A locking free numerical method for the poroelasticity–Forchheimer model

Abstract

In this paper, we consider a locking-free numerical method for the poroelasticity-Forchheimer problem, which exists locking phenomenon when directly using the continuous Galerkin mixed finite element method (MFEM) to be solved due to the influence of special physical parameters. To overcome the locking phenomenon, we reformulate the original problem into a new problem by introducing some new variables. The new problem can be taken as a Stokes-diffusion coupled model, which exists a built-in mechanism to circumvent the locking phenomenon for continuous Galerkin MFEM. Moreover, we prove the existence and uniqueness of weak solution with the help of a priori error estimate, some invariant quantities and the discussion on nonlinear term. After that, we propose a fully discrete numerical scheme to solve the reformulated problem, where the DL-scheme and L-scheme are designed to deal with the nonlinear term, and prove the optimal convergence in both time and space. Finally, numerical tests are presented to verify the theoretical results.