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This paper deals with a priori pointwise error estimates for the finiteelement solution of boundary value problems with Neumann boundary conditions inpolygonal domains. Due to the corners of the domain, the convergence rate ofthe numerical solutions can be lower than in case of smooth domains. As aremedy the use of local mesh refinement near the corners is investigated. Inorder to prove quasi-optimal a priori error estimates regularity results inweighted Sobolev spaces are exploited. This is the first work on the Neumannboundary value problem where both the regularity of the data is exactlyspecified and the sharp convergence order $h^2 \lvert \ln h \rvert $ in thecase of piecewise linear finite element approximations is obtained. As anextension we show the same rate for the approximate solution of a semilinearboundary value problem. The proof relies in this case on the superclosenessbetween the Ritz projection to the continuous solution and the finite elementsolution.

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