  In this paper, we design and analyze new methods based on Additive Average
Schwarz--AAS Method introduced in [3]. The new methods are designed for
elliptic problems with highly heterogeneous coefficients. The methods are of
non-overlapping type and the subdomain interactions are obtained via the coarse
space. The first method is the Minimum Energy Schwarz--MES Method, it has the
best coarse space with one degree of freedom per subdomain. The condition
number of MES is always smaller than AAS methods. The second class of methods
is the Non-overlapping Spectral Additive Schwarz--NOSAS methods, they are based
in low-rank discrete energy harmonic extension in each subdomain. We prove that
the condition number of NOSAS methods do not depend on the coefficients, and
with a specific choice of the threshold of the eigenvalues, we can obtain
condition numbers for the preconditioned system similar to the AAS method with
constant coefficients. Additionally, the NOSAS methods have good
parallelization properties, and the size of the coarse problem is equal to the
total number of eigenvalues chosen in each subdomain and which might be related
to the number of islands or channels with high coefficient that touch the
boundary of the subdomains.
