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We consider finite element methods of multiscale type to approximatesolutions for two-dimensional symmetric elliptic partial differential equationswith heterogeneous $L^\infty$ coefficients. The methods are of Galerkin typeand follows the Variational Multiscale and Localized OrthogonalDecomposition--LOD approaches in the sense that it decouples spaces intomultiscale and fine subspaces. In a first method, the multiscale basisfunctions are obtained by mapping coarse basis functions, based on corners usedon primal iterative substructuring methods, to functions of global minimalenergy. This approach delivers quasi-optimal a priori error energyapproximation with respect to the mesh size, however it deteriorates withrespect to high-contrast coefficients. In a second method, edge modes based onlocal generalized eigenvalue problems are added to the corner modes. As aresult, optimal a priori error energy estimate is achieved which is mesh andcontrast independent. The methods converge at optimal rate even if the solutionhas minimum regularity, belonging only to the Sobolev space $H^1$.

Adaptive ACMS: A robust localized Approximated Component Mode Synthesis Method