Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model

¹ GIMNAP-Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
² Department of Applied Mathematics, University of Waterloo, Waterloo, Canada
³ Centro de Investigación en Ingeniería Matemática, CI 2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

^* Corresponding author: pzuniga@ci2ma.udec.cl

Received: 2 December 2019
Accepted: 29 June 2020

Abstract

We present an a priori and a posteriori error analysis of a conforming finite element method for a four-field formulation of the steady-state Biot’s consolidation model. For the a priori error analysis we provide suitable hypotheses on the corresponding finite dimensional subspaces ensuring that the associated Galerkin scheme is well-posed. We show that a suitable choice of subspaces is given by the Raviart–Thomas elements of order k ≥ 0 for the fluid flux, discontinuous polynomials of degree k for the fluid pressure, and any stable pair of Stokes elements for the solid displacements and total pressure. Next, we develop a reliable and efficient residual-based a posteriori error estimator. Both the reliability and efficiency estimates are shown to be independent of the modulus of dilatation. Numerical examples in 2D and 3D verify our analysis and illustrate the performance of the proposed a posteriori error indicator.