An a posteriori error estimate for finite element approximations of a singularly perturbed advection-diffusion problem

In this paper the author presents an a posteriori error estimator for approximations of the solution to an advection-diffusion equation with a non-constant, vector-valued diffusion coefficient ϵ in a conforming finite element space. Based on the complementary variational principle, we show that the error of an approximate solution in an associated energy norm is bounded by the sum of the weighted L2-norms of solutions to a set of independent complementary variational problems, each defined on only one element of the partition. This error bound guarantees the over-estimation of the true error and does not depend unfavourably on ϵ as ∥ϵ∥∞ goes to zero. Although the original equation is a non-self-adjoint problem, the strong form of each local variational problem is always a Poisson equation with Neumann boundary conditions. The approximation of these local problems is then discussed and it is shown that, omitting a higher order term, the finite element solutions of these local complementary variational problems provide a computable upper error bound for the original finite element approximation in the energy norm. Numerical results, presented to validate the theoretical results, show that the computed error bounds are tight for a wide range of values of ϵ and always over-estimate the true errors.