Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs

Two-level overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully choose the coarse approximation. Recent theoretical results by the authors have shown that bases for such robust coarse spaces should be constructed such that the energy of the basis functions is minimized. We give a simple derivation of a method that finds such a minimum energy basis using one local solve per coarse space basis function and one global solve to enforce a partition of unity constraint. Although this global solve may seem prohibitively expensive, we demonstrate that a one-level overlapping Schwarz method is an effective and scalable preconditioner and we show that such a preconditioner can be implemented efficiently using the Sherman-Morrison-Woodbury formula. The result is an elegant, scalable, algebraic method for constructing a robust coarse space given only the supports of the coarse space basis functions. Numerical experiments on a simple two-dimensional model problem with a variety of binary and multiscale coefficients confirm this. Numerical experiments also show that, when used in a twolevel preconditioner, the energy minimizing coarse space gives better results than other coarse space constructions, such as the multiscale finite element approach.