Fast-convergent two-level restricted additive Schwarz methods based on optimal local approximation spaces

This paper proposes a two-level restricted additive Schwarz (RAS) method for multiscale PDEs, built on top of a multiscale spectral generalized finite element method (MS-GFEM). The method uses coarse spaces constructed from optimal local approximation spaces, which are based on local eigenproblems posed on (discrete) harmonic spaces. We rigorously prove that the method, used as an iterative solver or as a preconditioner for GMRES, converges at a rate of $\Lambda$, where $\Lambda$ represents the error of the underlying MS-GFEM. The exponential convergence property of MS-GFEM, which is indepdendent of the fine mesh size $h$ even for highly oscillatory and high contrast coefficients, thus guarantees convergence in a few iterations with a small coarse space. We develop the theory in an abstract framework, and demonstrate its generality by applying it to various elliptic problems with highly heterogeneous coefficients, including $H({\rm curl})$ elliptic problems. The performance of the proposed method is systematically evaluated and illustrated via applications to two and three dimensional heterogeneous PDEs, including challenging elasticity problems in realistic composite aero-structures.