Sharp estimates for finite element approximations to elliptic problems with Neumann boundary data of low regularity

Consider a second order homogeneous elliptic problem with smooth coefficients, $Au = 0$, on a smooth domain, $\Omega$, but with Neumann boundary data of low regularity. Interior maximum norm error estimates are given for $C^0$ finite element approximations to this problem. When the Neumann data is not in $L^1(\partial\Omega)$, these local estimates are not of optimal order but are nevertheless shown to be sharp. A method for ameliorating this sub-optimality by preliminary smoothing of the boundary data is given. Numerical examples illustrate the findings.