A Posteriori Error Analysis for Linear Convection-Diffusion Problems Under Weak Mesh Regularity Assumptions

Abstract

In this paper we consider the generalisation of standard a posteriori error estimates, derived for unsteady problems, to arbitrary space-time meshes. In particular, we derive an a posteriori error bound for the discontinuity capturing Lagrange-Galerkin method applied to an unsteady (linear) convection-diffusion problem, assuming only that the underlying mesh is non-degenerate. The proof of this error estimate will be based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method.