Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes

This paper investigates lowest-order weak Galerkin finite element methods for solving linear elasticity problems on rectangular and brick meshes. Specifically, constant vectors are used in element interiors and on element interfaces respectively for approximating displacement. For these constant basis functions, their discrete weak gradients are calculated in the local Raviart–Thomas spaces \( RT_{[0]}^d \) (\( d=2 \) or 3), whereas their discrete weak divergences are calculated as elementwise constants. Discrete weak strains are calculated accordingly. Then these quantities are used to develop finite element schemes in both strain-div and grad-div formulations, on both rectangular and brick meshes. A theoretical analysis supported by numerical experiments in both 2-dim and 3-dim reveal that the methods are locking-free and have optimal 1st order convergence in displacement, stress, and dilation (divergence of displacement), when the exact solution has full regularity. The methods can also capture low-regularity solutions very well. Strategies for efficient implementation including Schur complement are presented. Extension to quadrilateral and hexahedral meshes, in both theoretical analysis and numerical experiments, is also examined.