A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot's model. Part I -- Theoretical aspects

TL;DR

This paper introduces a novel nonsymmetric variational approach and a robust, quasi-optimal discretization method for the Biot's model, ensuring stability and accuracy across various material parameters.

Contribution

It develops a new nonsymmetric variational framework and a nonconforming discretization for Biot's model that are robust and quasi-optimal regardless of material parameters.

Findings

The method is proven to be stable uniformly across parameters.

The discretization achieves quasi-optimal error bounds.

The approach enhances robustness in numerical simulations of Biot's model.

Abstract

We consider the system of partial differential equations stemming from the time discretization of the two-field formulation of the Biot's model with the backward Euler scheme. A typical difficulty encountered in the space discretization of this problem is the robustness with respect to various material parameters. We deal with this issue by observing that the problem is uniformly stable, irrespective of all parameters, in a suitable nonsymmetric variational setting. Guided by this result, we design a novel nonconforming discretization, which employs Crouzeix-Raviart and discontinuous elements. We prove that the proposed discretization is quasi-optimal and robust in a parameter-dependent norm and discuss the consequences of this result.