An inherent regularization approach to parameter-free preconditioning for nearly incompressible linear poroelasticity and elasticity

An inherent regularization strategy and block Schur complement preconditioning are studied for linear poroelasticity problems discretized using the lowest-order weak Galerkin FEM in space and the implicit Euler scheme in time. At each time step, the resulting saddle point system becomes nearly singular in the locking regime, where the solid is nearly incompressible. This near-singularity stems from the leading block, which corresponds to a linear elasticity system. To enable efficient iterative solution, this nearly singular system is first reformulated as a saddle point problem and then regularized by adding a term to the (2,2) block. This regularization preserves the solution while ensuring the non-singularity of the new system. As a result, block Schur complement preconditioning becomes effective. It is shown that the preconditioned MINRES and GMRES converge essentially independent of the mesh size and the locking parameter. Both two- and three-field formulations are considered for the iterative solution of the linear poroelasticity. The efficient solution of the two-field formulation builds upon the effective iterative solution of linear elasticity. For this case, MINRES and GMRES achieve parameter-free convergence when used with block Schur complement preconditioning, where the inverse of the leading block leverages efficient solvers for linear elasticity. The poroelasticity problem can also be reformulated as a three-field system by introducing a numerical pressure variable into the linear elasticity part. The inherent regularization strategy extends naturally to this formulation, and preconditioned MINRES and GMRES also show parameter-free convergence for the regularized system. Numerical experiments in both two and three dimensions confirm the effectiveness of the regularization strategy and the robustness of the block preconditioners.