Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity

¹ Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram 695547, Kerala, India
² GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile
³ Centro de Investigación en Ingeniería Matemática (CI 2MA), Universidad de Concepción, Concepción, Chile
⁴ Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
⁵ Universidad Adventista de Chile, Casilla 7-D, Chillán, Chile
⁶ Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany

^* Corresponding author: ruizbaier@maths.ox.ac.uk

Received: 1 November 2018
Accepted: 29 August 2019

Abstract

We introduce a numerical method for the approximation of linear poroelasticity equations, representing the interaction between the non-viscous filtration flow of a fluid and the linear mechanical response of a porous medium. In the proposed formulation, the primary variables in the system are the solid displacement, the fluid pressure, the fluid flux, and the total pressure. A discontinuous finite volume method is designed for the approximation of solid displacement using a dual mesh, whereas a mixed approach is employed to approximate fluid flux and the two pressures. We focus on the stationary case and the resulting discrete problem exhibits a double saddle-point structure. Its solvability and stability are established in terms of bounds (and of norms) that do not depend on the modulus of dilation of the solid. We derive optimal error estimates in suitable norms, for all field variables; and we exemplify the convergence and locking-free properties of this scheme through a series of numerical tests.