A coupled multipoint stress -- multipoint flux mixed finite element method for the Biot system of poroelasticity

Abstract

We present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure-displacement system on simplicial and quadrilateral grids. A mixed stress-displacement-rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity-pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi-Douglas-Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, piecewise constant displacement and pressure, and continuous piecewise linear or bilinear rotation. A vertex quadrature rule is applied to the velocity, stress, and stress-rotation bilinear forms, which block-diagonalizes the corresponding matrices and allows for local velocity, stress, and rotation elimination. This leads to a cell-centered positive-definite system for pressure and displacement at each time step. We perform error analysis for the semidiscrete and fully discrete formulations, establishing first order convergence for all variables in their natural norms. The numerical tests confirm the theoretical convergence rates and illustrate the locking-free property of the method.

Locking free staggered DG method for the Biot system of poroelasticity on general polygonal meshes

“…Very recently, a stabilized hybrid mixed finite element method uniformly stable with respect to the physical parameters is proposed in [34]. Alternatively, fully-mixed formulations of the Biot system has also drawn great attention [24,46,26,1]. Typical difficulty encountered in the design of numerical schemes for the Biot system is to avoid locking and to get estimates independent of the storativity coefficient.…”

“…As such, we can exploit local elimination to reduce the global saddle point system to displacement-pressure system. Since this procedure is similar to that proposed in [1], where block diagonal matrices can be generated thanks to the vertex quadrature rule utilized, we will not repeat here. Instead, we will introduce fixed stress splitting scheme to reduce the size of the global system.…”