Preprint 46/2007

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

Andrew Lorent

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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
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Abstract:
Let formula33 where formula35 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 formula39. Let
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and let formula43 denote the subspace of functions in formula45 that satisfy the affine boundary condition Du=F on formula49 (in the sense of trace), where formula51. We consider the scaling (with respect to formula53) of
displaymath55

Secondly the finite element approximation to the N-well problem without surface energy.

We will show there exists a space of functions formula59 where each function formula61 is piecewise affine on a regular (non-degenerate) h-triangulation and satisfies the affine boundary condition formula65 on formula49 (where formula69 is affine with formula71) such that for
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there exists positive constants formula75 (depending on formula77, formula79, p) for which the following holds true
displaymath83

23.06.2018, 02:11