A Generalized Weak Galerkin Method for Linear Elasticity with Nonpolynomial Approximations

TL;DR

This paper introduces a flexible generalized weak Galerkin finite element method for linear elasticity that supports nonpolynomial approximations on complex meshes, offering robustness, efficiency, and convergence comparable to traditional methods.

Contribution

It develops a novel gWG framework that allows nonpolynomial approximation spaces and arbitrary meshes, improving flexibility and computational efficiency in linear elasticity simulations.

Findings

The method is locking-free and mesh-robust.

Activation-based spaces achieve polynomial-like convergence.

Numerical results confirm theoretical error estimates.

Abstract

This paper presents a generalized weak Galerkin (gWG) finite element method for linear elasticity problems on general polygonal and polyhedral meshes. The proposed framework is flexible and efficient, allowing for the use of nonpolynomial approximating functions. The generalized weak differential operators are defined as an element-level correction of the classical differential operators accounting for boundary discontinuities. This construction reduces computational cost and provides greater flexibility than standard weak Galerkin formulations. The gWG framework naturally accommodates arbitrary finite-dimensional approximation spaces, including nonpolynomial activation-based spaces with randomly selected parameters. Error equations and error estimates are established for the proposed method. Numerical experiments demonstrate that the method is locking-free, robust with respect to mesh geometry, and effective on general polygonal and polyhedral partitions. In particular, activation-based interior approximation spaces exhibit convergence behavior comparable to that of classical polynomial spaces.