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A Generalized Weak Galerkin Method for Linear Elasticity with Nonpolynomial Approximations

Junping Wang, Yue Wang

TL;DR

This work develops a generalized weak Galerkin (gWG) finite element method for linear elasticity on general polygonal and polyhedral meshes, enabling nonpolynomial approximation spaces. The method defines generalized weak gradient $\nabla_{g}$ and generalized weak divergence $\nabla_{g}\cdot$ through local boundary-correction problems, reducing computational cost and increasing space flexibility compared to standard WG methods. The authors establish well-posedness, derive error equations, and prove a priori error estimates, showing that the discretization error is governed by projection defects and boundary stabilization terms, with stability independent of mesh geometry. Numerical tests on triangular and rectangular meshes demonstrate locking-free performance and show that activation-based spaces with randomly chosen parameters can achieve convergence rates comparable to polynomial spaces, underscoring the method's robustness and versatility for practical elasticity simulations.

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.

A Generalized Weak Galerkin Method for Linear Elasticity with Nonpolynomial Approximations

TL;DR

This work develops a generalized weak Galerkin (gWG) finite element method for linear elasticity on general polygonal and polyhedral meshes, enabling nonpolynomial approximation spaces. The method defines generalized weak gradient and generalized weak divergence through local boundary-correction problems, reducing computational cost and increasing space flexibility compared to standard WG methods. The authors establish well-posedness, derive error equations, and prove a priori error estimates, showing that the discretization error is governed by projection defects and boundary stabilization terms, with stability independent of mesh geometry. Numerical tests on triangular and rectangular meshes demonstrate locking-free performance and show that activation-based spaces with randomly chosen parameters can achieve convergence rates comparable to polynomial spaces, underscoring the method's robustness and versatility for practical elasticity simulations.

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.
Paper Structure (5 sections, 6 theorems, 59 equations, 1 figure, 6 tables, 1 algorithm)

This paper contains 5 sections, 6 theorems, 59 equations, 1 figure, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Numerical Scheme
  3. Error Estimates
  4. Numerical Examples
  5. Conclusions

Key Result

Lemma 3

wang_locking-free_2016 (The second Korn's inequality) Assume $\Omega$ is connected, bounded, and has a Lipschitz boundary. Let $\Gamma_{1} \subset \partial \Omega$ be a nontrivial portion of the boundary $\partial \Omega$ with dimension $d-1$. For any $1 \leq p\leqslant p^*$, there exists a constant where $p^*=\infty$ if $d=2$ and $p^*=4$ if $d=3$.

Figures (1)

  • Figure 1: Two neighboring elements $T$ and $T^*$ sharing an interface $e=\partial T\cap\partial T^\in\mathcal{E}_h$, where $\bm{v}_b$ is single-valued.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Lemma 3
  • Theorem 4
  • Proof 1
  • Lemma 1
  • Proof 2
  • Lemma 2
  • Proof 3
  • Lemma 3
  • ...and 5 more