Analysis of Multiphysics Finite Element Method for a Nonlinear Poroelasticity Model

Abstract

In this paper, we propose and analyze a multiphysics finite element method for a nonlinear poroelasticity model. To more effectively capture the deformation and diffusion processes, we reformulate the original nonlinear fluid-solid coupling problem into a fluid-fluid coupling problem using a multiphysics approach. We then establish the growth, coercivity, and monotonicity properties of the nonlinear stress-strain relation, derive energy estimates, and use Schaefer’s fixed point theorem to prove the existence and uniqueness of the weak solution. Furthermore, we design a fully discrete time-stepping scheme – multiphysics finite element method with $P_2−P_1−P_1$ elements for spatial variables and the backward Euler method for time variable. To handle nonlinearity, we employ the Newton iterative method, establish discrete energy laws and give the optimal convergence order error estimates. Finally, we show some numerical examples to verify the rationality of theoretical analysis and the proposed method has no “locking phenomenon”.