Robust multilevel solvers for high-contrast anisotropic multiscale problems

A robust multilevel method for computing the solution of a scalar elliptic equation with anisotropic highly varying tensor coefficients is presented. The method, which belongs to the class of nonlinear algebraic multilevel iterations (AMLIs), uses an abstract framework for general symmetric positive definite bilinear forms previously presented in another publication by the author. The crucial ingredient for obtaining robustness with respect to the variations in the coefficients and the anisotropies is the design of a nested sequence of spaces based on local generalized eigenvalue problems. A discussion on how to achieve large coarsening factors in this sequence of spaces, which is desirable in terms of computational complexity, is included. Particular emphasis is put on how to handle the situation when the computed generating sets of the nested spaces are not minimal, i.e., do not constitute bases. Several numerical examples are provided verifying the theoretically established robustness results.