Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems

TL;DR

This paper introduces a divergence-conforming discontinuous Galerkin finite element method for solving Stokes eigenvalue problems, providing error estimates, a reliable a posteriori error estimator, and numerical verification of convergence and estimator efficiency.

Contribution

It develops a novel divergence-conforming DG method for Stokes eigenproblems with proven error bounds and an effective residual-based a posteriori error estimator.

Findings

Error estimates for eigenvalues and eigenfunctions are established.

The a posteriori error estimator is proven reliable and efficient.

Numerical examples confirm theoretical convergence and estimator performance.

Abstract

In this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a robust residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient in a mesh-dependent velocity-pressure norm. We finally present some numerical examples that verify the a priori convergence rates and the reliability and efficiency of the residual based a posteriori error estimator.