Two-level overlapping Schwarz methods for elliptic equations combine one level of 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 space used to construct this approximation.

Inspired by methods for the one-dimensional problem formulated using energy minimization, Wan Chan and Smith [1] proposed a coarse space based on a constrained energy minimization problem. This approach is supported by recent theory on the convergence of Schwarz methods that takes into account coefficient variation [2]. The Lagrange multipliers for the constrained minimization problem are found by solving a large linear system. The coarse basis functions are then found by solving local systems.

Xu and Zikatanov [3] showed that, for multigrid methods, the system for the Lagrange multipliers can be solved efficiently using an appropriate preconditioner. They hint at a preconditioner for the domain decomposition case, but only present a Jacobi preconditioner, which is not robust for two-level Schwarz methods applied to equations with highly varying coefficients. We propose a preconditioner for the domain decomposition case, discuss how it can be implemented efficiently and demonstrate that it is robust. The resulting approach provides an algebraic way to construct a robust two-level overlapping Schwarz method given only the system matrix and the matrices representing the local spaces for the one-level method.