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 residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient. 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.
Highlights
In fluid mechanics, eigenvalue problems are of great importance because of their role for the stability analysis of fluid flow problems
For example in [22], several stabilized finite element methods for the Stokes eigenvalue problem are considered by Huang et al A finite element analysis of a pseudo stress formulation for the Stokes eigenvalue problem is proposed by Meddahi et al [33]
We introduce an H div-DG finite element method for Stokes eigenvalue problems
Summary
Eigenvalue problems are of great importance because of their role for the stability analysis of fluid flow problems. Some superconvergence results and the related recovery type a posteriori error estimators for the Stokes eigenvalue problem is presented by Liu et al [31] based on a projection method. We have developed an a posteriori error analysis for the Arnold-Winther mixed finite element method of the Stokes eigenvalue problem in [16] using the stress-velocity formulation. We present a residual based a posteriori error analysis for the H div-DG finite element method and derive upper and local lower bounds for the eigenvalue error and the velocity-pressure error which is measured in terms of the mesh-dependent DG norm. The refined a posteriori error analysis which requires more smootheness of the right hand side than L2 and which involves the calculation of third order derivatives for higher order methods is Divergence-conforming discontinuous Galerkin finite.
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