The regularisation of the N-wellproblem by finite elements and by singular perturbation are scalingequivalent in two dimensions

Abstract

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\dots SO\left(2\right)A_{N}$ where $A_1,A_2,\dots, A_{N}$ 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 $p\in\left[1,2\right]$, $\Omega\subset \mathbb{R}^2$ be a convex polytopal region. Define $$ I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2 u\left(z\right)\right|^2 {\rm d}L^2 z $$ and let A_F denote the subspace of functions in $W^{2,2}\left(\Omega\right)$ that satisfy the affine boundary condition Du=F on $\partial \Omega$ (in the sense of trace), where $F\not\in K$. We consider the scaling (with respect to ϵ) of $$ m^p_{\epsilon}:=\inf_{u\in A_F} I^p_{\epsilon}\left(u\right). $$ Secondly the finite element approximation to the N-well problem without surface energy. We will show there exists a space of functions $\mathcal{D}_F^{h}$ where each function $v\in \mathcal{D}_F^{h}$ is piecewise affine on a regular (non-degenerate) h-triangulation and satisfies the affine boundary condition v=l_F on $\partial \Omega$ (where l_F is affine with $Dl_F=F$) such that for $$ \alpha_p\left(h\right):=\inf_{v\in \mathcal{D}_F^{h}} \int_{\Omega}d^p\left(Dv\left(z\right),K\right) {\rm d}L^2 z $$ there exists positive constants $\mathcal{C}_1<1<\mathcal{C}_2$ (depending on $A_1,\dots, A_{N}$, p) for which the following holds true $$ \mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq \mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0. $$