An Adaptive Coarse Space for P.L. Lions Algorithm and Optimized Schwarz Methods
Optimized Schwarz methods (OSM) are very popular methods which were introduced in Lions (On the Schwarz alternating method. III: a variant for nonoverlapping subdomains. In: Chan TF, Glowinski R, Périaux J, Widlund O (eds) Third international symposium on domain decomposition methods for partial differential equations, Houston, TX, March 20–22, 1989. SIAM, Philadelphia, PA, 1990) for elliptic problems and in Després (C R Acad Sci Paris 1(6):313–316, 1990) for propagative wave phenomena. We build here a coarse space for which the convergence rate of the two-level method is guaranteed regardless of the regularity of the coefficients. We do this by introducing a symmetrized variant of the ORAS (Optimized Restricted Additive Schwarz) algorithm (St-Cyr et al., SIAM J Sci Comput 29(6):2402–2425 (electronic), 2007) and by identifying the problematic modes using two different generalized eigenvalue problems instead of only one as in Spillane et al. (C R Math Acad Sci Paris 351(5–6):197–201, 2013) and Spillane et al. (Numer Math 126(4):741–770, 2014) for the ASM (Additive Schwarz method), BDD [balancing domain decomposition (Mandel, Comm Appl Numer Methods 9:233–241, 1992)] or FETI [finite element tearing and interconnection (Farhat and Roux, Int J Numer Meth Eng 32:1205–1227, 1991)] methods.