The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent
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Submission date: 03. May. 2007
published in: RAIRO / Mathematical modelling and numerical analysis, 15 (2008) 2, p. 322-366
DOI number (of the published article): 10.1051/cocv:2008039
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
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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
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