The Lowest Equal-Order Multiphysics Finite Element Methods for a Biot Model with Secondary Consolidation

Source abstract

In this paper, we propose two lowest equal-order multiphysics finite element methods for a Biot model with secondary consolidation. To reveal the underlying multiphysics processes and design some stable schemes, we reformulate the original fluid-solid coupled problem into a fluid-fluid coupled problem by a new multiphysics approach. As for the case of $c_0>0$ and a finite number $λ,$ we propose a multiphysics finite element method (MFEM) for the lowest equal-order element pairs without adding any stabilized term and prove that the MFEM is stable for any finite Lamé constant λ (whenever small or large enough); as for the case of $c_0>0$ and $λ→+∞$ or the case of $c_0=0,$ it is necessary to choose finite element spaces satisfying inf-sup condition, thus, we design a stabilized multiphysics finite element method (SMFEM) for the lowest equal-order $\mathbf{P}_1−P_1−P_1$ element pairs (not satisfying inf-sup condition) by adding a stabilized term. Moreover, we give the stability analysis and optimal convergence order estimates of MFEM and SMFEM for $λ^∗≥0.$ Also, we show some examples to verify the theoretical results and there are no locking phenomenon for the displacement and pressure oscillation. Thus, the proposed methods give a built-in mechanism to overcome the locking phenomenon of displacement for the case of $λ→+∞$ and pressure oscillation for the case of $c_0=0.$ Finally, we draw conclusions to summarize the main results of this paper.