A Generalized Weak Galerkin Method for Linear Elasticity with Nonpolynomial Approximations

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.