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In this paper, we study the numerical algorithm for a nonlinearporoelasticity model with nonlinear stress-strain relations. By using variablesubstitution, the original problem can be reformulated to a new coupledfluid-fluid system, that is, a generalized nonlinear Stokes problem ofdisplacement vector field related to pseudo pressure and a diffusion problem ofother pseudo pressure fields. A new technique is used to get the existence anduniqueness of the solution of the reformulated model and a fully discretenonlinear finite element method is proposed to solve the model numerically. Themultiphysics finite element is used to get the discretization of the spacevariable and the backward Euler method is taken as the time-stepping method inthe fully discrete case. Stability analysis and the error estimation are givenfor the fully discrete case and numerical test are taken to verify thetheoretical results.

Optimal error estimates of multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relations