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

Andrew Lorent (2009)

ESAIM: Control, Optimisation and Calculus of Variations

Let K : = S O 2 A 1 S O 2 A 2 S O 2 A 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 p 1 , 2 , Ω 2 be a convex polytopal region. Define I ϵ p u = Ω d p D u z , K + ϵ D 2 u z 2 d L 2 z and let A F denote the subspace of functions in W 2 , 2 Ω that satisfy the affine boundary condition D u = F on Ω (in the sense of trace), where F K . We consider the scaling (with respect to ϵ ) of m ϵ p : = inf u A F I ϵ p u . Secondly the finite element approximation to the N -well problem without surface...

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

Andrew Lorent (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Let K : = S O 2 A 1 S O 2 A 2 S O 2 A 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 p 1 , 2 , Ω 2 be a convex polytopal region. Define I ϵ p u = Ω d p D u z , K + ϵ D 2 u z 2 d L 2 z and let AF 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 F K . We consider the scaling (with respect to ϵ) of m ϵ p : = inf u A F I ϵ p u . Secondly the finite element approximation to the N-well problem without...

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