An a posteriori error estimate for finite element approximations of a singularly perturbed advection-diffusion problem
This paper deals with an a posteriori error estimator for an important nonselfadjoint problem, i.e. a singularly perturbed advection-diffusion problem in two dimensions. Based on a complementary variational principle, the author shows that the error in an energy norm is bounded by the sum of the weighted \(L^2\)-norms of solutions to a set of independent variational problems, each defined on only one element of a given partition. These local problems can be solved numerically by a finite element method on a refined mesh of the original. It is shown that the numerical solutions of the local variational problems define an approximate upper error bound which differs from the theoretical upper bound by a higher-order term than the true error. The numerical results presented here show that the computed error bounds are tight for a wide range of values of \(\varepsilon\) and always over-estimate the true errors.
scientific article; zbMATH DE number 1121351
scientific article