Well-balanced any order numerical solvers for linear and nonlinear poroelasticity based on combination of BDF and weak Galerkin

This paper develops a family of any order finite element solvers for both linear and nonlinear poroelasticity problems. The backward differentiation formulas (BDFs) are used for temporal discretizations, whereas the weak Galerkin (WG) finite elements on quadrilaterals are utilized for spatial discretizations. For the latter, the discrete weak gradients of shape functions are established in the vector- or matrix- Arbogast–Correa spaces. Such combinations of BDF and WG discretizations produce numerical solutions that have well-balanced spatial and temporal errors in all six quantities, namely, displacement, dilation (divergence of displacement), stress, pressure, velocity and normal flux. For nonlinear poroelasticity problems in which permeability depends on solid dilation or the mean stress, Picard iterations are employed to solve the resulting nonlinear algebraic systems. Rigorous analysis together with numerical experiments demonstrate that these new solvers have optimal-order convergence and are free of locking.