Discontinuous Galerkin approximation for a Stokes-Brinkman-type formulation for the eigenvalue problem in porous media

TL;DR

This paper develops and analyzes a discontinuous Galerkin method for approximating eigenvalues and eigenfunctions of a Stokes-Brinkman problem in porous media, providing stability, convergence, and computational insights.

Contribution

It introduces a novel DG approach for the eigenvalue problem in porous media and establishes stability and convergence results under standard assumptions.

Findings

The method is stable under mesh assumptions.

Convergence and error estimates are derived using non-compact operators.

Computational analysis shows the influence of stabilization parameters on spectrum approximation.

Abstract

We introduce a family of discontinuous Galerkin methods to approximate the eigenvalues and eigenfunctions of a Stokes-Brinkman type of problem based in the interior penalty strategy. Under the standard assumptions on the meshes and a suitable norm, we prove the stability of the discrete scheme. Due to the non-conforming nature of the method, we use the well-known non-compact operators theory to derive convergence and error estimates for the method. We present an exhaustive computational analysis where we compute the spectrum with different stabilization parameters with the aim of study its influence when the spectrum is approximated.