Nonconforming finite-element discretization of convex variational problems

Abstract

The Lavrentiev gap phenomenon is a well-known effect in the calculus of variations, related to singularities of minimizers. In its presence, conforming finite-element methods are incapable of reaching the energy minimum. By contrast, it is shown in this work that for convex variational problems the nonconforming Crouzeix–Raviart finite-element discretization always converges to the correct minimizer and that the discrete energy converges to the correct limit.