A Recursive Variant of Schwarz Type Domain Decomposition Methods
DOI:
https://doi.org/10.23755/rm.v30i1.4Keywords:
Domain Decomposition, Finite Element Method, Linear Systems.Abstract
In this paper a slightly different approach to the use of the domain decomposition method of the Schwarz type is proposed. Instead of the standard coarse space construction we propose to use a recursive solution on each domain. Thus we do not need to construct a coarse space but nevertheless we are still keeping O(1) convergence speed. For local problems we use the standard iterative solvers for which the amount of the work for one step is O(N), where N is the number of equations. Due to the fact that the overlapping is under our control we can keep total work in O(N^(1+γ) ) operations with arbitrary positive γ.
References
H. A. Schwarz, Gessamelte Mathematische Abhandlungen, volume 2, pages 133 − 143, Springer, Berlin, (1890). First published in Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich,¨ Uber einen Grenzübergang durch alternierendes Verfahren, volume 15, (1870), pp. 272 − 286.
A. Toselli and O. Widlund, Domain Decomposition Methods - Algorithms and Theory. Springer-Verlag, Berlin Heidelberg, (2005).
M. Brezina and P. Vanˇ ek, A black-box iterative solver based on a twolevel Schwarz method. Computing 63(3), (1999), 233 − 263.
R. Varga, Matrix Iterative Analysis. Prentice Hall, first edition, (1962).
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