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additive Schwarz algorithm algorithm 4.1 Anal analysis basis functions bilinear form boundary layer bounded Cauchy-Schwarz inequality Comp computed constant cubes curve search method defined denote derivatives difference scheme discontinuous Galerkin method discrete domain decomposition algorithm elliptic problems exists finite element method finite volume element finite volume method Galerkin method grid Helmholtz equation integral interpolation K€Th L2 projection L2-norm Lemma linear Math matrix Meth methods for elliptic monotone Multigrid Methods Navier-Stokes nonlinear norm notation numbers of iterations numerical experiments obtain operator optimal overlaps paper parabolic parameter piecewise polynomials posteriori error estimation preconditioner Proof properties prove reaction-diffusion refined right-hand side satisfies Schwarz method Section Shishkin SIAM singularly perturbed Sobolev space solving Stokes problem subdomain superconvergence Theorem 4.1 tions transform triangle triangle inequality two-scale finite volume uniform convergence uniform mesh variational inequalities vector volume element method wave number wavelet