A new DG method for a pure--stress formulation of the Brinkman problem with strong symmetry

TL;DR

This paper introduces a strongly symmetric discontinuous Galerkin method for the Brinkman equations that accurately approximates stress, ensures divergence-free velocities, and provides optimal convergence rates with proven stability.

Contribution

The paper develops a novel DG method for the Brinkman problem that achieves strong stress symmetry, divergence-free velocities, and explicit error estimates, advancing numerical solutions for porous media flow.

Findings

Method is stable with respect to a DG-energy norm.

Achieves optimal convergence rates for stress and post-processed variables.

Numerical examples confirm theoretical error estimates in 2D and 3D.

Abstract

A strongly symmetric stress approximation is proposed for the Brinkman equations with mixed boundary conditions. The resulting formulation solves for the Cauchy stress using a symmetric interior penalty discontinuous Galerkin method. Pressure and velocity are readily post-processed from stress, and a second post-process is shown to produce exactly divergence-free discrete velocities. We demonstrate the stability of the method with respect to a DG-energy norm and obtain error estimates that are explicit with respect to the coefficients of the problem. We derive optimal rates of convergence for the stress and for the post-processed variables. Moreover, under appropriate assumptions on the mesh, we prove optimal $L^2$-error estimates for the stress. Finally, we provide numerical examples in 2D and 3D.