Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System

In this paper, we study a mixed discontinuous Galerkin-finite element method (DG-FEM) for solving the semi-stationary compressible Stokes system in a bounded domain. The approximation of continuity equation is obtained by a piecewise constant discontinuous Galerkin method. The discretization of momentum equation is obtained by conforming Bernardi–Raugel finite elements. The convergence of mixed DG-FEM for nonlinear, isentropic stokes problem is rigorously established by compactness arguments and the existence analysis of Lions on the discrete level. Employing the continuous relative energy functional method and a detailed consistency analysis, we derive two error estimates for the numerical solution of the semi-stationary isentropic stokes system. In particular, we establish the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} error estimates for the pressure. All convergence results do not require the boundedness of numerical solutions.