Analysis of a discontinuous Galerkin method for the Biot's consolidation problem

In this paper, a fully discrete stabilized discontinuous Galerkin method is proposed to solve the Biot's consolidation problem. The existence and uniqueness of the finite element solution are obtained. The stability of the fully discrete solution is discussed. The corresponding error estimates for the approximation of displacement and pressure in a mesh dependent norm are obtained. The error estimate for the approxima tion of pressure in L 2 norm is also obtained. Variational principles for the Biot's consolidati on problem and finite element approximat ions based on the Galerkin method were presented in [17,30,31]. With this formulation , certain combinati ons of finite element interpolations (including equal order for both fields) were discarded, due to the incompress ibility constrain t on the displacement field in the initial state. In [22], the authors analyzed mixed Galerkin methods for the Biot's equations. Error estimate s for the semidiscrete and fully discrete approximat ions, with a first order backward scheme in time, were presente d. Asympto tic behavior of semi-discrete finite element approximat ions for the Biot's consolidaton problem was discussed in [23]. All those methods are based on the continuo us finite element spaces, and require that the discrete displacemen t and pressure satisfy the Babus ˘ka-Brezz i stability condition. To avoid the stability condition, some stabilized methods are proposed for the Stokes and Navier-Stokes problems in [7,9,14,32]. The aim of this paper is to provide a discontinuo us Galerkin method for the Biot's consolidation problem. The discontinuo us Galerkin (DG) method was firstly introduced by Reed and Hill in [26] for hyperboli c equations in 1973, but less attention has been paid to it. Since the late 1980s, the Runge–Kutta DG (RKDG) method was developed by Cockburn and Shu in [11–13], and extended to conservation law and system of conservation laws, respectively. The mathematical analysis of its convergence behavior has been conducted. From that time on, more attention has been paid to the DG method. In general, the DG method keeps the good propertie s of the finite element method (FEM) and the finite volume method (FVM). The DG method is locally conservative, stable, and high-order accurate method which can easily handle complex geometries, irregular meshes with hanging nodes, and approximat ions that have polynomials of different degrees in different elements. Because of these advantag es, the DG method has become a very active Unified analysis of the discontinuo us Galerkin …