Multiphysics finite element method for a nonlinear poroelasticity model with finite strain

In this paper, we propose a fully discrete multiphysics finite element method to solve a nonlinear poroelasticity model with finite strain. To reveal the multi-physical processes of deformation and diffusion and propose a stable numerical method, we reformulate the original model into a fluid-fluid coupled problem-a generalized nonlinear Stokes problem of displacement vector field and pseudo pressure field and a diffusion problem of other pseudo pressure field by a new technique. Then, we propose a multiphysics finite element method to approximate the spatial variables and use Newton method to solve the nonlinear problem, prove that the proposed numerical method is stable and has the optimal convergence orders, and give some numerical tests to show that the proposed numerical method is stable and has no oscillation for displacement and pressure. Finally, we draw conclusions to summary the main results of this work.