A mixed discontinuous Galerkin method for the Biot equations

Abstract

The Biot model is a coupling problem between the elastic media material with small deformation and porous media fluid flow, its mixed formulation uses the pore pressure, fluid flux, displacement as well as total stress tensor as the primary unknown variables. In this article, combining the discontinuous Galerkin method and the backward Euler method, we propose a mixed discontinuous Galerkin (MDG) method for the mixed Biot equations, it is based on coupling two MDG methods for each subproblem: the MDG method for the porous media fluid flow subproblem and the Hellinger-Reissner formulation of linear elastic subproblem. Then, we prove the well-posedness and the optimal priori error estimates for the MDG method under suitable norms. In particular, the optimal convergence rate of the pressure, displacement and stress tensor in discrete $L^{\infty }\left(L^2\right)$ norm and the fluid flux in discrete $L^2\left(L^2\right)$ norm are proved when the storage coefficient $c_0$ is strictly positive. Similarly, we deduce the optimal convergence rate of all variables in discrete $L^2\left(L^2\right)$ norm when $c_0$ is nonnegative. Finally, some numerical experiments are given to examine the convergence analysis.