Convergent finite elements on arbitrary meshes, the WG method (2407.19382v1)

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.

• The paper introduces a novel WG finite element method that achieves two order superconvergence on arbitrary meshes, overcoming convergence issues due to maximum angle violations.
• The methodology reduces complexity by omitting inter-element integrals and penalty terms, resulting in solutions nearly equivalent to L²-projections even on degenerate meshes.
• Numerical experiments on 2D and 3D Poisson's equations validate the WG method’s robust convergence, offering potential for advanced structural and fluid dynamic applications.

Convergence of the WG Finite Element Method on Arbitrary Meshes

The research by Ran Zhang and Shangyou Zhang presents a significant contribution to the understanding of the convergence properties of finite element methods (FEM) on arbitrary and non-standard meshes, addressing longstanding issues associated with the violation of the maximum angle condition. The paper revisits the limitations of traditional finite element techniques—standard conforming, nonconforming, and discontinuous Galerkin (DG) finite elements—particularly when these methods do not converge to the true solution as mesh size approaches zero in dimensions higher than one.

Finite Element Approaches and Limitations

The paper begins by discussing the fundamental limitations seen in finite element solutions on triangular and tetrahedral meshes that violate the maximum angle condition. Classic FEM theories have shown that convergence fails on such meshes, a phenomenon highlighted by the Baska and Aziz example. This limitation is akin to the discrepancy observed when approximating curved geometries, such as cylinders, with linear elements. The approximation could result in increased errors due to angular misalignment with the tangent planes, impacting the convergence towards the true geometrical or solution properties.

Weak Galerkin Finite Element Method (WG FEM)

The core of this research builds upon the weak Galerkin (WG) finite element methods, which are developed to address these issues by producing superconvergent solutions even on meshes violating traditional conditions. These methods advantageously omit inter-element integrals and penalty terms, making the WG solutions almost equal to the L²-projection, unlike the classic techniques which introduce inconsistencies through inter-element discontinuities.

The authors propose a novel WG FEM strategy that ensures two order superconvergence on essentially any triangulation or tetrahedral grid, circumventing the impediments presented by poor mesh shapes. The work crucially explores ensuring solution viability without the need for maximum angle adherence—a process supported by rigorous mathematical frameworks, including the successful implementation of a gradient-preserving Zhang-Zhang transformation.

Numerical Experiments and Validation

To substantiate their theoretical claims, Zhang and Zhang provide extensive numerical analyses. These involve solving 2D and 3D Poisson’s equations on both quasi-uniform and degenerate meshes. The comparative results between standard P1 conforming and P1 weak Galerkin finite element methods reveal the superiority of the latter in terms of convergence rates on both uniform and degenerate meshes. Notably, on degenerate meshes, the P1 WG FEM displays an order of two superconvergence in both L² and energy norms, attesting to its robustness and effectiveness over traditional methods which falter in such testing conditions.

Implications and Future Directions

The findings lay the groundwork for future finite element methodologies that can operate without stringent mesh quality prerequisites, potentially offering computational efficiencies by minimizing the need for refined, condition-compliant meshes. The WG method stands as a pivotal development for applications in structural analysis, fluid dynamics, and other domains requiring complex geometric mesh configurations.

This work invites further exploration into the application of WG techniques to other complex equations and in facilitating adaptive mesh refinement strategies while persisting within computational constraints. By advancing the understanding of mesh robustness and convergence in FEM, it paves the way for the development of next-generation computational models in engineering and scientific computing.