A stabilized finite element method for the two-field and three-field Stokes eigenvalue problems

TL;DR

This paper introduces a stabilized finite element method for solving two-field and three-field Stokes eigenvalue problems, utilizing a subgrid scale approach that simplifies residual computation and ensures convergence with validated numerical results.

Contribution

The study presents a novel residual-based stabilization technique using orthogonal projections to maintain linear eigenvalue problems in Stokes formulations.

Findings

Method converges for both two-field and three-field formulations.

Error estimates are provided for eigenvalues and eigenfunctions.

Numerical tests confirm theoretical convergence and accuracy.

Abstract

In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement-pressure) and the three-field (stress-displacement-pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated.