Fully Discrete Finite Element Method And Multirate Iterative Scheme For A Nonlinear Poroelasticity Model

In the paper,the fully discrete finite element method and multirate iterative scheme are proposed for nonlinear poroelasticity model.Firstly,the paper reformulates the original problem to the new coupling system,that is,a generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure and a diffusion problem of other pseudo pressure fields.Then this paper presents fully discrete finite element method and multirate iterative scheme to solve the problem.Fully discrete finite element method is discretized in space by the multiphysics finite element method and discretized in time by the backward Euler method,the two subproblems are decoupled at each time step of the algorithm.Due to the generalized Stokes problem is nonlinear,we use the Newton iterative method to solve.The stability of the method is proved by theoretical analysis,and the error estimation is made in the paper.Also,numerical examples are given to verify the theoretical results.It has been clearly shown that this method can overcome the numerical oscillation.Next,multirate iterative scheme is proposed in the paper,that is,multiphysics finite element method is still used for spatial discretization,but in terms of time discretization,the generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure field is solved on a large time step,and the diffusion problem of the other pseudo pressure fields is solved on a small time step.Then,the stability of the method is proved by theoretical analysis,and the error estimation is made in the paper.Finally,numerical examples are given to verify the theoretical results and prove that this method doesn't decrease the precision of the solution.It has been clearly shown that the method can reduce the computation time and overcome the numerical oscillation.