Arnold-Winther mixed finite elements for Stokes eigenvalue problems
Provides improved error control and convergence for Stokes eigenvalue problems in computational mechanics, but the contribution is incremental as it extends existing methods.
The paper develops a posteriori error estimates and a local post-processing technique for Arnold-Winther mixed finite element methods applied to 2D Stokes eigenvalue problems, achieving higher-order convergence for eigenvalues on convex domains and optimal convergence on nonconvex domains with adaptive meshes.
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.