A Posteriori Error Analysis of $hp$-FEM for singularly perturbed problems

Abstract

We consider the approximation of singularly perturbed linear second-order boundary value problems by $hp$-finite element methods. In particular, we include the case where the associated differential operator may not be coercive. Within this setting we derive an a posteriori error estimate for a natural residual norm. The error bound is robust with respect to the perturbation parameter and fully explicit with respect to both the local mesh size $h$ and the polynomial degree $p$.