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Zbl 0891.65104
Tai, Xue-Cheng
A space decomposition method for parabolic equations. (English)
[J] Numer. Methods Partial Differ. Equations 14, No.1, 27-46 (1998). ISSN 0749-159X; ISSN 1098-2426/e

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 $O(|\log\tau|)$ steps of iteration at each time level are needed, where $\tau$ 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.
[S.F.McCormick (Boulder)]

MSC 2000:: *65M55 Multigrid methods; domain decomposition (IVP of PDE)
65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
35K15 Second order parabolic equations, initial value problems

Keywords: convergence; parabolic equation; domain decomposition method; multilevel method; multigrid method; Crank-Nicolson scheme

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Zentralblatt MATH Berlin [Germany]