Optimal-order error estimates for the finite element approximation of the solution of a nonconvex variational problem

Abstract

Nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials. The solution of these variational problems must be described in terms of a microstructure rather than in terms of a deformation. Moreover, the numerical approximation of the deformation gradient often does not converge strongly as the mesh is refined. Nevertheless, the probability distribution of the deformation gradients near each material point does converge. Recently we introduced a metric to analyze this convergence. In this paper, we give an optimal-order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier.