A New Multiphysics Finite Element Method for a Biot Model with Secondary Consolidation (CSIAM Transactions on Applied Mathematics, 2024)
In this paper, we propose a new multiphysics finite element method fora Biot model with secondary consolidation in soil dynamics. To better describe themultiphysical processes underlying in the original model and propose stable numerical methods to overcome “locking phenomenon” of pressure and displacement, wereformulate the swelling clay model with secondary consolidation by a new multiphysics approach, which transforms the fluid-solid coupling problem to a fluid coupled problem. Then, we give the energy law and prior error estimate of the weaksolution. Also, we design a fully discrete time-stepping scheme to use multiphysicsfinite element method with $P_2−P_1−P_1$ element pairs for the space variables and backward Euler method for the time variable, and we derive the discrete energy laws andthe optimal convergence order error estimates. Also, we show some numerical examples to verify the theoretical results and there is no “locking phenomenon”. Finally, wedraw conclusions to summarize the main results of this paper.