$L^{\infty}$-error estimates for Neumann boundary value problems on graded meshes

TL;DR

This paper establishes sharp pointwise error estimates for finite element solutions of Neumann boundary problems on polygonal domains with corner singularities, using graded meshes and weighted Sobolev space regularity.

Contribution

It provides the first precise regularity and convergence order results for Neumann problems, including for semilinear cases, using local mesh refinement and supercloseness techniques.

Findings

Achieves the convergence rate $h^2 | ext{ln} h|$ for linear finite elements.

Extends error estimates to semilinear boundary value problems.

Demonstrates the effectiveness of graded meshes near corners.

Abstract

This paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in case of smooth domains. As a remedy the use of local mesh refinement near the corners is investigated. In order to prove quasi-optimal a priori error estimates regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order $h^2 \lvert \ln h \rvert $ in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution.