The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

MIS MPG, Inselstrasse 22, 04103 Leipzig, Germany. andrew.lorent@sns.it

Received: 7 June 2007
Revised: 15 January 2008

Abstract

Let where are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N-well problem with surface energy. Let , be a convex polytopal region. Define and let A_F denote the subspace of functions in that satisfy the affine boundary condition Du=F on (in the sense of trace), where . We consider the scaling (with respect to ϵ) of Secondly the finite element approximation to the N-well problem without surface energy. We will show there exists a space of functions where each function is piecewise affine on a regular (non-degenerate) h-triangulation and satisfies the affine boundary condition v=l_F on (where l_F is affine with ) such that for there exists positive constants (depending on , p) for which the following holds true