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

Andrew Lorent

Contact the author: Please use for correspondence this email.
Submission date: 03. May. 2007
Pages: 50
published in: ESAIM / Mathematical modelling and numerical analysis, 15 (2008) 2, p. 322-366 
DOI number (of the published article): 10.1051/cocv:2008039
Bibtex
with the following different title: The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions
MSC-Numbers: 74N, 15
Download full preprint: PDF (574 kB), PS ziped (411 kB)

Abstract:
Let where are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems.

Firstly the N-well problem with surface energy. Let . Let

and let 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 ) 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 on (where is affine with ) such that for

there exists positive constants (depending on , , p) for which the following holds true