A two-level multiphysics finite element method for a nonlinear poroelasticity model

Abstract

In this paper, we propose a two-level multiphysics finite element method for a nonlinear poroelasticity model. To clearly reveal the multiphysics processes and overcome the “locking phenomenon”, we reformulate the original fluid-solid coupled problem into a fluid-fluid coupled problem–a generalized nonlinear Stokes problem coupled with a diffusion problem. To reduce the cost of computation, we propose two fully discrete coupling and decoupling two-level multiphysics finite element methods and use the backward Euler method for time variable of the reformulated nonlinear poroelasticity model. In computation, the generalized nonlinear poroelasticity (Stokes in Algorithm 2) problem is solved by using the Newton iterative method on a coarse mesh, then a linearized poroelasticity problem is solved on a fine mesh by using the solution of the coarse mesh. Then, we give the stability analysis and error estimates of the proposed methods. Finally, we show some numerical examples to verify the theoretical results–overcoming “locking phenomenon” and reducing the computational cost. • It is the first time to propose a two-level multiphysics finite element method for the nonlinear poroelasticity model. • The optimal order error estimate of the proposed two-level method is given. • To reduce the cost of computation, a poroelasticity problem is solved by using the Newton method on a coarse mesh.