Issue |
ESAIM: COCV
Volume 15, Number 2, April-June 2009
|
|
---|---|---|
Page(s) | 322 - 366 | |
DOI | https://doi.org/10.1051/cocv:2008039 | |
Published online | 24 June 2008 |
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
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 AF 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=lF on
(where lF is affine with
) such that for
there exists positive constants
(depending on
, p) for which the following holds true
Mathematics Subject Classification: 74N15
Key words: Two wells / surface energy
© EDP Sciences, SMAI, 2008
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