Analysis of a Mixed DG Method for Stress-Velocity Formulation of the Stokes Equations

摘要

Abstract In this paper we propose and analyze a novel mixed DG scheme for stress-velocity formulation of the Stokes equations with arbitrary polynomial orders on simplicial meshes and the symmetry of stress is strongly imposed. The optimal convergence error estimates are proved for stress and velocity measured in L2documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^2$$end{document} errors. The primary difficulty is to prove L2documentclass12pt{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 of stress, and standard techniques will lead to sub-optimal convergence error estimates. As such, some new ingredients are adopted to recover the optimal convergence rates. The proposed scheme is also extended to solve the Brinkman problem, aiming to get a uniformly robust scheme for both the Stokes and Darcy limits. Finally, several numerical experiments are carried out to verify the performances of the proposed scheme. In particular, the numerical results demonstrate that the proposed scheme is robust with respect to the values of the viscosity.