Error analysis of discontinuous Galerkin methods for the Stokes problem under minimum regularity

Abstract

In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs to $[{H}_{0}^{1}(\Omega )]^{d}$ and the pressure $p\in {L}_{0}^{2}(\Omega )$. First, we analyse standard DG methods assuming that the right-hand side f belongs to $[H^{-1}(\Omega )\cap L^{1}(\Omega )]^{d}$. A DG method that is well defined for f belonging to $[H^{-1}(\Omega )]^{d}$ is then investigated. The methods under study include stabilized DG methods using equal-order spaces and inf–sup stable ones where the pressure space is one polynomial degree less than the velocity space.