The regularisation of the N-wellproblem by finite elements and by singular perturbation are scalingequivalentin 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. Weestablish a sharp relation between the following two minimisationproblems 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 $$I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2u\left(z\right)\right|^2 {\rm d}L^2 z$$ and let A _F denote the subspace of functions in $W^{2,2}\left(\Omega\right)$ that satisfy the affine boundary conditionDu=F on $\partial \Omega$ (in the sense of trace), where $F\not\inK$ . 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 problemwithout surface energy. We will show there exists a space of functions $\mathcal{D}_F^{h}$ whereeach function $v\in \mathcal{D}_F^{h}$ is piecewise affine on a regular(non-degenerate) h-triangulation and satisfies the affine boundarycondition 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 $$\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq\mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0.$$