Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error estimation for finite element method on meshes that contain thin elements

Kenta Kobayashi, Takuya Tsuchiya

TL;DR

This work addresses error estimation for finite element solutions of the Poisson equation on meshes containing thin or degenerate elements that violate the usual shape-regularity conditions. By introducing a virtual covering approach, the authors show that the standard $H^1$-error estimate can be preserved if every bad element is encompassed by a simplex satisfying a minimum angle condition, formalized through a general Assumption A2 and a 2D simplification A5. They derive explicit bounds $|u - Π^*u|_{H^1(Ω)} ≤ E h |u|_{H^2(Ω)}$ with a computable constant $E$ depending on mesh-geometry constants, cluster overlap $M$, and local cover properties, and validate the theory with a Poisson problem demonstrating $O(h)$ convergence even with many bad elements. The results extend prior approaches (Kučera; Duprez–Lleras–Lozinski) to broader configurations and dimensions, offering practical guidelines for handling irregular meshes in finite element computations. Potential extensions to higher-order elements are discussed as future work.

Abstract

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements elements that violate the shape regularity or maximum angle condition are covered virtually by simplices that satisfy the minimum angle condition. A numerical experiment illustrates the theoretical result.

Error estimation for finite element method on meshes that contain thin elements

TL;DR

This work addresses error estimation for finite element solutions of the Poisson equation on meshes containing thin or degenerate elements that violate the usual shape-regularity conditions. By introducing a virtual covering approach, the authors show that the standard -error estimate can be preserved if every bad element is encompassed by a simplex satisfying a minimum angle condition, formalized through a general Assumption A2 and a 2D simplification A5. They derive explicit bounds with a computable constant depending on mesh-geometry constants, cluster overlap , and local cover properties, and validate the theory with a Poisson problem demonstrating convergence even with many bad elements. The results extend prior approaches (Kučera; Duprez–Lleras–Lozinski) to broader configurations and dimensions, offering practical guidelines for handling irregular meshes in finite element computations. Potential extensions to higher-order elements are discussed as future work.

Abstract

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements elements that violate the shape regularity or maximum angle condition are covered virtually by simplices that satisfy the minimum angle condition. A numerical experiment illustrates the theoretical result.
Paper Structure (9 sections, 2 theorems, 37 equations, 14 figures, 1 table)

This paper contains 9 sections, 2 theorems, 37 equations, 14 figures, 1 table.

Table of Contents

  1. Céa's lemma and interpolations
  2. Conditions and constants related to meshes
  3. Sobolev embedding theorem
  4. Example of error estimation on a bad mesh.
  5. Assumptions and the main theorem
  6. Simpler formulation for the case $n=2$.
  7. Comparison of the results with the results of prior works
  8. Numerical experiment
  9. Concluding remarks and future works

Key Result

Theorem 6

Suppose that the mesh $\mathcal{T}_h$ satisfies Assumption A2. The interpolation $\Pi^* u$ is then well defined for $u \in H^2(\Omega)$, and we have

Figures (14)

  • Figure 1: Mesh $\mathcal{T}_h^1$ (left) and its refinement $\mathcal{T}_h^2$ (right).
  • Figure 2: Examples of $V({T_1},{T_2})$.
  • Figure 3: Examples of $V_{\partial\Omega}({T_1})$.
  • Figure 4: Two situations regarding Assumption 3 (3).
  • Figure 5: A situation that makes $\Pi^* u$ be not well-defined.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 4
  • Theorem 6
  • Corollary 8