Auto-Stabilized Weak Galerkin Finite Element Methods for Biot's consolidation model on Non-Convex Polytopal Meshes

TL;DR

This paper introduces an auto-stabilized weak Galerkin finite element method for Biot's consolidation model that is stable, accurate, and flexible on complex polytopal meshes without traditional stabilizers.

Contribution

The proposed WG scheme achieves stability and optimal convergence on non-convex polytopal meshes without the need for stabilizers, using bubble functions for analysis.

Findings

The method is stable and oscillation-free for pressure approximations.

Optimal-order convergence is theoretically established.

Numerical experiments confirm robustness and efficiency.

Abstract

This paper presents an auto-stabilized weak Galerkin (WG) finite element method for the Biot's consolidation model within the classical displacement-pressure two-field formulation. Unlike traditional WG approaches, the proposed scheme achieves numerical stability without the requirement of traditional stabilizers. Spatial discretization is performed using weak Galerkin finite elements for both displacement and pressure approximations, while a backward Euler scheme is employed for temporal discretization to ensure a fully implicit and stable formulation. We establish the well-posedness of the resulting linear system at each time step and provide a rigorous error analysis, deriving optimal-order convergence. A significant merit of this WG scheme is its flexibility on general shape-regular polytopal meshes, including those with non-convex geometries. By utilizing bubble functions as a primary analytical tool, the method produces stable, oscillation-free pressure approximations without specialized treatment. Numerical experiments are presented to validate the theoretical convergence rates and demonstrate the computational efficiency and robustness of the auto-stabilized formulation.