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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-error estimates for the stress. Finally, we provide numerical examples in 2D and 3D.</p>