Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework

Abstract

This paper develops a general framework for a posteriori error estimates in numerical approximations of the Laplace eigenvalue problem, applicable to all standard numerical methods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on the energy error in the approximation of the associated eigenvector. The bounds are valid under the sole condition that the approximate i-th eigenvalue lies between the exact $(i-1)$-th and $(i+1)$-th eigenvalue, where the relative gaps are sufficiently large. We give a practical way how to check this; the accuracy of the resulting estimates depends on these relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a further explicit, a posteriori, minimal resolution condition, the multiplicative constant in our estimates can be reduced by a fixed factor; moreover, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of 1 with mesh refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along with numerical illustrations. Our key ingredients are equivalences between the i-th eigenvalue error, the associated eigenvector energy error, and the dual norm of the residual. We extend them in an appendix to the generic class of bounded-below self-adjoint operators with compact resolvent.

Appendices

Appendix

The current analysis was presented for the Laplace operator of (1.1). The generic equivalences can, however, be extended to a larger class of operators that we show in part A of this appendix, for a conforming approximation. We next complement in part B the estimate of Theorem 2 by a further possible improvement of the first eigenvalue upper bound.

Appendix A: Extension to a generic operator

We formulate here the results of [17, Theorems 3.4 and 3.5] for conforming approximations and any bounded-below self-adjoint operator with compact resolvent, see, e.g., Helffer [44]. This comprises for example the operator $A {:}{=}-\varDelta +w$ with domain $D(A){:}{=}\{v \in H^1_0(\varOmega ); \varDelta v \in L^2(\varOmega )\}$, which is self-adjoint on $L^2(\varOmega )$ whenever $w \in L^\infty (\varOmega )$. It appears that only the operator considered ($-\varDelta$) and the norms (${\Vert }\cdot {\Vert }, {\Vert }\nabla \cdot {\Vert }$, and ${\Vert }\cdot {\Vert }_{-1}$) need to be changed.

Let ${\mathcal {H}}$ be a separable Hilbert space endowed with a scalar product denoted by $(\cdot , \cdot )_{{\mathcal {H}}}$. Now let A be a bounded-below self-adjoint operator on ${\mathcal {H}}$ with domain D(A) and compact resolvent. There exists a non-decreasing sequence of real numbers $(\lambda _k)_{k \ge 1}$ such that $\lambda _k \rightarrow \infty$ and an orthonormal basis $(u_k)_{k \ge 1}$ of ${\mathcal {H}}$ consisting of vectors of D(A) such that

$$\begin{aligned} \forall k \ge 1, \quad A \, u_k = \lambda _k u_k. \end{aligned}$$

Making the additional assumption that the k-th eigenvalue of A is simple, that is $\lambda _{k-1}<\lambda _k<\lambda _{k+1}$, the k-th eigenvector is unique up to the sign. Up to shifting the operator A by a constant $c\in {\mathbb {R}}^+$ such that $c+A$ is a positive definite operator, we can suppose that A is a positive definite operator, in which case $(\lambda _k)_{k \ge 1}$ is a sequence of positive numbers. This enables to define an operator $A^\frac{1}{2}$ analogous to the operator $|\nabla |$ in the previous case (recall that ${\Vert }|\nabla v|{\Vert } = {\Vert }\nabla v{\Vert }$ for $v \in H^1(\varOmega )$) by its domain

$$\begin{aligned} D(A^\frac{1}{2}) {:}{=}\left\{ v\in {\mathcal {H}}; \quad \sum _{k \ge 1} \lambda _k |(v,u_k)_{{\mathcal {H}}}|^2 < +\infty \right\} \end{aligned}$$

and its expression

$$\begin{aligned} A^\frac{1}{2}: v\in D(A^\frac{1}{2}) \mapsto \sum _{k \ge 1} \sqrt{\lambda _k} (v, u_k)_{{\mathcal {H}}} u_k. \end{aligned}$$

Replace now $-\varDelta$ by A; for the norms, the scalar product $(\cdot , \cdot )_{{\mathcal {H}}}$ of the Hilbert space ${\mathcal {H}}$ substitutes the $L^2$ scalar product $(\cdot , \cdot )$, and naturally the norm of ${\Vert }\cdot {\Vert }_{{\mathcal {H}}}$ replaces the $L^2$-norm ${\Vert }\cdot {\Vert }$. The energy norm ${\Vert }\nabla \cdot {\Vert }$ is changed into ${\Vert }A^\frac{1}{2} \cdot {\Vert }_{{\mathcal {H}}}$, and the duality pairing becomes $\langle \cdot , \cdot \rangle _{D(A^\frac{1}{2})', D(A^\frac{1}{2})}$.

Let $(w_i,\lambda _{ih}) \in D(A^\frac{1}{2}) \times \mathbb {R}^+$ with ${\Vert }w_i{\Vert }_{{\mathcal {H}}}=1$ and $(w_i,\chi _i)_{{\mathcal {H}}} > 0$ be given, for $\chi _i \in {\mathcal {H}}$, $i \ge 1$ fixed. Its residual ${\text {Res}}_{\theta }(w_i,\lambda _{ih})\in D(A^\frac{1}{2})'$ is now defined by

$$\begin{aligned} \langle {\text {Res}}_{\theta }(w_i,\lambda _{ih}), v\rangle _{D(A^\frac{1}{2})',D(A^\frac{1}{2})} {:}{=}\lambda _{ih}(w_i, v)_{{\mathcal {H}}} - (A^\frac{1}{2} w_i, A^\frac{1}{2} v)_{{\mathcal {H}}} \qquad \forall v \in D(A^\frac{1}{2}), \end{aligned}$$

with the dual norm

$$\begin{aligned} {\Vert }{\text {Res}}_{\theta }(w_i,\lambda _{ih}){\Vert }_{D(A^\frac{1}{2})'} {:}{=}\sup _{v \in D(A^\frac{1}{2}), \, {\Vert }A^\frac{1}{2} v{\Vert }_{{\mathcal {H}}} = 1} \langle {\text {Res}}_{\theta }(w_i,\lambda _{ih}), v\rangle _{D(A^\frac{1}{2})',D(A^\frac{1}{2})}. \end{aligned}$$

The Riesz representation of the residual is given by

Let

$$\begin{aligned} \lambda _{i-1}< \lambda _{ih}\quad \text { when } i>1, \quad \lambda _{ih}< \lambda _{i+1}, \end{aligned}$$

(A.1)

and

(A.2)

where

$$\begin{aligned} C_{ih} {:}{=}\min \left\{ \left( 1 - \frac{\lambda _{ih}}{\lambda _{i-1}}\right) ^2, \left( 1 - \frac{\lambda _{ih}}{\lambda _{i+1}}\right) ^2 \right\} . \end{aligned}$$

The generalizations of [17, Theorems 3.4 and 3.5] then read:

Theorem 5

(Eigenvalue bounds) Let $(w_i,\lambda _{ih}) \in D(A^\frac{1}{2}) \times \mathbb {R}^+$ with ${\Vert }w_i{\Vert }_{{\mathcal {H}}}=1$ and $(w_i,\chi _i)_{{\mathcal {H}}} > 0$, $i \ge 1$. Let assumptions (A.1) and (A.2) be satisfied. Then

$$\begin{aligned} {\Vert }A^\frac{1}{2}(u_i- w_i){\Vert }^2_{{\mathcal {H}}} - \lambda _i\alpha _{ih}^2 \le {\Vert }A^\frac{1}{2}w_i{\Vert }^2_{{\mathcal {H}}} - \lambda _i\le {\Vert }A^\frac{1}{2}(u_i- w_i){\Vert }^2_{{\mathcal {H}}}. \end{aligned}$$

(A.3a)

If, moreover $\alpha _{1h} \le \sqrt{2}$, then, for $i=1$,

$$\begin{aligned} {\frac{1}{2}}\left( 1-\frac{\lambda _1}{\lambda _2}\right) \left( 1-\frac{\alpha _{1h}^2}{4}\right) {\Vert }A^\frac{1}{2}(u_1- w_1){\Vert }^2_{{\mathcal {H}}} \le {\Vert }A^\frac{1}{2}w_1{\Vert }^2_{{\mathcal {H}}} - \lambda _1. \end{aligned}$$

(A.3b)

Let

$$\begin{aligned} {\overline{C}}_{ih} {:}{=}1 \text { if } i=1, \quad {\overline{C}}_{ih} {:}{=}\max \left\{ \left( \frac{\lambda _{ih}}{\lambda _1}- 1\right) ^2, 1\right\} \text{ if } i>1 \end{aligned}$$

and

$$\begin{aligned} \gamma _{ih} {:}{=}{\left\{ \begin{array}{ll} {\Vert }A^\frac{1}{2}(u_i- w_i){\Vert }^2_{{\mathcal {H}}} &{} \text {if } \lambda _i\le {\Vert }A^\frac{1}{2} w_i{\Vert }^2_{{\mathcal {H}}} \text { is known to hold, }\\ \max \{{\Vert }A^\frac{1}{2}(u_i- w_i){\Vert }^2_{{\mathcal {H}}}, \lambda _i\alpha _{ih}^2\} &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(A.4)

Then we also have:

Theorem 6

(Eigenvector bounds) Let the assumptions of Theorem 5 be satisfied. Then

(A.5a)

(A.5b)

If, moreover $\alpha _{ih}^2 \le 2 \frac{\lambda _1}{\lambda _i}$, then

Appendix B: Further improvement of the first eigenvalue upper bound

In [17, Theorem 5.2], a further improvement of the eigenvalue upper bounds of type of Theorem 2 was possible. We now extend it to the present setting, for the first eigenvalue.

We first need to generalize the conforming local residual lifting from [17, Sect. 4.3] to the present setting. Let for each vertex ${\mathbf{a }}\in {\mathcal {V}}_h$, $X_h^{\mathbf{a }}$ be an arbitrary finite-dimensional subspace of the space $H^1_*({\omega _{\mathbf{a }}})$ defined in (4.1). Typically, $X_h^{\mathbf{a }}{:}{=}{\mathbb {P}}_{p+1}({\mathcal {T}}_{\mathbf{a }}) \cap H^1_*({\omega _{\mathbf{a }}})$, similarly as for $W_h^{\mathbf{a }}$ in Sect. 3.2. We will now solve homogeneous local Neumann (Neumann–Dirichlet close to the boundary) problems on the patches ${\omega _{\mathbf{a }}}$ via conforming primal counterparts of problems (3.3a):

Definition 4

(Conforming local Neumann problems) For each ${\mathbf{a }}\in {\mathcal {V}}_h$, define $r_{1h}^{\mathbf{a }}\in X_h^{\mathbf{a }}$ by

(B.1)

Then set

$$\begin{aligned} r_{1h} {:}{=}\sum _{{\mathbf{a }}\in {\mathcal {V}}_h} \psi _{\mathbf{a }}r_{1h}^{\mathbf{a }}\in V. \end{aligned}$$

The functions $r_{1h}^{\mathbf{a }}$ are discrete Riesz representations of the local residual of the pair $(s_{1h}, \lambda _{1h})$ with hat-weighted test functions. Note that the right-hand side in (B.1) does not necessarily satisfy the usually required Neumann compatibility condition $(\psi _{\mathbf{a }}\lambda _{1h}s_{1h}- \nabla s_{1h}{\cdot }\nabla \psi _{\mathbf{a }}, 1)_{\omega _{\mathbf{a }}}= 0$ for ${\mathbf{a }}\in {\mathcal {V}}^{{\text {int}}}_h$, so that (B.1) cannot hold for a constant function $v_h=1$ on ${\omega _{\mathbf{a }}}$. Assumption 1 is in particular not required for $s_{1h}$; this does not influence the existence and uniqueness of $r_{1h}^{\mathbf{a }}$ (the system matrix in (B.1) is regular). Note also that $r_{1h}^{\mathbf{a }}\not \in V$ (when extended by zero outside of ${\omega _{\mathbf{a }}}$) but $\psi _{\mathbf{a }}r_{1h}^{\mathbf{a }}\in H^1_0({\omega _{\mathbf{a }}})$, whence the sum $r_{1h}$ belongs to V. For this construction, we have:

Lemma 1

(Lower dual residual bound) Let $(u_{1h},\lambda _{1h}) \in {\mathbb {P}}_p({\mathcal {T}}_h) \times \mathbb {R}^+$ be arbitrary. Construct $s_{1h}$ by Definition 3 and $r_{1h}$ by Definition 4. Then

Proof

The proof is trivial from (2.7b) and from the fact that $r_{1h} \in V$ for Definition 4. Importantly, this bound is positive, see [57, proof of Theorem 2]. $\square$

Equipped with these tools, we can now hopefully improve the upper bound (6.15) in Theorem 2 (we actually only mimic the Case B of Theorem 1, the other cases can be treated similarly).

Proposition 2

(Possible improvement of the first eigenvalue upper bound) Let ${\underline{\lambda }}_1, {\underline{\lambda }}_2$ be as in Theorem 1. Let $(u_{1h},\lambda _{1h}) \in {\mathbb {P}}_p({\mathcal {T}}_h) \times \mathbb {R}^+$, $p \ge 1$, be arbitrary. Let $s_{1h}$ be constructed following Definition 3 and $r_{1h}$ following Definition 4. Let $(s_{1h},\chi _1)>0$ and

$$\begin{aligned} {\overline{\alpha }}_{1h} {:}{=}{}&\sqrt{2} \left( 1 - \frac{\lambda _{1h}}{{\underline{\lambda }}_2}\right) ^{-1} {\underline{\lambda }}_2^{\!\!{-1/2}} \frac{1}{{\Vert }s_{1h}{\Vert }}\left( \frac{\lambda _{1h}}{\sqrt{{\underline{\lambda }}_1}} {\Vert }u_{1h}-s_{1h}{\Vert } + {\Vert }\nabla s_{1h}+ {{\varvec{\sigma }}}_{1h}{\Vert }\right) \\ \le {}&\min \left\{ \sqrt{2}, {\Vert }\chi _1{\Vert }^{-1} ({\tilde{s}}_{1h},\chi _1)\right\} , \end{aligned}$$

with ${\tilde{s}}_{1h}{:}{=}\frac{s_{1h}}{{\Vert }s_{1h}{\Vert }}$. Then

$$\begin{aligned} \lambda _1\le {\Vert }\nabla {\tilde{s}}_{1h}{\Vert }^2 - {{\tilde{\eta }}}_1, \end{aligned}$$

where

Proof

Note first that all the assumptions of [17, Theorems 3.4 and 3.5] are satisfied. We start by the second bound in [17, Theorem 3.4] which immediately implies, using ${\underline{\lambda }}_1\le \lambda _1$, ${\underline{\lambda }}_2\le \lambda _2$, and ,

Similarly, the second bound in [17, Theorem 3.5] now takes the form

Denote , , as well as . Combined with Lemma 1 and $0 < {\underline{\lambda }}_1\le \lambda _1$, this last inequality implies

$$\begin{aligned} e_h^2 + e_h\left( {\underline{\lambda }}_1+ 2 l_h\right) - \left( {\underline{\lambda }}_1R_h - l_h^2\right) \ge 0. \end{aligned}$$

Note that the discriminant of this quadratic inequality is the term $d_h$ and that it is non-negative. Thus

$$\begin{aligned} e_h \ge \frac{-\left( {\underline{\lambda }}_1+ 2 l_h\right) + \sqrt{d_h}}{2} \end{aligned}$$

and the desired bound follows. Note finally that for this estimate to actually improve on (6.15), ${{\tilde{\eta }}}_1$ needs to be positive, which follows when ${\underline{\lambda }}_1R_h > l_h^2$ and ${\Vert }\nabla {\tilde{s}}_{1h}{\Vert }^2 < {\underline{\lambda }}_2$. $\square$