Matrix analysis on leading term of condition number for additive Schwarz methods (Q2725282)

The paper is concerned with the application of preconditioned conjugate gradient algorithm and efficiency of the parallel additive Schwarz method for the solution of 1D, 2D and 3D second order elliptic boundary value problems with self-adjoint positive operator in polyhedral domains. A subdomain partition with and without overlapping is used. A decomposition of the higher discrete domains, discrete operators, \(M\)-matrix and related preconditioner via lower a dimension case is assumed. For instance, this is possible for a block-rectangle or a strip subdomain partition with the considered type of equations. Under these conditions a dimension reducing procedure is proposed for estimation of the smallest eigenvalue of the preconditioned matrix. Some model problems and numerical results are given.