5h_{\max }$ is smaller or equal the estimated wavelength of $\lambda _k$. This triangulation serves as initial triangulation in ${\mathcal {T}}_0$ in the modified algorithm of Sect. 5.3 with some bulk parameter $\theta$ smaller than $(1-\Lambda _1^2\Lambda _3)^{-1}$. In this way, the pre-asymptotic range is (hopefully) kept small while the asymptotic convergence rate remains optimal.