A posteriori error estimates of mixed discontinuous Galerkin method for the Stokes eigenvalue problem

TL;DR

This paper develops residual-based a posteriori error estimates for mixed discontinuous Galerkin methods solving the Stokes eigenvalue problem in 2D and 3D, demonstrating their reliability, efficiency, and optimal convergence through numerical experiments.

Contribution

It introduces new a posteriori error estimators for the mixed DG method applied to the Stokes eigenvalue problem, with proven reliability and efficiency.

Findings

Error estimators are reliable and efficient.

Numerical results confirm optimal convergence order.

Method effectively approximates eigenvalues and eigenfunctions.

Abstract

In this paper, for the Stokes eigenvalue problem in $d$-dimensional case $(d=2,3)$, we present an a posteriori error estimate of residual type of the mixed discontinuous Galerkin finite element method using $P_{k}-P_{k-1}$ element $(k\geq 1)$. We give the a posteriori error estimators for approximate eigenpairs, prove their reliability and efficiency for eigenfunctions, and also analyze their reliability for eigenvalues. We implement adaptive calculation, and the numerical results confirm our theoretical predictions and show that our method can achieve the optimal convergence order $O(dof^{-2k/d})$.