[Paper Review] $L^\infty$- and $W^{1,\infty}$-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Takahito Kashiwabara, Tomoya Kemmochi|arXiv (Cornell University)|Apr 2, 2018

Advanced Numerical Methods in Computational Mathematics18 references3 citations

TL;DR

This paper establishes optimal pointwise error estimates for the linear finite element method applied to second-order elliptic Neumann problems in smooth domains, accounting for domain perturbation due to polyhedral approximations. By combining regularized Green’s functions with local $H^1$- and $L^2$-estimates in dyadic annuli, it proves $O(h^2|\ ext{log}~h|)$ and $O(h)$ convergence rates in the $L^\infty$- and $W^{1,\infty}$-norms, respectively, despite non-conformity from boundary approximation.

ABSTRACT

Pointwise error analysis of the linear finite element approximation for $-Δu + u = f$ in $Ω$, $\partial_n u = τ$ on $\partialΩ$, where $Ω$ is a bounded smooth domain in $\mathbb R^N$, is presented. We establish $O(h^2|\log h|)$ and $O(h)$ error bounds in the $L^\infty$- and $W^{1,\infty}$-norms respectively, by adopting the technique of regularized Green's functions combined with local $H^1$- and $L^2$-estimates in dyadic annuli. Since the computational domain $Ω_h$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $Ω_h eq Ω$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.

Motivation & Objective

• To provide pointwise error estimates for the linear finite element method in smooth domains with Neumann boundary conditions.
• To address the challenge of non-conformity arising from polyhedral domain approximation ($\Omega_h \neq \Omega$) and boundary mismatch ($\Gamma_h \neq \Gamma$).
• To extend standard $L^\infty$ and $W^{1,\infty}$ error analysis to Neumann problems under domain perturbation.
• To establish optimal convergence rates despite the breakdown of Galerkin orthogonality due to domain discrepancy.
• To validate the theoretical findings with a numerical example confirming the derived error bounds.

Proposed method

• Utilizes regularized Green’s functions to reduce $L^\infty$ and $W^{1,\infty}$ error analysis to $W^{1,1}$-error estimation of the Green’s function approximation.
• Employs a dyadic decomposition of the domain into annuli $A_j$ centered at the point of interest, enabling localized error control.
• Applies local $H^1$- and $L^2$-estimates within each annulus to bound the error in the regularized Green’s function approximation.
• Treats domain perturbation terms (boundary-skin effects) arising from $\Omega_h \neq \Omega$ and $\Gamma_h \neq \Gamma$ by carefully estimating their contributions in weighted norms.
• Uses the Galerkin orthogonality relation with correction terms to account for non-conformity, ensuring stability and convergence.
• Applies scaling heuristics and dyadic decomposition to derive sharp bounds on the $L^1$-norm of the gradient of the error in the Green’s function.

Experimental results

Research questions

• RQ1 What is the optimal pointwise convergence rate of the linear finite element method for the Neumann problem in a smooth domain when the computational domain is polyhedral?
• RQ2 How does domain perturbation from boundary approximation affect $L^\infty$ and $W^{1,\infty}$ error estimates in the finite element method?
• RQ3 Can regularized Green’s functions be effectively combined with dyadic annuli to derive sharp $L^\infty$ and $W^{1,\infty}$ error bounds under non-conformity?
• RQ4 What is the role of boundary-skin terms in the error analysis when $\Omega_h \neq \Omega$ and $\Gamma_h \neq \Gamma$?
• RQ5 How do the derived error estimates compare to standard results in the conforming case, and what is the impact of the logarithmic factor in the $L^\infty$-error?

Key findings

• The paper establishes an $O(h^2|\text{log}~h|)$ error bound in the $L^\infty$-norm for the linear finite element solution of the Neumann problem.
• An $O(h)$ error bound is proven in the $W^{1,\infty}$-norm, which matches the optimal rate for smooth solutions.
• The convergence rates are achieved despite non-conformity due to domain approximation, with careful treatment of boundary-skin terms from $\Omega_h \neq \Omega$.
• The analysis relies on regularized Green’s functions and local estimates in dyadic annuli to control pointwise error near the singularity.
• Corollaries confirm that boundary-skin layer contributions are controlled by $\delta = C_0 E h^2$, with $L^1$-norms of gradients decaying appropriately.
• A numerical example confirms the theoretical convergence rates, validating the derived error estimates.