Domain decomposition preconditioners for multiscale problems

In this paper, we study domain decomposition preconditioners for multiscale elliptic problems in high contrast media. We construct preconditioners such that the condition number of the preconditioned system is independent of media contrast. For this purpose, multiscale spaces for the interpolation on the coarse grid is developed using a local weighted spectral problem. A main observation is that the eigenvalues of this spectral problem control the condition number. In the presence of high-contrast inclusions, there are small, asymptotically vanishing, eigenvalues, i.e., these eigenvalues decrease as we increase the contrast. We propose the coarse space that includes the eigenfunctions corresponding to these small, asymptotically vanishing, eigenvalues. We prove that domain decomposition preconditioners with this coarse space result to the convergence of the methods independent of the contrast. The coarse space constructed using the eigenfunctions of local spectral problem can be large if there are many isolated inclusions. We propose approaches to reduce the dimension of the coarse space. Numerical results are presented. We compare the proposed methods with domain decomposition methods where multiscale finite element basis or energy minimizing basis functions are used in constructing coarse spaces. We show that the number of iterations is smaller with proposed methods and they remain bounded as the contrast increases.