A posteriori error analysis for the mixed Laplace eigenvalue problem: investigations for the BDM‐element

Abstract

A posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem can be derived using a reconstruction in the standard H 1 0 -conforming space for the primal variable of the mixed problem. In the case of Raviart-Thomas finite elements of arbitrary polynomial degree the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. This paper shows that the extension for the BDM-element is not straightforward.Let Ω ⊂ R 2 be a polygon. Based on the hypercircle identity dating back at least to [1] and [2], guaranteed, easily, fully, and locally computable upper bounds of the error measured in the energy norm have been derived using a flux reconstruction of the primal variable in Σ := H(div; Ω), see e.g. [3,4]. A unified framework for a posteriori error estimation based on stress reconstruction for the Stokes system is carried out in [5] and an extension to the linear elasticity problem is presented in [6]. For the mixed scheme, the hypercircle identity can be used to obtain a postprocessed better approximation to the scalar variable of the source problem, see [7] and [8]. In order to obtain a continuous reconstruction of the gradient, this postprocessed solution can be averaged (see [9][10][11]). In [12] this approach was extended to the Laplace eigenvalue problem using Raviart-Thomas element and in this paper, the extension to the Brezzi-Douglas-Marini element is investigated. We consider a solution (λ, u, σ) ∈ R × L 2 (Ω)\{0} × Σ of the mixed formulation of the Laplace eigenvalue problem, that satisfiesGiven a triangulation T h of Ω, P k (T h ) denotes the space of discontinuous piecewise polynomials of degree k ≥ 0, while the Raviart-Thomas and Brezzi-Douglas-Marini space are given by Σ 0 h := Σ RT h = {τ ∈ Σ : τ = τ 1 + xτ 2 with τ 1 ∈ [P k (T h )] 2 , τ 2 ∈ P k (T h )} and Σ 1 h := Σ BDM h = [P k+1 (T h )] d ∩ Σ .

Some Remarks on the a Posteriori Error Analysis of the Mixed Laplace Eigenvalue Problem

“…It could be extended to the BDM scheme as well, but in this case the term hot(t) is not of higher order. Numerical experiments in [2] show that indeed it is of the same order as the estimator.…”

“…Fleurianne Bertrand, Daniele Boffi, Joscha Gedicke, and Arbaz Khan Corollary 1 (reliability for the error in the eigenvalue). Under the same hypotheses of the previous theorem, for h small enough it holds |λ − λ h | ≤η2 RT + hot(h) withη…”

“…More precisely, the results translate almost verbatim to the approximation of the mixed Laplacian based on BDM finite elements. It is then quite natural to compare the obtained results with the discussion in [3] and [2]. The a posteriori estimator of [12] is based on the first step only: a local postprocessing procedure in the spirit of [14].…”

“…In [2] it was shown that the same result doesn't apply to the case when Brezzi-Douglas-Marini (BDM) elements are used for the approximation of the gradients. Actually, the nonlinear term has been shown to converge at the same order as the other quantities of interest.…”

A note on asymptotically exact a posteriori error estimates for mixed Laplace eigenvalue problems

“…In [4] (using ideas from [20]) the same authors (and collaborators) were able to achieve an optimal approximation by using the Brezzi-Douglas-Marini (BDM) Finite element instead. However, this was only possible by paying the price of unknown constants in the a posteriori estimates since the additional term in (1) is not of higher order any more as was also observed in [5].…”