$L^{\infty}$-error estimates for Neumann boundary value problems on graded meshes
It provides the first sharp error estimates for Neumann problems with explicit regularity conditions, advancing numerical analysis for non-smooth domains.
The paper proves sharp $L^\\infty$-error estimates for finite element solutions of Neumann boundary value problems on polygonal domains with graded meshes, achieving $h^2 |\\ln h|$ convergence for piecewise linear elements, and extends the result to semilinear problems.
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.