Arnold-Winther mixed finite elements for Stokes eigenvalue problems

TL;DR

This paper analyzes the Arnold-Winther mixed finite element method for 2D Stokes eigenvalue problems, introducing a post-processing technique that enhances approximation accuracy and provides reliable error estimation.

Contribution

It presents a novel local post-processing approach for improved eigenvalue and eigenfunction approximation in the Arnold-Winther method, along with an a posteriori error estimator.

Findings

Higher order convergence of post-processed eigenvalues on convex domains

Numerically optimal convergence on nonconvex domains with adaptive meshes

Empirical efficiency of the error estimator

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.