On a posteriori error estimation in the maximum norm

In this thesis we consider residual-based a posteriori error estimates in the maximum norm for the finite element solution of some partial differential equations. The thesis consists of three papers. The first paper concerns a pointwise a posteriori error estimate for the time dependent obstacle problem. The analysis is based on a penalty formulation of the problem, where the penalty parameter is allowed to vary in space and time. For the discretisation we use the Discontinuous Galerkin method. The proof is based on a maximal regularity estimate for parabolic equations. In the second paper we consider a stationary convection-diffusion problem. For the space discretisation we use the Streamline Diffusion method. We prove a global error estimate in the maximum norm. We also prove a localized version of this result in a special case. In the third paper we combine techniques from the first two papers to prove an a posteriori error estimate in the maximum norm for a time dependent convection-diffusion problem. For the discretisation in time and space we use the Discontinuous Galerkin method combined with the Streamline Diffusion method.