A new multiphysics finite element method for a Biot model with secondary consolidation

TL;DR

This paper introduces a novel multiphysics finite element method for the Biot model with secondary consolidation, reformulating it into coupled Stokes and diffusion problems, with proven energy laws and optimal error estimates.

Contribution

It presents a new multiphysics reformulation of the Biot model and develops a fully discrete scheme with proven stability and convergence properties.

Findings

Energy law and error estimates established

Optimal convergence order demonstrated

Numerical examples verify theoretical results

Abstract

In this paper, we propose a new multiphysics finite element method for a Biot model with secondary consolidation in soil dynamics. To better describe the processes of deformation and diffusion underlying in the original model, we reformulate Biot model by a new multiphysics approach, which transforms the fluid-solid coupled problem to a fluid coupled problem--a generalized Stokes problem and a diffusion problem. Then, we give the energy law and prior error estimate of the weak solution. And we design a fully discrete time-stepping scheme to use mixed finite element method for $P_2-P_1-P_1$ element pairs to approximate the space variables and backward Euler method for the time variable, and we prove the discrete energy laws and the optimal convergence order error estimates. Also, we show some numerical examples to verify the theoretical results. Finally, we draw a conclusion to summarize the main results of this paper.