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We study solutions of the 2D Ginzburg–Landau equation $-\Delta u+{\epsilon }^{-2}{u\left(\right|u|}^2-1)=0$ subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, $\left|u\right|=1$, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small $\epsilon$. For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy...

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ( ) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution ) minimizing the ...