Weak Galerkin finite element method for Biot’s consolidation problem

Abstract

In this paper, a fully discrete weak Galerkin (WG) finite element method is proposed to solve Biot’s consolidation problem, where weakly defined gradient and divergence operators over discontinuous functions are introduced. ${\mathbb{P}}_l$–${\mathbb{P}}_l$ $(l\ge 1)$ finite element combination is used for the displacement and pressure approximations in the interior of the elements, and ${\mathbb{P}}_l$–${\mathbb{P}}_{l-1}$ combination for the corresponding trace approximations on the interfaces of the finite element partition. The existence and uniqueness of the discrete linear system at each time step is derived, and error estimates for the approximation of displacement and pressure are obtained. Numerical experiments confirm the theoretical results and show that the proposed WG method is capable of overcoming pressure oscillations.