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Nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems. (English) Zbl 1204.65140
The authors propose a substructuring preconditioner for solving three-dimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by J. H. Bramble, J. E. Pasciak and A. H. Schatz [Math. Comput. 53, No. 187, 1–24 (1989; Zbl 0668.65082)], but with much simpler coarse solvers. It is shown that the preconditioned conjugate gradient method with such substructuring preconditioner has a nearly optimal convergence rate, although the condition numbers of the preconditioned systems are not quasi-optimal yet.
65N55Multigrid methods; domain decomposition (BVP of PDE)
65F10Iterative methods for linear systems
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
65F08Preconditioners for iterative methods
35J25Second order elliptic equations, boundary value problems
35R05PDEs with discontinuous coefficients or data