A parameter-robust and decoupled discretization scheme for nonlinear Biot’s model in poroelasticity

Abstract

We present a new robust and decoupled scheme for Biot’s model with displacement-dependent nonlinear permeability. The scheme combines a low-order H(div)-conforming element pair $P_1\oplus {\mathrm{RT}}_0-P_1$ approximation with a first-order semi-explicit time discretization. Two stabilization terms are incorporated into the modified conforming-like formulation to achieve the scheme’s stability. One term penalizes the nonconformity of the H(div)-conforming component, and the other stabilizes pressure oscillations. The constructed scheme sequentially solves the displacement and pressure, which naturally linearizes the nonlinear terms without additional internal iterations, thus enhancing the computational efficiency. We improve the weak coupling condition and then prove optimal convergence in space and time under this condition. Meanwhile, our scheme provides robust solutions that do not suffer from volumetric locking as the Lamé coefficient tends to infinity and spurious pressure oscillations as the constrained specific storage coefficient and the permeability or time step go to zero. Finally, we verify these theoretical results with several numerical examples of nonlinear displacement–permeability relationships.