A Stokes-Brinkman-type formulation for the eigenvalue problem in porous media

TL;DR

This paper develops a finite element method to accurately approximate the natural frequencies of flow systems in porous media governed by Stokes-Brinkman equations, with proven convergence and verified through numerical tests.

Contribution

It introduces a novel finite element approach for eigenvalue problems in porous media and provides rigorous convergence analysis and error estimates.

Findings

Convergence of the finite element method is proven.

Error estimates for eigenvalues and eigenfunctions are established.

Numerical tests confirm theoretical results.

Abstract

In this paper we introduce and analyze, for two and three dimensions, a finite element method to approximate the natural frequencies of a flow system governed by the Stokes-Brinkman equations. Here, the fluid presents the capability of being within a porous media. Taking advantage of the Stokes regularity results for the solution, and considering inf-sup stable families of finite elements, we prove convergence together with a priori and a posteriori error estimates for the eigenvalues and eigenfunctions with the aid of the compact operators theory. We report a series of numerical tests in order to confirm the developed theory.