Can a finite element method perform arbitrarily badly?

Abstract

In this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and show that adaptive procedures cannot improve this slow convergence. We also show that the L 2-norm and the nodal point errors converge arbitrarily slowly. With the L 2-norm two cases need to be distinguished, and the usual duality principle does not characterize the error completely. The constructed elliptic problems are one dimensional.