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Cascadic Conjugate Gradient Methods for Elliptic Partial Differential Equations I. Algorithm and Numerical Results.

Please always quote using this URN: urn:nbn:de:0297-zib-1197
  • Cascadic conjugate gradient methods for the numerical solution of elliptic partial differential equations consists of Galerkin finite element methods as outer iteration and (possibly preconditioned) conjugate gradient methods as inner iteration. Both iterations are known to minimize the energy norm of the arising iterations errors. A simple but efficient strategy to control the discretization errors versus the PCG iteration errors in terms of energy error norms is derived and worked out in algorithmic detail. In a unified setting, the relative merits of different preconditioners versus the case of no preconditioning is compared. Surprisingly, it appears that the cascadic conjugate gradient method without any preconditioning is not only simplest but also fastest. The numerical results seem to indicate that the cascade principle in itself already realizes some kind of preconditioning. A theoretical explanation of these observations will be given in Part II of this paper.

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Metadaten
Author:Peter Deuflhard
Document Type:ZIB-Report
Date of first Publication:1993/09/30
Series (Serial Number):ZIB-Report (SC-93-23)
ZIB-Reportnumber:SC-93-23
Published in:Appeared in: D. E. Keyes, J. Xu (eds.) Domain Decomposition Methods in Science and Engineering. Proc. of the 7th int. conf. on domain decomposition, Oct. 27-30, 1993. Providence, RI: AMS 1994. (Contemporary Math., vol. 180) pp. 29-42
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