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In this paper we discuss robust two-level domain decomposition preconditioners for highly anisotropic heterogeneous multiscale problems. We present a construction of several coarse spaces that employ standard finite element  and multiscale basis functions and discuss techniques to reduce the dimensions of coarse spaces without sacrificing the robustness. We experimentally study the performance of the preconditioner on a variety two-dimensional test problems with channels of high anisotropy. The numerical tests confirm  the robustness of the perconditioner with respect to the underlying physical parameters.

Robust Two-level Domain Decomposition Preconditioners for High-contrast Anisotropic Flows in Multiscale Media