From Additive Average Schwarz Methods to Non-overlapping Spectral Additive Schwarz Methods

In this paper, we design and analyze two new methods based on additive average Schwarz -- AAS method introduced in \cite{MR1943457}. The new methods design for elliptic problems with highly heterogeneous coefficients. The methods are of the non-overlapping type, and the subdomain interactions obtain via the coarse space. The first method is the minimum energy Schwarz -- MES method. MES has the minimum energy for the coarse space with constant extension inside each subdomain. The condition number of the MES method is always smaller than in the AAS method. The second class of methods is the non-overlapping spectral additive Schwarz -- NOSAS methods based on low-rank discrete energy harmonic extension in each subdomain. To achieve the low-rank, we solve a generalized eigenvalue problem in each subdomain. NOSAS have the minimum energy for a given rank of the coarse space. The condition number of the NOSAS methods does not depend on the coefficients. Additionally, the NOSAS methods have good parallelization properties. The size of the global problem is equal to the total number of eigenvalues chosen in each subdomain. It is only related to the number of high-permeable islands that touch the subdomains' interface.