Convergent finite elements on arbitrary meshes, the WG method

TL;DR

This paper demonstrates that a specific weak Galerkin finite element method converges on arbitrary meshes, including those violating the maximum angle condition, unlike standard methods, with numerical tests confirming the theoretical results.

Contribution

The paper introduces a weak Galerkin finite element method that converges on arbitrary meshes, expanding applicability beyond traditional mesh restrictions.

Findings

Convergence of the WG method on arbitrary meshes.

Standard finite elements fail to converge under maximum angle condition violations.

Numerical tests validate the theoretical convergence results.

Abstract

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e., on arbitrary meshes. Numerical tests confirm the theory.