A Mixed finite element method for the velocity-pseudostress formulation of the Oseen eigenvalue problem

TL;DR

This paper develops a mixed finite element method for the Oseen eigenvalue problem using pseudostress, eliminating pressure and providing convergence analysis and numerical validation.

Contribution

It introduces a novel mixed formulation with pseudostress, enabling pressure elimination and rigorous numerical analysis for the Oseen eigenvalue problem.

Findings

Convergence and error estimates are established.

Numerical tests confirm theoretical predictions.

Method effectively approximates eigenvalues in 2D and 3D.

Abstract

In this paper, we introduce and analyze a mixed formulation for the Oseen eigenvalue problem by introducing the pseudostress tensor as a new unknown, allowing us to eliminate the fluid pressure. The well-posedness of the solution operator is established using a fixed-point argument. For the numerical analysis, we use the tensorial versions of Raviart-Thomas and Brezzi-Douglas-Marini elements to approximate the pseudostress, and piecewise polynomials for the velocity. Convergence and a priori error estimates are derived based on compact operator theory. We present a series of numerical tests in two and three dimensions to confirm the theoretical findings.