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 AF 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=lF on $\partial \Omega$ (where lF 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
$$
\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq
\mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0.
$$