Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method

Abstract

In this paper, we consider a stabilized finite element method for the approximation of the Stokes eigenvalue problem on triangular domains. The method depends on orthogonal subscales that has proved to be an appropriate means for approximating eigenvalue problems in the framework of residual based approaches. We consider several isosceles triangular domains with various apex angles to investigate the characteristics of the eigensolutions in regards to the variation of the domain properties. This study presents the first finite element approximation to the solutions of the Stokes eigenvalue problem on triangular domains, to the best of our knowledge. We provide plots of several velocity and pressure fields corresponding to the fundamental eigenmodes to analyze the flow characteristics in detail. Furthermore, we consider the problem on triangular domains including a crack, and investigate the influence of the length of the slit on the fundamental mode to some extent. The results reveal the correlation between the domain properties and the eigenpairs, and the fact that there are various critical lengths of the slit where the eigenspace is notably affected.