Some useful techniques for pointwise and local error estimates of the quantities of interest in the finite element approximation

Abstract

In this paper we review some existing techniques to obtain pointwise and local a posteriori error estimates for the quantities of interest in finite element approximations by using duality arguments. We also present a new approach to obtain computable error bounds for the recovered pointwise quantities. The new method is extended to include the practically important case of non-homogeneous Dirichlet data. Existing methods require purely Neumann data, or the Dirichlet data to be homogeneous. The new techniques are developed here to provide computable error bounds on the genuine pointwise quantities and allow the use of non-homogeneous Dirichlet data. The strength and weakness of each technique will be analysed and compared. The numerical experiments to justify our analysis will be presented.