1 Introduction

In many examples from physics to industrial applications, the solution of eigenvalue problems plays an essential role. Similar as for standard source problems, the finite element method seems to be a very promising method to discretize these problems due to its flexibility and good approximation properties. Numerous works deal with the analysis concerning stability, convergence properties and a priori error estimates, see [3, 9].

Since in general one can not assume high regularity of the eigenfunctions on arbitrary domains [27], the requirement for an adaptive mesh refinement strategy is obvious. Central to this approach is the derivation of an efficient and reliable a posteriori error estimator, as already developed for finite element methods in general [1, 35], and for eigenvalue problems in particular in [18].

In this work we consider the Laplace eigenvalue problem and approximate it using a mixed method. Several examples can be found in the literature using this approach, see [10, 17, 24, 26, 30], where adaptivity by means of residual error estimators (and using an \(H({\text {div}}) \times L^2\)-norm analysis) is discussed. We particularly want to refer to [14] where a unified framework for (guaranteed) a posteriori bounds (using a proper discrete \(H^1\)-energy norm) and a detailed overview of the literature is presented. A fundamental observation when using a mixed method is that it gives access to the hypercircle theory, see [28, 32], eventually leading to asymptotically exact upper bounds and local efficiency. However, unlike for standard source problems, see [13, 19, 23, 25, 36], a more profound approach is needed since the orthogonality of the corresponding errors is no longer exactly satisfied.

For eigenvalue problems this was first introduced in the work [6], by means of the Raviart-Thomas finite element. To discuss details, note that we have

$$\begin{aligned} \Vert \sigma _h - \sigma \Vert _0^2 + \Vert \nabla (u - u_h^{**}) \Vert _0^2&= \Vert \sigma _h - \nabla u_h^{**}\Vert _0^2 - 2(\sigma _h - \sigma , \nabla (u - u_h^{**}) ), \end{aligned}$$
(1)

where \(\lambda , u, \sigma \) are the exact eigenvalue, eigenfunction and its gradient, \(\lambda _h, u_h, \sigma _h\) are the corresponding approximations and \(u_h^{**}\) denotes some \(H^1\)-conforming post-processed function of \(u_h\). The first term on the right-hand side of (1) is computable and can therefore be used to define an a posteriori estimator \(\eta \). The astonishing observation in [6] was that in the case of an approximation using the Raviart-Thomas finite element, the second term \( 2(\sigma _h - \sigma , \nabla (u - u_h^{**}) )\) converges with higher order. Consequently, \(\eta \) is an asymptotically exact upper bound for the errors on the left-hand side of (1).

The question was whether the same ideas can be applied when using the Brezzi-Douglas-Marini (BDM) finite element instead. Surprisingly, as observed in [5] this is not the case. However, in [4] (using ideas from [21]) the authors were able to derive optimal upper bounds but with unknown constants (in contrast to the asymptotic bounds provided by \(\eta \) above).

The goal of this work is to derive asymptotically exact upper bounds (for the eigenfunction) as in [6] when using the BDM finite element method. For this we use the post-processing techniques for the eigenvalue and the eigenfunction as in [4], and consider modifications of the approaches from [6]. We introduce an additional (local) post-processing for the flux variable \(\sigma _h\), where we correct its divergence to fit the additional term in (1), which consequently converges again with higher order. Note, that the proposed method of this work is defined for all polynomial orders \(k \ge 1\), but the convergence results are only improved (compared to the Raviart-Thomas finite element) for \(k \ge 1\), see Remark 2.

The rest of the paper is organized as follows. Section 3 discusses the problem setting and its approximation. In Sect. 4 we present the local post-processing technique for the eigenfunction and the eigenvalue. The main results are then discussed in Sect. 5. While we first recapture the standard a posteriori error analysis based on (1) and reveal its breakdown due to a slow convergence of the additional terms, we then introduce the novel post-processing of the flux and derive the asymptotically exact upper bound. In the last Sect. 6 we present two numerical examples to validate our findings. The appendix, see Sect. 1, considers some additional results needed in the analysis.

2 Notation

We use the established notation for Sobolev spaces, i.e. \(L^2(\varOmega ), H^1(\varOmega )\) and \(H({\text {div}}, \varOmega )\) for a given domain \(\varOmega \). An additional zero subscript (for the latter two) indicates a vanishing trace. Further \(H^1(\varOmega , \mathbb {R}^d)\) (and similarly for other spaces) denotes a corresponding vector-valued version with d components. For \(\omega \subset \varOmega \) we use \((\cdot , \cdot )_\omega \) and \(\Vert \cdot \Vert _{0,\omega }\) for the inner product and the norm on \(L^2(\omega )\), respectively, and \(\vert \cdot \vert _{s,\omega }\) as the standard Sobolev seminorm of order s. If \(\omega = \varOmega \) we omit the additional subscript. We write \(A \lesssim B\) when there is a positive constant C, that is independent of the mesh parameter h (see below) such that \(A \le C B\). Analogously we define \(A \gtrsim B\).

3 Problem setting

Let \(\varOmega \subset \mathbb {R}^d\) be a polygon or polyhedron for \(d = 2, 3\), respectively. We consider the mixed formulation of the Laplace eigenvalue problem with homogeneous Dirichlet boundary conditions, i.e. we want to find a \(\lambda \in \mathbb {R}, u \in L^2(\varOmega )\) and \(\sigma \in H({\text {div}}, \varOmega )\) such that

$$\begin{aligned} (\sigma , \tau ) {+} ({\text {div}}\tau , u)&= 0 \quad \quad \quad \quad \forall \tau \in H({\text {div}}, \varOmega ), \end{aligned}$$
(2a)
$$\begin{aligned} - ({\text {div}}\sigma , v)&= \lambda (u,v) \quad \forall v \in L^2(\varOmega ). \end{aligned}$$
(2b)

We approximate (2) by a mixed method using the \(\text {BDM}\) finite element for the approximation of \(\sigma \) and a piece-wise polynomial approximation of u. To this end let \({\mathcal {T}_h}\) be a regular triangulation of \(\varOmega \) into triangles and tetrahedrons in two and three dimensions, respectively. Let \(k \ge 1\) be a fixed integer (see Remark 2 for a comment regarding the lowest order case). We introduce the spaces

$$\begin{aligned} U_h&:= \{v_h \in L^2(\varOmega ): v_h\vert _K \in \mathbb {P}^k(K) ~\forall K \in {\mathcal {T}_h}\},\\ \varSigma _h&:= \{\tau _h \in H({\text {div}}, \varOmega ): {\tau _h}\vert _K \in \mathbb {P}^{k+1}(K, \mathbb {R}^d) ~\forall K \in {\mathcal {T}_h}\}, \end{aligned}$$

where \(\mathbb {P}^l(K)\) denotes the space of polynomials of order \(l \ge 0\) on K, and \(\mathbb {P}^l(K, \mathbb {R}^d)\) denotes the corresponding vector-valued version. An approximation of (2) then seeks \(\lambda _h \in \mathbb {R}\), \(u_h \in U_h\) and \(\sigma _h \in \varSigma _h\) such that

$$\begin{aligned} (\sigma _h, \tau _h) {+} ({\text {div}}\tau _h, u_h)&= 0 \quad \quad \quad \quad \quad \quad \! \forall \tau _h \in \varSigma _h, \end{aligned}$$
(3a)
$$\begin{aligned} - ({\text {div}}\sigma _h, v_h)&= \lambda _h (u_h,v_h) \quad \forall v_h \in U_h. \end{aligned}$$
(3b)

Review article [9] (for example) states that problem (3) defines a good approximation of the continuous eigenvalue problem (2) in the sense that it does not produce any spurious modes and that eigenfunctions are approximated with the proper multiplicity. The approximation results are summarized in the following. To this end let \(s > 1/2\) and let \((\lambda , u, \sigma )\) be a solution of the eigenvalue problem (2) with the regularity \(u \in H^{1 + s}(\varOmega )\) and \(\sigma \in H({\text {div}}, \varOmega ) \cap H^s(\varOmega , \mathbb {R}^d)\) (for the regularity results see [20, 22]). Then there exists a discrete solution of (3) such that (see [3, 9])

$$\begin{aligned} \Vert u - u_h \Vert _0&\lesssim h^{r} \vert u \vert _{r+1}, \end{aligned}$$
(4a)
$$\begin{aligned} \Vert \sigma - \sigma _h \Vert _0&\lesssim h^{r'} \vert u \vert _{r'+1}, \end{aligned}$$
(4b)
$$\begin{aligned} \Vert {\text {div}}(\sigma - \sigma _h) \Vert _0&\lesssim h^{r} (\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}), \end{aligned}$$
(4c)

where \(r = \min \{s, k+1\}\) and \(r' = \min \{s, k+2\}\), and \(h=\max \limits _{K \in {\mathcal {T}_h}} h_K\) where \(h_K\) is the diameter of an element K. If s is big enough we have \(r' = r+1\). Above estimates follow from the abstract theory from [9, 17] and [30], and the approximation results of the source problem, see [10]. It is worth mentioning, that the constants in (4) are non-trivial as they depend, beside the discrete stability constants of (3), particularly on the spectrum of the associated solution operator of the continuous eigenvalue problem (2). In addition we have

$$\begin{aligned} \Vert u - u_h \Vert _{1,h} \lesssim h^{r-1} \vert u \vert _{r+1}, \end{aligned}$$
(5)

where

$$\begin{aligned} \Vert u - u_h \Vert _{1,h}^2 := \sum _{K \in {\mathcal {T}_h}} \Vert \nabla (u - u_h) \Vert _{0,K}^2 + \sum _{F \in \mathcal {F}_h} \frac{1}{h_F} \Vert {[\![ u_h ] \!]} \Vert _{0,F}^2. \end{aligned}$$

Here \( {[\![ \cdot ] \!]}\) denotes the standard jump operator, \(\mathcal {F}_h\) the set of facets of the triangulation \({\mathcal {T}_h}\), and \(h_F\) the diameter of a facet \(F \in \mathcal {F}_h\). Note that above results demand a careful choice of the approximated eigenfunction \(u_h\) and the approximated gradient \(\sigma _h\). An example, well established in the literature, is given by a normalization such that \(\Vert u_h \Vert _0 = \Vert u \Vert _0 = 1\) and by choosing the sign \((u,u_h) > 0\). Note that this also fixes \(\sigma \) and \(\sigma _h\) by (2a) and (3a), respectively. For simplicity, we assume for the rest of this work that \(\lambda \) is a simple eigenvalue and that the above choice of sign and scaling of the continuous and the discrete eigenfunctions is applied. Further, for simplicity, we will call \((\lambda , u, \sigma )\) the solution of (2), keeping in mind that a different scaling and sign can be chosen.

Remark 1

The case of eigenvalues with a higher multiplicity demands more carefulness, particularly if an a posteriori analysis is considered. We particularly want to refer to [11] where the authors considered eigenvalue clusters using a mixed formulation. For the convergence of the adaptive scheme they used a residual based error estimator and provided a detailed analysis. An extension to estimators that are based on identity (1) is still open and is discussed in future works.

Remark 2

Although the schemes proposed in this work are computable also for the lowest order case \(k=0\), one does not observe any high-order convergence of the post processed variables defined later in the work. The reason for this is that the Aubin-Nitsche technique, needed in the analysis, can not be applied for this case.

4 Local post-processing for \(u_h\) and \(\lambda _h\)

For a sufficiently smooth solution, estimates (4) and (5) show that there is a gap of two between the order of convergence of \(\Vert \sigma - \sigma _h \Vert _0\) and \(\Vert u - u_h \Vert _{1,h}\). In [17] the following identity is proven

$$\begin{aligned} \lambda - \lambda _h = \Vert \sigma - \sigma _h\Vert _0^2 - \lambda _h \Vert u - u_h \Vert _0^2, \end{aligned}$$
(6)

which, due to (4), gives the estimate (using \(r \le r'\))

$$\begin{aligned} \vert \lambda - \lambda _h \vert&\lesssim h^{2r} \vert u \vert ^2_{r+1} + h^{2r'} \vert u \vert ^2_{r'+1} \lesssim h^{2r} (\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}). \end{aligned}$$
(7)

We see that the order of convergence of \(\vert \lambda - \lambda _h \vert \) is dominated by the order of the \(L^2\)-error of the eigenfunction. The reduced convergence of \(u_h\) (compared to the \(L^2\)-error of \(\sigma _h\)) is well known for mixed methods and can be improved by means of a local post-processing, see [2, 34], and particularly for eigenfunctions in [15]. Consequently, using the ideas from [21], we can then also get an improved eigenvalue.

For a given integer \(l \ge 0\) let \(\varPi ^l\) denote the \(L^2\)-projection onto element-wise polynomials of order l. Consider the spaces

$$\begin{aligned} U_h^* := \{v_h \in L^2(\varOmega ): v_h\vert _K \in \mathbb {P}^{k+2}(K) ~\forall K \in {\mathcal {T}_h}\}, \quad \text {and} \quad U_h^{**} := U_h^* \cap H^1_0(\varOmega ), \end{aligned}$$

then we define \(u^*_h \in U_h^*\) by

$$\begin{aligned} (\nabla u_h^*, \nabla v_h^*)_K&= (\sigma _h, \nabla v^*_h)_K \quad \forall v_h^* \in ({\text {id}}- \varPi ^k)\vert _K \mathbb {P}^{k+2}(K), \forall K \in {\mathcal {T}_h}, \end{aligned}$$
(8a)
$$\begin{aligned} \varPi ^k u_h^*&= u_h. \end{aligned}$$
(8b)

For the discretization of the standard source problem (i.e. the Poisson problem), it is known that the kernel inclusion property \({\text {div}}\varSigma _h \subseteq U_h\) (see [10]) and commuting interpolation operators yield a super convergence property of the projected error \(\Vert \varPi ^k u - u_h \Vert _0\) given by \(\rho (h)\mathcal {O}(h^{r'})\). Here \(\rho (h)\) is a function that depends on the regularity of the problem and for which we have \(\rho (h) \rightarrow 0\) as \(h \rightarrow 0\). For convex domains we have \(\rho (h) = \mathcal {O}(h)\). This super convergence of the projected error is the fundamental ingredient to derive the enhanced approximation properties of \(u_h^*\).

Unfortunately the same technique, i.e. the one from the standard source problem, does not work for the eigenvalue problem and an improved convergence estimate of \(\Vert \varPi ^k u - u_h \Vert _0\) is more involved. This has been discussed for the lowest order case in [20], for a more general setting including eigenvalue clusters in [11], for Maxwell eigenvalue problems in [12] and for the Stokes problem for example in [21]. Unfortunately, these results are only presented using the full \(\Vert \cdot \Vert _{{\text {div}}}\)-norm (or the corresponding mixed norm) for \(\varSigma \) and \(\varSigma _h\). While such an estimate is applicable for an approximation of (3) using Raviart-Thomas finite elements, the \(\text {BDM}\) case is not covered since (4c) and (4b) show different convergence orders which would spoil the estimate. As the author is not aware of a detailed proof that can be found in the literature, it will be given in the appendix in Sect. 1. Note however, that these results are already used for example in [4] (without proof). The resulting super convergence reads as

$$\begin{aligned} \Vert \varPi ^k u - u_h \Vert _0 \lesssim \rho (h)( h \Vert u - u_h \Vert _0 + \Vert \sigma - \sigma _h\Vert _0), \end{aligned}$$
(9)

which in combination with the techniques from [34], then yields the approximation properties (see also [29])

$$\begin{aligned} \Vert u - u_h^* \Vert _0&\lesssim \rho (h) h^{r'} (\vert u \vert _{r+1} + \vert u\vert _{r'+1}), \end{aligned}$$
(10a)
$$\begin{aligned} \Vert u - u_h^*\Vert _{1,h}&\lesssim h^{r'} (\vert u \vert _{r+1} + \vert u\vert _{r'+1}). \end{aligned}$$
(10b)

Since \(u_h^*\) is not \(H^1\)-conforming the final post-processing step consists of the application of an averaging operator \(I^a: U_h^* \rightarrow U_h^{**}\) often also called Oswald operator, see [31] and [16] for details on the approximation properties. We set \(u_h^{**}:= I^a(u_h^*)\) for which we have by (10)

$$\begin{aligned} \Vert u - u_h^{**} \Vert _0&\lesssim \rho (h) h^{r'} (\vert u \vert _{r+1} + \vert u\vert _{r'+1}), \end{aligned}$$
(11a)
$$\begin{aligned} \Vert \nabla (u - u_h^{**}) \Vert _0&\lesssim h^{r'} (\vert u \vert _{r+1} + \vert u\vert _{r'+1}). \end{aligned}$$
(11b)

We conclude this section by introducing a post-processing of the eigenvalue. As in [4, 21] we define

$$\begin{aligned} \lambda _h^* := \frac{-({\text {div}}\sigma _h, u_h^*)}{(u_h^*,u_h^*)}. \end{aligned}$$
(12)

The following lemma was given in [21]. Since we need some intermediate steps in the sequel, we include the proof.

Lemma 1

Let \(s > 1/2\) and let \((\lambda , u, \sigma )\) be the solution of (2) with the regularity \(u \in H^{1 + s}(\varOmega )\) and \(\sigma \in H({\text {div}}, \varOmega ) \cap H^s(\varOmega , \mathbb {R}^d)\). Further let \(\Vert u_h^* \Vert _0 \ne 0\). There holds

$$\begin{aligned} \vert \lambda - \lambda _h^* \vert \lesssim (\rho (h) h^{r'+r} + h^{2r'}) (\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}), \end{aligned}$$

where \(r = \min \{s,k+1\}\) and \(r' = \min \{s,k+2\}\).

Proof

Since \(\Vert u\Vert _0 = 1\) we have using that \({\text {div}}\varSigma _h \subseteq U_h\) and (8b)

$$\begin{aligned} (\sigma ,\sigma )&= -({\text {div}}\sigma ,u) = \lambda (u,u) = \lambda , \\ (\sigma _h,\sigma _h)&= -({\text {div}}\sigma _h,u_h) = -(\varPi ^k {\text {div}}\sigma _h,u_h) = -({\text {div}}\sigma _h, u^*_h) = \lambda _h^* (u_h^*,u_h^*),\\ \Vert \sigma - \sigma _h \Vert ^2_0&= (\sigma - \sigma _h ,\sigma - \sigma _h ) = (\sigma ,\sigma ) + (\sigma _h,\sigma _h) - 2(\sigma ,\sigma _h) \\ {}&= \lambda + \lambda _h^* (u_h^*,u_h^*) + 2 ({\text {div}}\sigma _h, u). \end{aligned}$$

Using \(\lambda _h^*\Vert u - u_h^* \Vert ^2_0 = \lambda _h^*(u,u) + \lambda _h^*(u_h^*,u_h^*) - 2 \lambda _h^*(u,u_h^*)\) we have in total

$$\begin{aligned} \lambda -&\lambda _h^* \\ {}&= \Vert \sigma - \sigma _h \Vert ^2_0 - \lambda _h^* (u_h^*,u_h^*) - 2 ({\text {div}}\sigma _h, u) - \lambda _h^*,\\&=\Vert \sigma - \sigma _h \Vert ^2_0 + \lambda _h^* (u,u) - 2\lambda _h^* (u,u_h^*) - \lambda _h^* \Vert u - u_h^* \Vert ^2_0 - 2 ({\text {div}}\sigma _h, u) - \lambda _h^*, \end{aligned}$$

and thus again with \(\Vert u \Vert _0 = 1\) this gives

$$\begin{aligned} \lambda - \lambda _h^*&= \Vert \sigma - \sigma _h \Vert ^2_0 - \lambda _h^* \Vert u - u_h^* \Vert ^2_0 - 2 ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u). \end{aligned}$$
(13)

Since \(({\text {div}}\sigma _h + \lambda _h^* u_h^*, u_h^*) = 0\) (according to the definition of \(\lambda _h^*\)), the last term can be written as

$$\begin{aligned} ({\text {div}}\sigma _h&+ \lambda _h^* u_h^*, u) \\ {}&= ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u - u_h^*), \\ {}&= ({\text {div}}(\sigma _h - \sigma ), u - u_h^*) + ({\text {div}}\sigma + \lambda _h^* u_h^*, u - u_h^*),\\ {}&= ({\text {div}}(\sigma _h - \sigma ), u - u_h^*) + (-\lambda u + \lambda _h^* u_h^*, u - u_h^*)\\ {}&= ({\text {div}}(\sigma _h - \sigma ), u - u_h^*) + \lambda _h^* (u_h^* - u, u - u_h^*) - (\lambda - \lambda _h^*) (u, u - u_h^*). \end{aligned}$$

By the Cauchy-Schwarz inequality we finally get

$$\begin{aligned} \vert \lambda - \lambda _h^* \vert \le&\Vert \sigma - \sigma _h \Vert ^2_0 + \lambda _h^* \Vert u - u_h^* \Vert ^2_0 \\ {}&+ 2 \Vert {\text {div}}(\sigma _h - \sigma ) \Vert _0 \Vert u - u_h^*\Vert _0 + 2\vert \lambda - \lambda _h^*\vert \Vert u - u_h^*\Vert _0 \\ \lesssim&\Vert \sigma - \sigma _h \Vert ^2_0 + \Vert u - u_h^* \Vert ^2_0 + \Vert {\text {div}}(\sigma _h - \sigma ) \Vert _0\Vert u - u_h^*\Vert _0 + \vert \lambda - \lambda _h^* \vert ^2. \end{aligned}$$

Thus, for h small enough, the last term can be moved to the left hand side, and we can conclude the proof using (10) and (4). \(\square \)

5 A posteriori analysis

In this section we provide an a posteriori error analysis and define an appropriate error estimator. We follow [6] where the authors derived an error estimator using the variables \(\sigma _h\) and \(u_h^{**}\). While this works for a mixed approximation of (4) using the Raviart-Thomas finite element of order k (as was done in [6]), this does not work for our setting. To discuss the problematic terms and to motivate our modification, we present more details in the following. Since \(\sigma = \nabla u\) we have

$$\begin{aligned} \Vert \sigma _h - \nabla u_h^{**}\Vert _0^2&= \Vert \sigma _h - \sigma + \sigma - \nabla u_h^{**}\Vert _0^2 \\ {}&= \Vert \sigma _h - \sigma \Vert _0^2 + \Vert \nabla (u - u_h^{**}) \Vert _0^2 + 2(\sigma _h - \sigma , \nabla (u - u_h^{**}) ). \end{aligned}$$

Using integration by parts, \(u_h^{**} \in H^1_0(\varOmega )\) and \(-{\text {div}}\sigma _h = \lambda _h u_h\), the last term can be written as

$$\begin{aligned} (\sigma _h - \sigma , \nabla (u - u_h^{**}) )&= -({\text {div}}(\sigma _h - \sigma ), u - u_h^{**} ) \\ {}&= -(\lambda _h u_h - \lambda u, u - u_h^{**} ) \\ {}&= -(\lambda _h u_h + \lambda u_h - \lambda u_h - \lambda u, u - u_h^{**} ) \\ {}&= -(\lambda _h - \lambda ) ( u_h, u - u_h^{**} ) - \lambda ( u_h - u, u - u_h^{**}). \end{aligned}$$

In total this gives the guaranteed upper bound

$$\begin{aligned}&\Vert \sigma _h - \sigma \Vert _0^2 + \Vert \nabla (u - u_h^{**}) \Vert _0^2 \\&\quad \le \Vert \sigma _h - \nabla u_h^{**}\Vert _0^2 + 2\vert \lambda _h - \lambda \vert \Vert u - u_h^{**}\Vert _0 + 2 \lambda \Vert u_h - u\Vert _0 \Vert u - u_h^{**}\Vert _0. \end{aligned}$$

In [6] the first term on the right hand side is the (computable) proposed error estimator, whereas the second and third are high-order terms. Compared to our setting we can see the problem since

$$\begin{aligned} \Vert \sigma _h - \nabla u_h^{**}\Vert _0^2&\lesssim h^{2k+4}, \\ \vert \lambda _h - \lambda \vert \Vert u - u_h^{**}\Vert _0&\lesssim h^{3k+4}, \\ \Vert u_h - u\Vert _0 \Vert u - u_h^{**}\Vert _0&\lesssim h^{2k+4}, \end{aligned}$$

where for simplicity, i.e. to allow a simpler comparison, we assumed a smooth solution. Whereas the second term converges with an increased rate (compared to \(2k+4\)), the reduced convergence order of \(\Vert u - u_h\Vert _0\), see equation (4a), spoils the estimate of the last term. Note that the error of \(u_h\) appears in the estimates because we used the identity \(-{\text {div}}\sigma _h = \lambda _h u_h\) in the above proof.

To fix this problem we propose another post-processing. Whereas the first two post-processing routines were used to increase the convergence rate of the error of the eigenfunction and eigenvalue i.e. \(u_h^{*}\) (and \(u_h^{**}\)) and \(\lambda _h^*\), respectively, we now aim to construct a \(\sigma _h^*\) with a fixed divergence constraint rather than improving its approximation properties measured in the \(L^2\)-norm. To this end we define the space

$$\begin{aligned} \varSigma _h^* := \{ \tau _h \in H({\text {div}}, \varOmega ): \tau _h\vert _K \in \mathbb {P}^{k+3}&(K, \mathbb {R}^d) ~\forall K \in {\mathcal {T}_h}, \\ {}&\tau _h\cdot n\vert _F \in \mathbb {P}^{k+1}(F)~\forall F \in \mathcal {F}_h\}. \end{aligned}$$

The space \(\varSigma _h^*\) reads as a \(\text {BDM}\) space of order \(k+3\) with a reduced polynomial order of the normal traces. Note that other choices of \(\varSigma _h^*\) are possible, see Remark 3. The basic idea now is to find a \(\sigma _h^* \in \varSigma _h^*\) being as "close" as possible to \(\sigma _h\) (i.e. being a good approximation) such that the divergence is modified appropriately using the additional high-order normal-bubbles (i.e. functions with a zero normal component along the boundary of each element). Since these bubbles are defined locally, this can be done in an element-wise procedure. Now let \(\xi _h \in \varSigma _h^*\) be arbitrary. Proposition 2.3.1 in [10] shows that the following degrees of freedom (here applied to \(\xi _h\))

$$\begin{aligned} \text {Facet moments:}&~~ \int _F \xi _h \cdot n r_h {\text {ds}}\quad \forall r_h \in \mathbb {P}^{k+1}(F) ~ \forall F \in \mathcal {F}_h, \end{aligned}$$
(14a)
$$\begin{aligned} \text {Div moments:}&~~ \int _K {\text {div}}\xi _h q_h {\text {dx}}\quad \forall q_h \in \mathbb {P}^{k+2}(K) / \mathbb {P}^{0}(K) ~\forall K \in {\mathcal {T}_h}, \end{aligned}$$
(14b)
$$\begin{aligned} \text {Vol moments:}&~~ \int _K \xi _h \cdot l_h {\text {dx}}\quad \forall l_h \in \mathbb {H}^{k+3}(K) ~\forall K \in {\mathcal {T}_h}, \end{aligned}$$
(14c)

where \(\mathbb {H}^{k+3}(K):= \{l_h \in \mathbb {P}^{k+3}(K, \mathbb {R}^d): {\text {div}}l_h = 0, l_h\cdot n\vert _{\partial K} = 0\}\), are unisolvent. By that we can define the post processed flux \(\sigma ^*_h \in \varSigma _h^*\) by

$$\begin{aligned}&\int _F (\sigma ^*_h - \sigma _h) \cdot n r_h {\text {ds}}= 0 \qquad \qquad \qquad \ \quad \qquad \forall r_h \in \mathbb {P}^{k+1}(F)~ \forall F \in \mathcal {F}_h, \end{aligned}$$
(15a)
$$\begin{aligned}&\int _K ({\text {div}}\sigma ^*_h + \lambda _h u_h^*) q_h {\text {dx}}= 0 \qquad \qquad \qquad \ \quad \forall q_h \in \mathbb {P}^{k+2}(K) / \mathbb {P}^{0}(K) ~\forall K \in {\mathcal {T}_h}, \end{aligned}$$
(15b)
$$\begin{aligned}&\int _K (\sigma ^*_h - \sigma _h) \cdot l_h {\text {dx}}= 0 \quad \!\! \forall l_h \in \mathbb {H}^{k+3}(K)\quad ~ \forall K \in {\mathcal {T}_h}. \end{aligned}$$
(15c)

Note that since \(\sigma _h\) is normal continuous, i.e. the normal trace coincides on a common facet of two neighboring elements, the boundary constraints (15a) of \(\sigma _h^*\) can be set locally on each element (boundary) separately. Further, since \(\sigma _h \cdot n\) and \(\sigma _h^* \cdot n \) have the same polynomial degree \(k+1\), the moments from (15a) result in \(\sigma _h \cdot n = \sigma ^*_h \cdot n\). In total this shows that one can solve for \(\sigma _h^*\) on each element independently, i.e. this can be done computationally very efficient. In Remark 4 we also make a comment on the choice of (15b).

Theorem 1

Let \(\sigma _h^* \in \varSigma _h^*\) be the function defined by (15), then there holds

$$\begin{aligned} -{\text {div}}\sigma _h^* = \lambda _h u_h^*. \end{aligned}$$

Let \(s > 1/2\) and \(\sigma \in H({\text {div}}, \varOmega ) \cap H^s(\varOmega , \mathbb {R}^d)\) be the solution of the eigenvalue problem (2). There holds the a priori error estimate

$$\begin{aligned} \Vert \sigma - \sigma _h^* \Vert _0 \lesssim h^{r'} (\vert u \vert _{r+1} + \vert u\vert _{r'+1}), \end{aligned}$$

where \(r' = \min \{s,k+2\}\) and \(r = \min \{s,k+1\}\).

Proof

We start with the proof of the divergence identity. Let \(K \in {\mathcal {T}_h}\) and \(q_h \in \mathbb {P}^{k+2}(K)\) be arbitrary, then we have

$$\begin{aligned} -\int _K {\text {div}}\sigma _h^* q_h {\text {dx}}&= -\int _K {\text {div}}\sigma _h^* (q_h - \varPi ^0q_h) {\text {dx}}- \int _K {\text {div}}\sigma _h^* \varPi ^0q_h {\text {dx}}\\ {}&= \int _K \lambda _h u_h^* (q_h - \varPi ^0q_h) {\text {dx}}- \varPi ^0q_h \int _{\partial K} \sigma _h^* \cdot n {\text {ds}}, \end{aligned}$$

where the second step followed due to (15b) and the Gauss theorem. Using (15a) and (3b), the last integral can be written as

$$\begin{aligned} -\varPi ^0q_h \int _{\partial K} \sigma _h^* \cdot n {\text {ds}}&= -\varPi ^0q_h \int _{\partial K} \sigma _h \cdot n {\text {ds}}= -\int _K \varPi ^0q_h {\text {div}}\sigma _h {\text {dx}}\end{aligned}$$
(16a)
$$\begin{aligned}&= \int _K \varPi ^0q_h \lambda _h u_h {\text {dx}}=\int _K \varPi ^0q_h \lambda _h u^*_h {\text {dx}}, \end{aligned}$$
(16b)

where we used (8b) in the last step. All together this gives

$$\begin{aligned} -\int _K {\text {div}}\sigma _h^* q_h {\text {dx}}= \int _K \lambda _h u^*_h q_h {\text {dx}}, \end{aligned}$$

from which we conclude the proof as \({\text {div}}\sigma _h^* - \lambda _h u^*_h \in \mathbb {P}^{k+2}(K)\) and \(q_h\) was arbitrary.

Now let \(I^*_h\) be the canonical interpolation operator into \(\varSigma _h^*\) with respect to the moments (14), and let \(I_h\) be the interpolation operator into \(\varSigma _h\) which is defined using the same moments (14) but with \(q_h \in \mathbb {P}^{k}(K) / \mathbb {P}^{0}(K)\) and \(l_h \in \mathbb {H}^{k+1}(K)\) instead. First, the triangle inequality gives \( \Vert \sigma - \sigma _h^* \Vert _0 \le \Vert \sigma - I^*_h\sigma \Vert _0 + \Vert I^*_h\sigma - \sigma _h^* \Vert _0\). Since the first term can be bounded by the properties of \(I^*_h\), we continue with the latter which can be written as

$$\begin{aligned} \Vert I^*_h\sigma - \sigma _h^*\Vert _0 = \Vert I^*_h(\sigma - \sigma _h^*)\Vert _0 \le \Vert (I^*_h- I_h) (\sigma - \sigma _h^*) \Vert _0 + \Vert I_h(\sigma - \sigma _h^*)\Vert _0. \end{aligned}$$

By the definition of the interpolation operators and similar steps as above we have \(I_h(\sigma _h^*) = \sigma _h\), and thus the term most to the right simplifies to

$$\begin{aligned} \Vert I_h(\sigma - \sigma _h^*)\Vert _0 = \Vert I_h\sigma - \sigma _h\Vert _0 \le \Vert I_h\sigma - \sigma \Vert _0 + \Vert \sigma - \sigma _h\Vert _0. \end{aligned}$$

We continue with the other term. For this let \(\psi ^{{\text {div}}}_i\) be the hierarchical dual basis functions of the highest order divergence moments from (14b) given by \(\int _K {\text {div}}(\cdot ) q_i {\text {dx}}\) with \(q_i \in \mathbb {P}^{k+2}(K) / \mathbb {P}^{k}(K)\). Similarly let \(\psi _i^{\mathbb {H}}\) be the hierarchical dual basis functions of the highest order vol moments from (14c) given by \(\int _K (\cdot ) \cdot l_i {\text {dx}}\) with \(l_i \in \mathbb {H}^{k+3}(K) / \mathbb {H}^{k+1}(K)\). An explicit construction of these basis functions can be found for example in [8, 37]. Also let \(N_{{\text {div}}}\) and \(N_\mathbb {H}\) be the corresponding index sets. Using (2b), (15b) and (15c), this then gives

$$\begin{aligned} (I^*_h- I_h) (\sigma&- \sigma _h^*) \\ {}&= \sum _{i \in N_{{\text {div}}}} \int _K {\text {div}}(\sigma - \sigma _h^*) q_i {\text {dx}}\psi ^{{\text {div}}}_i + \sum _{i \in N_{\mathbb {H}}} \int _K (\sigma - \sigma _h^*) l_i {\text {dx}}\psi ^\mathbb {H}_i\\ {}&= - \sum _{i \in N_{{\text {div}}}} \int _K (\lambda u - \lambda _h u_h^*) q_i {\text {dx}}\psi ^{{\text {div}}}_i + \sum _{i \in N_{\mathbb {H}}} \int _K (\sigma - \sigma _h) l_i {\text {dx}}\psi ^\mathbb {H}_i, \end{aligned}$$

which implies that (using that the norms of the \(q_i, l_i\) and \(\psi ^{{\text {div}}}_i,\psi ^{\mathbb {H}}_i\) are bounded)

$$\begin{aligned} \Vert (I^*_h- I_h) (\sigma - \sigma _h^*)\Vert _0&\lesssim \Vert \lambda u - \lambda _h u_h^* \Vert _0 + \Vert \sigma - \sigma _h \Vert _0 \\ {}&\lesssim \vert \lambda \vert \Vert u - u_h^*\Vert _0 + \vert \lambda - \lambda _h \vert \Vert u_h^*\Vert _0 + \Vert \sigma - \sigma _h \Vert _0. \end{aligned}$$

Since \(\Vert u_h^*\Vert _0 \le \Vert u_h^* - u\Vert _0 + \Vert u \Vert _0\), we can conclude the proof by the approximation properties of \(I_h\) and \(I^*_h\) (see Proposition 2.5.1 in [10]), estimates (7) and (10) and by \(\rho (h) h^{r'} \le h^{r'}\) and \(h^{2r} \le h^{r'}\). \(\square \)

Remark 3

Instead of choosing \(\varSigma _h^*\) as above, one can for example also use the standard Raviart-Thomas space of order \(k+2\) denoted by \(RT^{k+2}\). Since \({\text {div}}RT^{k+2} = U_h^*\) it is again possible to set \(-{\text {div}}\sigma _h^* = \lambda _h u_h^*\) (using the appropriate degrees of freedom). However, since the normal trace of \(\sigma ^*_h\) is now in \(\mathbb {P}^{k+2}(F)\) on each facet \(F \in \mathcal {F}_h\), one has to be more careful defining the edge moments. Precisely, we would now set

$$\begin{aligned} \varPi ^{k+1} (\sigma ^*_h \cdot n) = \sigma _h \cdot n, \quad \text {and} \quad ({\text {id}}- \varPi ^{k+1})( \sigma ^*_h \cdot n) = 0. \end{aligned}$$

where the projection has to be understood as the \(L^2\)-projection on the facets.

Remark 4

One might be curious why we do not use \(\lambda _h^*\) instead of \(\lambda _h\) in the definition of \({\text {div}}\sigma ^*_h\) in (15b). Indeed, as can be seen in the proof this is a crucial choice since we used in (16) that the mean value of the divergence is fixed by the constant normal moments (first equal sign) and thus coincides with \(\varPi ^0 (\lambda _h u_h)\) (third equal sign). Choosing \(\lambda _h^*\) in (15a) would then lead to a mismatch of the low-order and high-order parts of the divergence.

We are now in the position of defining the local error estimator on each element \(K \in {\mathcal {T}_h}\) by

$$\begin{aligned} \eta (K) := \Vert \nabla u_h^{**} - \sigma _h^* \Vert _K, \end{aligned}$$

and the corresponding global estimator by

$$\begin{aligned} \eta := \Big (\sum _{K \in {\mathcal {T}_h}} \eta (K)^2\Big )^{1/2} = \Vert \nabla u_h^{**} - \sigma _h^* \Vert _0. \end{aligned}$$

Theorem 2

Let \((\lambda ,u,\sigma )\) be the solution of (2). Let \((\lambda _h,u_h,\sigma _h)\) be the solution of (4) and let \(u_h^{**}\) and \(\sigma _h^*\) be the post-processed solutions. There holds the reliability estimate

$$\begin{aligned} \Vert \nabla u - \nabla u_h^{**} \Vert _0^2 + \Vert \sigma - \sigma ^*_h\Vert ^2_0 \le \eta ^2 + {\text {hot}}(h) \end{aligned}$$

where \({\text {hot}}(h):= 2 \vert (\sigma _h^* - \sigma , \nabla (u-u_h^{**}))\vert \) with

$$\begin{aligned} {\text {hot}}(h) \lesssim \rho (h) (h^{2r + r'} + \rho (h)h^{2r'}) (\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}), \end{aligned}$$

is a high-order term compared to \(\mathcal {O}(h^{2r'})\) as \(h \rightarrow 0\). Further, there holds the efficiency

$$\begin{aligned} \eta \le \Vert \nabla u - \nabla u_h^{**} \Vert _0 + \Vert \sigma - \sigma ^*_h\Vert _0. \end{aligned}$$

Proof

Following the same steps as at the beginning of this section we arrive at

$$\begin{aligned} \Vert \nabla u - \nabla u_h^{**} \Vert _0^2 + \Vert \sigma - \sigma ^*_h\Vert ^2_0 = \Vert \nabla u_h^{**} - \sigma _h^* \Vert ^2_0 + 2 (\sigma _h^* - \sigma , \nabla (u-u_h^{**})). \end{aligned}$$

For the last term we now have

$$\begin{aligned} (\sigma ^*_h - \sigma , \nabla (u - u_h^{**}) )&= -({\text {div}}(\sigma ^*_h - \sigma ), u - u_h^{**} ) \\ {}&= -(\lambda _h u^*_h - \lambda u, u - u_h^{**} ) \\ {}&= -( \lambda _h - \lambda ) ( u^*_h, u - u_h^{**} ) - \lambda ( u^*_h - u, u - u_h^{**}). \end{aligned}$$

Whereas the first term converges of order

$$\begin{aligned} \vert \lambda _h - \lambda \vert \vert ( u^*_h, u - u_h^{**} )\vert&\le \vert \lambda _h - \lambda \vert \Vert u_h^*\Vert _0 \Vert u - u_h^{**}\Vert _0 \\ {}&\lesssim \rho (h)h^{2r+r'} (\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}), \end{aligned}$$

we have for the second term

$$\begin{aligned} \vert \lambda \vert \vert ( u^*_h - u, u - u_h^{**})\vert&\le \vert \lambda \vert \Vert u_h^* - u \Vert _0 \Vert u - u_h^{**} \Vert _0\\ {}&\lesssim \rho (h)^2 h^{2r'}(\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}). \end{aligned}$$

It remains to show that \({\text {hot}}(h) \lesssim \rho (h) (h^{2r + r'} + \rho (h)h^{2r'})\) is of higher order compared to \(h^{2r'}\). Due to the additional \(\rho (h)\) in the upper bound of \({\text {hot}}(h)\), we only have to show that \(2r' \le 2r + r'\). For the low regularity case, i.e. \(s = r = r'\), and the case of full regularity, i.e. \(r = k+1\) and \(r' = k + 2\), this follows immediately. For the case where \(r = k+1\) and \(r' = s\) with \(k+1< s < k+2\), we also have

$$\begin{aligned} 2 r' = 2 s< k + 2 + s < 2(k+1) + s = 2r + r', \end{aligned}$$

from which we conclude the proof of the reliability.

The efficiency estimate follows by the triangle inequality and \(\sigma = \nabla u\). \(\square \)

Using the estimator from above we are now also able to derive an upper bound for \(\lambda _h^*\). To this end let

$$\begin{aligned} \eta _\lambda := \eta ^2 + \Vert \sigma _h - \sigma _h^* \Vert ^2_0 + \vert ( \lambda _h^* u_h^* - \lambda _h u_h, u_h^{**}) \vert . \end{aligned}$$

The last two terms from the estimator \(\eta _\lambda \) are needed to measure the difference between the quantities used in \(\eta \) and the functions used in the definition of \(\lambda _h^*\). Unfortunately the authors do not see how the definition of \(\lambda _h^*\) can be changed such that only \(\sigma _h^*\) and \(u_h^{**}\) are used, which would allow a direct estimate by \(\eta \).

Theorem 3

Let \((\lambda ,u,\sigma )\) be the solution of (2). Let \((\lambda _h,u_h,\sigma _h)\) the the solution of (4) and let \(u_h^{**}\) and \(\sigma _h^*\) be the post-processed solutions. There holds the estimate

$$\begin{aligned} \vert \lambda - \lambda _h^* \vert \lesssim \eta _\lambda + {\text {hot}}(h) + \widetilde{{\text {hot}}}(h), \end{aligned}$$

where \(\widetilde{{\text {hot}}}(h):= \Vert u_h^* - u\Vert _0 \Vert u - u_h^{**}\Vert _0 + \Vert u - u_h^{**}\Vert ^2_0\) with

$$\begin{aligned} \widetilde{{\text {hot}}}(h) \lesssim \rho (h)^2h^{2r'} (\vert u \vert ^2_{r+1} + \vert u \vert ^2_{r'+1}), \end{aligned}$$

and \({\text {hot}}(h)\) are both higher order terms compared to \(\mathcal {O}(\rho (h)h^{r + r'} + h^{2r'})\) as \(h \rightarrow 0\).

Proof

Following (13) we have the equation

$$\begin{aligned} \lambda - \lambda _h^*&= \Vert \sigma - \sigma _h \Vert ^2_0 - \lambda _h^* \Vert u - u_h^* \Vert ^2_0 - 2 ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u). \end{aligned}$$
(17)

Note that the second term on the right side is already of higher order, thus we only consider the remaining terms. The idea is to modify the terms including \(\sigma _h\) such that we can use the results from the previous theorem. By the triangle inequality we have \(\Vert \sigma - \sigma _h \Vert _0 \le \Vert \sigma - \sigma ^*_h \Vert _0 + \Vert \sigma _h^* - \sigma _h \Vert _0\). Since the error \(\Vert \sigma _h^* - \sigma _h \Vert _0\) is computable and \(\Vert \sigma - \sigma ^*_h \Vert _0\) can be bounded by the estimator from the previous theorem, we are left with an estimate for the last term on the right hand side of (17).

In contrast to the proof of Lemma 1 we now add and subtract \(u_h^{**}\) (and not \(u_h^*\)) which gives

$$\begin{aligned} ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u)&= ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u-u_h^{**}) + ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u_h^{**}) \\&= ({\text {div}}\sigma _h + \lambda _h^* u_h^*, u-u_h^{**}) + ( \lambda _h^* u_h^* - \lambda _h u_h, u_h^{**}). \end{aligned}$$

The last term is computable and will be used in the estimator. For the first one we have using that \(u_h^{**} \in H_0^1(\varOmega )\), \(\Vert u \Vert _0=1\) and integration by parts

$$\begin{aligned}&({\text {div}}\sigma _h + \lambda _h^* u_h^*, u - u_h^{**})\\ {}&\quad = ({\text {div}}(\sigma _h - \sigma ), u - u_h^{**}) + ({\text {div}}\sigma + \lambda _h^* u_h^*, u - u_h^{**}),\\&\quad =-(\sigma _h - \sigma , \nabla (u - u_h^{**})) + (-\lambda u + \lambda _h^* u_h^*, u - u_h^{**}),\\ {}&\quad \le \Vert \sigma _h - \sigma \Vert _0^2 + \Vert \nabla (u - u_h^{**})\Vert _0^2 \\ {}&\qquad + \lambda _h^* \Vert u_h^* - u\Vert _0 \Vert u - u_h^{**}\Vert _0 + \vert \lambda - \lambda _h^*\vert \Vert u - u_h^{**}\Vert _0, \\ {}&\quad \le \Vert \sigma _h - \sigma \Vert _0^2 + \Vert \nabla (u - u_h^{**})\Vert _0^2 \\ {}&\qquad + \lambda _h^* \Vert u_h^* - u\Vert _0 \Vert u - u_h^{**}\Vert _0 + \vert \lambda - \lambda _h^*\vert ^2 + \Vert u - u_h^{**}\Vert ^2_0. \end{aligned}$$

The first term can be estimated as before, thus for h small enough we have

$$\begin{aligned} \vert \lambda - \lambda _h^* \vert&\lesssim \Vert \sigma - \sigma _h^* \Vert ^2_0 + \Vert \nabla (u - u_h^{**})\Vert _0^2 + \Vert \sigma _h - \sigma _h^* \Vert ^2_0 \\ {}&\quad + \vert ( \lambda _h^* u_h^* - \lambda _h u_h, u_h^{**})\vert + \widetilde{{\text {hot}}}(h),\\ {}&\lesssim \eta ^2 + \Vert \sigma _h - \sigma _h^* \Vert ^2_0 + \vert ( \lambda _h^* u_h^* - \lambda _h u_h, u_h^{**}) \vert + {\text {hot}}(h) + \widetilde{{\text {hot}}}(h). \end{aligned}$$

To show that \({\text {hot}}(h)\) and \(\widetilde{{\text {hot}}}(h)\) are of higher order compared to \(\mathcal {O}(\rho (h)h^{r + r'} + h^{2r'})\), one follows the same steps as in the proof of Theorem 2.

\(\square \)

6 Numerical examples

In this section we discuss some numerical examples to validate our theoretical findings. All methods were implemented in the Finite Element library Netgen/NGSolve, see www.ngsolve.org and [33].

6.1 Convergence on a unit square

The first example considers the unit square domain \(\varOmega = (0,1)^2\). The eigenfunction and the smallest eigenvalue of (2) is given by \(u = 2 \sin (2\pi x) \sin (2\pi y)\) and \(\lambda = 2 \pi ^2\), respectively. We start with an initial mesh with \(\vert {\mathcal {T}_h}\vert = 32\) elements and use a uniform refinement. Note that for simplicity we used a structured mesh for this example, thus we have \(h \sim (0.5 \vert {\mathcal {T}_h}\vert )^{-1/2}\). In Tables 1 and 2 we present several errors and their convergence rate (given in brackets) for different polynomial orders \(k = 1\) and \(k = 2\). Beside the errors we also plot the high-order term from Theorem 2, and the efficiencies

$$\begin{aligned} {\text {eff}}:= \frac{\eta ^2}{\Vert \nabla u - \nabla u_h^{**} \Vert _0^2 + \Vert \sigma - \sigma ^*_h\Vert ^2_0}, \qquad \text {and} \qquad {\text {eff}}_{\lambda } := \frac{\eta _\lambda }{\vert \lambda - \lambda _h^*\vert }. \end{aligned}$$

Since \(\varOmega \) is convex we have for this example that \(\rho (h) \sim h\), thus we expect the following convergence orders (for simplicity recalled here)

$$\begin{aligned}{} & {} \Vert u - u_h^{**} \Vert _0 \lesssim h^{k+3}, \quad \qquad \Vert \nabla (u - u_h^{**}) \Vert _0 \lesssim h^{k+2}, \\ {}{} & {} \quad \Vert \sigma - \sigma _h^* \Vert _0 \lesssim h^{k+2}, \quad \quad \quad \qquad \vert \lambda - \lambda _h^*\vert \lesssim h^{2(k+2)}. \end{aligned}$$

In accordance to the theory all errors converge with the optimal orders. Further the high-order term \({\text {hot}}(h)\) converges faster than the estimator \(\eta \) as predicted by Theorem 2. Note that this results in an efficiency \({\text {eff}}\) converging to one, i.e. the error estimator is asymptotically exact. Also the estimator for the error of the eigenvalue converges appropriately and shows a good efficiency \({\text {eff}}_\lambda \). The same conclusions can be made for \(k=2\), however, the error of the eigenvalues \(\lambda _h\) and \(\lambda _h^*\) converge so fast that they are too small and rounding errors dominate on the finest meshes. For the same reason we also do not present any numbers for \(\widetilde{{\text {hot}}}(h)\) since this term converges even faster resulting in very small numbers already on coarse meshes.

Table 1 Convergence of several errors for the example on the unit square with \(k=1, 2\)
Full size table
Table 2 Convergence of several errors for the example on the unit square with \(k=1, 2\)
Full size table

6.2 Adaptive refinement on the L-shape

For the second example we choose the L-shape domain \(\varOmega = (-1,1)^2 {\setminus } ([0,1] \times [-1,0])\) where the first eigenvalue reads as \(\lambda \approx 9.63972384402\), see [7]. In this example the corresponding eigenfunction is singular, thus we expect a suboptimal convergence on a uniform refined mesh. To this end we solve the problem using an adaptive mesh refinement. The refinement loop is defined as usual by

$$\begin{aligned} \textrm{SOLVE} \rightarrow \textrm{ESTIMATE} \rightarrow \textrm{MARK} \rightarrow \textrm{REFINE} \rightarrow \textrm{SOLVE} \rightarrow \ldots \end{aligned}$$

and is based on the local contributions \(\eta (K)\) as element-wise refinement indicators. In the marking step we mark an element if \(\eta (K) \ge \frac{1}{4} \max \limits _{K \in {\mathcal {T}_h}} \eta (K)\). The refinement routine then refines all marked elements plus further elements in a closure step to guarantee a regular triangulation. In Fig. 1 we present the error history of the post processed eigenvalue \(\lambda _h^*\), its estimator \(\eta _\lambda \) and the estimator for the eigenfunction error \(\eta \) for polynomial order \(k=2,3\). We can observe an optimal convergence \(\mathcal {O}(N^{-2(k+2)})\), \(\mathcal {O}(N^{-2(k+2)})\) and \(\mathcal {O}(N^{-(k+2)})\), for \(\vert \lambda - \lambda _h^*\vert \), \(\eta _\lambda \) and \(\eta \), respectively, where N denotes the number of degrees of freedom. Further \(\eta _\lambda \) shows a good efficiency.

Fig. 1
figure 1

Convergence history of the L-shape example using an adaptive refinement for \(k=2,3\)

Full size image