Discontinuous Galerkin methods for the Stokes equations with nonlinear damping term on general meshes

Abstract

This paper presents the interior penalty discontinuous Galerkin (IPDG) methods to solve the stationary Stokes equations with nonlinear damping term. The IPDG discrete schemes are established on general meshes. The corresponding consistency and stabilization of those schemes are proved. Subsequently, we analyze the existence, boundedness and uniqueness of the discrete solutions. Then the optimal error estimates are derived in the $L^2$-norm and $H^1$-like DG-norm for the velocity variable and $L^2$-norm for the pressure variable, respectively. Finally, some numerical experiments in two dimensional are reported to demonstrate the theoretical results and show the robustness of our discrete schemes.