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Arnold--Winther Mixed Finite Elements for Stokes Eigenvalue Problems

2018; Society for Industrial and Applied Mathematics; Volume: 40; Issue: 5 Linguagem: Inglês

10.1137/17m1162032

1095-7197

Joscha Gedicke, Arbaz Khan,

Numerical methods in engineering

This paper is devoted to studying 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 postprocessing. With the help of the local postprocessing, 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 postprocessed eigenvalues for convex domains with smooth eigenfunctions. On adaptively refined meshes we obtain numerically optimal higher orders of convergence of the postprocessed eigenvalues even on nonconvex domains.