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A space decomposition method for parabolic equations. (English) Zbl 0891.65104
Author’s summary: A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank-Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O(|logτ|) steps of iteration at each time level are needed, where τ is the time-stepsize. Applications to overlapping domain decomposition and to a two-level method are given for a second-order parabolic equation. The analysis shows that only a one-element overlap is needed. Discussions about iterative and noniterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically.
MSC:
65M55Multigrid methods; domain decomposition (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35K15Second order parabolic equations, initial value problems