Arnold--Winther Mixed Finite Elements for Stokes Eigenvalue Problems

Abstract

This paper is devoted to study the Arnold-Winther mixed finite element method for two dimensional Stokes eigenvalue problems using the stress-velocity formulation. A priori error estimates for the eigenvalue and eigenfunction errors are presented. To improve the approximation for both eigenvalues and eigenfunctions, we propose a local post-processing. With the help of the local post-processing, we derive a reliable a posteriori error estimator which is shown to be empirically efficient. We confirm numerically the proven higher order convergence of the post-processed eigenvalues for convex domains with smooth eigenfunctions. On adaptively refined meshes we obtain numerically optimal higher orders of convergence of the post-processed eigenvalues even on nonconvex domains.

Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

“…Proof. From the continuous inf-sup condition (12) we deduce that there exists a w ∈ H 1 0 (Ω) such that…”

“…In [18], Huang presents a posteriori lower and upper eigenvalue bounds for the Stokes eigenvalue problem for two stabilized finite element methods based on the lowest equal-order finite element pair. Recently, we have developed an a posteriori error analysis for the Arnold-Winther mixed finite element method of the Stokes eigenvalue problem in [12] using the stress-velocity formulation.…”

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

“…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]. Moreover, it provides in a natural way a postprocessed solution for the eigenvalues which converges faster than the original one.…”

“…The finite element discretization is based on the Arnold-Winther (AW) space for the approximation of the stresses which can be seen as a generalization of the BDM space. For this reason, the results of [12] prove very useful in this context. More precisely, the results translate almost verbatim to the approximation of the mixed Laplacian based on BDM finite elements.…”

“…The eigenvalue problem associated with the Stokes problem was considered in [12] by using a stressvelocity formulation related to linear elasticity. The finite element discretization is based on the Arnold-Winther (AW) space for the approximation of the stresses which can be seen as a generalization of the BDM space.…”