Mixed methods for the velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem

TL;DR

This paper develops and analyzes mixed finite element methods for solving the Stokes eigenvalue problem using velocity, pressure, and pseudostress variables, ensuring convergence and avoiding spurious modes.

Contribution

The paper introduces two finite element schemes for the Stokes eigenvalue problem with pseudostress, providing theoretical convergence analysis and numerical comparisons.

Findings

Methods do not produce spurious eigenvalues.

Convergence and error estimates are established.

Numerical results compare accuracy and robustness of schemes.

Abstract

In two and three dimensional domains, we analyze mixed finite element methods for a velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem. The methods consist in two schemes: the velocity and pressure are approximated with piecewise polynomial and for the pseudostress we consider two classic families of finite elements for $\mathrm{H}(\mathop{\mathrm{div}})$ spaces: the Raviart-Thomas and the Brezzi-Douglas Marini elements. With the aid of the classic spectral theory for compact operators, we prove that our method does not introduce spurious modes. Also, we obtain convergence and error estimates for the proposed methods. In order to assess the performance of the schemes, we report numerical results to compare the accuracy and robustness between both numerical schemes.