A Simple Weak Galerkin Finite Element Method for the Reissner-Mindlin Plate Model on Non-Convex Polytopal Meshes
Chunmei Wang, Shangyou Zhang
TL;DR
This work introduces a simple weak Galerkin finite element method for the Reissner–Mindlin plate model that works on non-convex polytopal meshes and removes a stabilizer by employing higher-order discrete weak gradients. It defines appropriate WG spaces and projections, derives error equations, and proves existence, uniqueness, and optimal-order error estimates in a discrete $H^1$-norm. Numerical experiments across various mesh families and polynomial degrees validate the theory, demonstrating locking-free behavior for thin plates and occasional super-convergence for displacement. The method offers geometric flexibility and broad applicability to PDEs, while maintaining the sparsity pattern of the stiffness matrix and enabling dimension-agnostic implementation. Overall, the approach provides a robust, flexible, and computationally efficient WG framework for Reissner–Mindlin plate analysis on complex meshes.
Abstract
This paper presents a simple weak Galerkin (WG) finite element method for the Reissner-Mindlin plate model that partially eliminates the need for traditionally employed stabilizers. The proposed approach accommodates general, including non-convex, polytopal meshes, thereby offering greater geometric flexibility. It utilizes bubble functions without imposing the restrictive conditions required by existing stabilizer-free WG methods, which simplifies implementation and broadens applicability to a wide range of partial differential equations (PDEs). Moreover, the method allows for flexible choices of polynomial degrees in the discretization and can be applied in any spatial dimension. We establish optimal-order error estimates for the WG approximation in a discrete H^1 norm, and present numerical experiments that validate the theoretical results.