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The regularisations of the \(N\)-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions. (English) Zbl 1161.74044

Summary: Let \(K:=SO(2)A_1\cup SO(2)A_2\dots SO(2)A)_N\), 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 \(p\in[1,2]\), \(\Omega\subset \mathbb R^2\), be a convex polytopal region. Define
\[ I^p_\varepsilon(u)=\int_\Omega d^p (Du(z),K) + \varepsilon |D^2u(z)|^2 \,dL^2z, \]
and let \(A_F\) denote the subspace of functions in \(W^{2,2}(\Omega)\) 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 \(\varepsilon\)) of
\[ m^P_\varepsilon:=\inf_{u\in A_F}I^p_\varepsilon(u). \]
Secondly, the finite element approximation to the \(N\)-well problem without surface energy. We show that there exists a space of functions \({\mathcal D}^h_F\) where each function \(v\in{\mathcal D}^h_F\) 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
\[ \alpha_p(h):=\inf_{v\in{\mathcal D}^h_F}\int_\Omega d^p(Dv(z),K)\,dL^2z \]
there exist positive constants \({\mathcal C}_1<1<{\mathcal C}_2\) (depending on \(A_1,\dots,A_N, p)\) for which the following inequality holds
\[ {\mathcal C}_1\alpha_p(\sqrt{\varepsilon})\leq m^p_\varepsilon \leq{\mathcal C}_2\alpha_p(\sqrt{\varepsilon})\text{ for all }\varepsilon>0. \]

MSC:

74N15 Analysis of microstructure in solids
74G65 Energy minimization in equilibrium problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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