Parabolic Singularly Perturbed Problems with Exponential Layers: Robust Discretizations Using Finite Elements in Space on Shishkin Meshes

Abstract

A parabolic initial-boundary value problem with solutions displaying exponential layers is solved using layer-adapted meshes. The paper combines finite elements in space, i.e., a pure Galerkin technique on a Shishkin mesh, with some standard discretizations in time. We prove error estimates as well for the $\theta$-scheme as for discontinuous Galerkin in time.