Computable error bounds for pointwise derivatives of a Neumann problem

In this paper we discuss the recovery of derivatives and thecomputation of rigorous and useful upper bounds for the pointwiseerror in the recovered derivatives, for finite element approximationsof the Laplace equation with Neumann boundary conditions, especiallyat points close to or on a smooth, curved boundary. We analyzethe dipole image technique for the case of curved boundaries,and show how to compute reliable recovered derivatives and errorbounds even in the limiting case of points lying on the curvedboundary. Numerical experiments show reasonably tight errorbounds for points both close to and away from a curved boundary.