The regularisation of the N-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 322-366.

Let K:=SO2A 1 SO2A 2 SO2A N where A 1 ,A 2 ,,A N 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 p1,2, Ω 2 be a convex polytopal region. Define

I ϵ p u= Ω d p Duz,K+ϵD 2 uz 2 dL 2 z
and let A F denote the subspace of functions in W 2,2 Ω that satisfy the affine boundary condition Du=F on Ω (in the sense of trace), where FK. We consider the scaling (with respect to ϵ) of
m ϵ p :=inf uA F I ϵ p u.
Secondly the finite element approximation to the N-well problem without surface energy. We will show there exists a space of functions 𝒟 F h where each function v𝒟 F h is piecewise affine on a regular (non-degenerate) h-triangulation and satisfies the affine boundary condition v=l F on Ω (where l F is affine with Dl F =F) such that for
α p h:=inf v𝒟 F h Ω d p Dvz,KdL 2 z
there exists positive constants 𝒞 1 <1<𝒞 2 (depending on A 1 ,,A N , p) for which the following holds true
𝒞 1 α p ϵm ϵ p 𝒞 2 α p ϵforallϵ>0.

DOI : https://doi.org/10.1051/cocv:2008039
Classification : 74N15
Mots clés : two wells, surface energy
@article{COCV_2009__15_2_322_0,
     author = {Lorent, Andrew},
     title = {The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {322--366},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {2},
     year = {2009},
     doi = {10.1051/cocv:2008039},
     zbl = {1161.74044},
     mrnumber = {2513089},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2009__15_2_322_0/}
}
Lorent, Andrew. The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 322-366. doi : 10.1051/cocv:2008039. http://www.numdam.org/item/COCV_2009__15_2_322_0/

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