Analysis of new mixed finite element method for a Barenblatt-Biot poroelastic model

Abstract

In this work, we study the locking-free numerical method for a Barenblatt-Biot poroelastic model. When solving by the continuous Galerkin mixed finite element method, the model exists two kind of locking phenomena for special physical parameters. To overcome these locking phenomena, we introduce new variables to reformulate the original problem into a new problem, which exists a built-in mechanism to keep the continuous Galerkin mixed finite element method stable. It can be regarded as a generalized Stokes problem for two given β and η and two diffusion problems for a given δ. The generalized Stokes problem can be adopted by some stable solver and the diffusion problems can be solved by continuous Galerkin finite element. Moreover, the existence and uniqueness of weak solution is proved by using the standard Galerkin method and combining with priori estimates and some invariant quantities. After that, we design fully discrete time-stepping schemes to use mixed finite element method with $P_2-P_1-P_1-P_1$ element pairs for space variables and backward Euler method for time variable, and analyze the coupled and decoupled time stepping methods based on the proposed scheme. The optimal convergence order is obtained in both space and time. Finally, some numerical examples are presented to show the optimal convergence rates about variables and the robustness of proposed method with respect to ν, and to verify that there is no locking phenomenon.