Virtual element methods for the three-field formulation of time-dependent linear poroelasticity

TL;DR

This paper develops a virtual element method for the three-field formulation of time-dependent linear poroelasticity, providing optimal error estimates and numerical validation for the approach.

Contribution

It introduces a novel virtual element discretisation for transient poroelasticity, extending previous methods to include time dependence and establishing rigorous error bounds.

Findings

Optimal a priori error estimates are proven.

Numerical tests confirm the method's accuracy.

The approach is locking-free and suitable for complex domains.

Abstract

A virtual element discretisation for the numerical approximation of the three-field formulation of linear poroelasticity introduced in [R. Oyarz\'ua and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity, SIAM J. Numer. Anal. 54 (2016) 2951--2973] is proposed. The treatment is extended to include also the transient case. Appropriate poroelasticity projector operators are introduced and they assist in deriving energy bounds for the time-dependent discrete problem. Under standard assumptions on the computational domain, optimal a priori error estimates are established. Furthermore, the accuracy of the method is verified numerically through a set of computational tests.