  We present an analysis of the additive average Schwarz preconditioner with
two newly proposed adaptively enriched coarse spaces which was presented at the
23rd International conference on domain decomposition methods in Korea, for
solving second order elliptic problems with highly varying and discontinuous
coefficients. It is shown that the condition number of the preconditioned
system is bounded independently of the variations and the jumps in the
coefficient, and depends linearly on the mesh parameter ratio H/h, that is the
ratio between the subdomain size and the mesh size, thereby retaining the same
optimality and scalablity of the original additive average Schwarz
preconditioner.
