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In this paper, we design and analyze two new methods based on additiveaverage Schwarz -- AAS method introduced in \cite{MR1943457}. The new methodsdesign for elliptic problems with highly heterogeneous coefficients. Themethods are of the non-overlapping type, and the subdomain interactions obtainvia the coarse space. The first method is the minimum energy Schwarz -- MESmethod. MES has the minimum energy for the coarse space with constant extensioninside each subdomain. The condition number of the MES method is always smallerthan in the AAS method. The second class of methods is the non-overlappingspectral additive Schwarz -- NOSAS methods based on low-rank discrete energyharmonic extension in each subdomain. To achieve the low-rank, we solve ageneralized eigenvalue problem in each subdomain. NOSAS have the minimum energyfor a given rank of the coarse space. The condition number of the NOSAS methodsdoes not depend on the coefficients. Additionally, the NOSAS methods have goodparallelization properties. The size of the global problem is equal to thetotal number of eigenvalues chosen in each subdomain. It is only related to thenumber of high-permeable islands that touch the subdomains' interface.

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