Parameter-free inexact block Schur complement preconditioning for linear poroelasticity under a hybrid Bernardi-Raugel and weak Galerkin finite element discretization

TL;DR

This paper develops a parameter-free inexact block Schur complement preconditioner for linear poroelasticity problems discretized with hybrid finite elements, ensuring mesh-independent convergence even in nearly incompressible regimes.

Contribution

It introduces a reformulation as a three-field problem with regularization, enabling robust iterative solutions for challenging boundary conditions and material parameters.

Findings

Preconditioners achieve convergence independent of mesh size and locking parameter.

Regularization improves solver robustness under pure Dirichlet boundary conditions.

Numerical experiments confirm effectiveness in 2D, 3D, and spinal cord simulations.

Abstract

This work investigates inexact block Schur complement preconditioning for linear poroelasticity problems discretized using a hybrid approach: Bernardi-Raugel elements for solid displacement and lowest-order weak Galerkin elements for fluid pressure. When pure Dirichlet boundary conditions are applied to the displacement, the leading block of the resulting algebraic system becomes almost singular in the nearly incompressible (locking) regime, hindering efficient iterative solution. To overcome this, the system is reformulated as a three-field problem with an inherent regularization that maintains the original solution while ensuring nonsingularity. Analysis shows that both the minimal residual (MINRES) and generalized minimal residual (GMRES) methods, when preconditioned with inexact block diagonal and triangular Schur complement preconditioners, achieve convergence independent of mesh size and the locking parameter for the regularized system. Similar theoretical results are established for the situation with displacement subject to mixed boundary conditions, even without regularization. Numerical experiments in 2D and 3D confirm the benefits of regularization under pure Dirichlet conditions and the robustness of the preconditioners with respect to mesh size and the locking parameter in both boundary condition scenarios. Finally, a spinal cord simulation with discontinuous material parameters further illustrates the effectiveness and robustness of the proposed iterative solvers.