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Optimal $L^\infty$-error estimate for isoparametric finite element method in a smooth domain

Takahito Kashiwabara

TL;DR

This work analyzes the isoparametric finite element method for the Dirichlet Poisson problem on a smooth domain, proving an optimal $L^{\infty}$-error rate of $O(h^{k+1})$ for quadratic or higher-order elements on a curved boundary approximation $\Omega_h$. The authors extend prior Neumann-boundary techniques by leveraging a Dirichlet Green function framework and a dyadic, localized error analysis to control the regularized Green function $\nabla(\tilde{g}-g_h)$ in weighted norms, avoiding reliance on exact boundary triangulations or maximum principles. The approach combines boundary-skin estimates, a stable reflection-based extension operator, and careful interpolation error analyses for the isoparametric space, culminating in global $L^{\infty}$-accuracy without a logarithmic loss. The results hinge on a detailed reduction to $W^{1,1}$-control of a regularized Green function and a sequence of weighted $H^1$ and $L^2$ estimates, with key technical contributions in Sections 4–6 and the Appendix. The findings are significant for high-accuracy FEM in curved domains, enabling optimal pointwise convergence for $k\ge 2$ under realistic geometric approximations.

Abstract

We consider the isoparametric finite element method (FEM) for the Poisson equation in a smooth domain with the homogeneous Dirichlet boundary condition. Because the boundary is curved, standard triangulated meshes do not exactly fit it. Thereby we need to introduce curved elements if better accuracy than linear FEM is desired, which necessitates the use of isoparametric FEMs. We establish optimal rate of convergence $O(h^{k+1})$ in the $L^\infty$-norm for $k \ge 2$, by extending the approach of our previous work [Kashiwabara and Kemmochi, Numer.\ Math.\ \textbf{144}, 553--584 (2020)] developed for Neumann boundary conditions and $k = 1$.

Optimal $L^\infty$-error estimate for isoparametric finite element method in a smooth domain

TL;DR

This work analyzes the isoparametric finite element method for the Dirichlet Poisson problem on a smooth domain, proving an optimal -error rate of for quadratic or higher-order elements on a curved boundary approximation . The authors extend prior Neumann-boundary techniques by leveraging a Dirichlet Green function framework and a dyadic, localized error analysis to control the regularized Green function in weighted norms, avoiding reliance on exact boundary triangulations or maximum principles. The approach combines boundary-skin estimates, a stable reflection-based extension operator, and careful interpolation error analyses for the isoparametric space, culminating in global -accuracy without a logarithmic loss. The results hinge on a detailed reduction to -control of a regularized Green function and a sequence of weighted and estimates, with key technical contributions in Sections 4–6 and the Appendix. The findings are significant for high-accuracy FEM in curved domains, enabling optimal pointwise convergence for under realistic geometric approximations.

Abstract

We consider the isoparametric finite element method (FEM) for the Poisson equation in a smooth domain with the homogeneous Dirichlet boundary condition. Because the boundary is curved, standard triangulated meshes do not exactly fit it. Thereby we need to introduce curved elements if better accuracy than linear FEM is desired, which necessitates the use of isoparametric FEMs. We establish optimal rate of convergence in the -norm for , by extending the approach of our previous work [Kashiwabara and Kemmochi, Numer.\ Math.\ \textbf{144}, 553--584 (2020)] developed for Neumann boundary conditions and .

Paper Structure

This paper contains 13 sections, 7 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Approximation by isoparametric FEM
  3. Assumptions on $\Omega$
  4. Construction of approximate domains
  5. Estimates on a boundary-skin layer
  6. Extension from $\Omega$ to $\Omega \cup \Gamma(\delta)$
  7. Finite element approximation
  8. Reduction to $W^{1,1}$-analysis of a regularized Green function
  9. Weighted $H^1$-estimate
  10. Weighted $L^2$-estimate
  11. Appendix
  12. Proof of Proposition \ref{['prop: regularized delta']} (cf. SSW96)
  13. Interpolation error estimates for $\mathring I_h$

Key Result

Theorem 1.1

Let $k \ge 2$ and $\tilde{u} \in W^{k+1, \infty}(\mathbb R^N)$ be an arbitrary extension of $u$ such that $\|\tilde{u}\|_{W^{k+1, \infty}(\mathbb R^N)} \le C \|u\|_{W^{k+1, \infty}(\Omega)}$. Then we have where the constants $C$ depend only on $N$, $\Omega$, and $k$.

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.1
  • Proposition 4.1
  • Remark 4.1
  • Proposition 5.1
  • Remark 5.1
  • ...and 2 more