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Classical and Cascadic Multigrid - A Methodical Comparison

Please always quote using this URN: urn:nbn:de:0297-zib-2368
  • Using the full multigrid method {\em without} any coarse grid correction steps but with an a posteriori control of the number of smoothing iterations was shown by Bornemann and Deuflhard [1996] to be an optimal iteration method with respect to the energy norm. They named this new kind of multigrid iteration the {\em cascadic multigrid method}. However, numerical examples with {\em linear} finite elements raised serious doubts whether the cascadic multigrid method can be made optimal with respect to the {\em $L^2$-norm}. In this paper we prove that the cascadic multigrid method cannot be optimal for linear finite elements and show that the case might be different for higher order elements. We present a careful analysis of the two grid variant of the cascadic multigrid method providing a setting where one can understand the methodical difference between the cascadic multigrid method and the classical multigrid $V$-cycle almost immediately. As a rule of thumb we get that whenever the cascadic multigrid works the classical multigrid will work too but not vice versa.

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Metadaten
Author:Folkmar A. Bornemann, Rolf KrauseORCiD
Document Type:ZIB-Report
Date of first Publication:1996/08/16
Series (Serial Number):ZIB-Report (SC-96-25)
ZIB-Reportnumber:SC-96-25
Published in:Appeared in: P. Bjorstad, M. Espedal, D. Keyes (eds.) Proc. 9th Int. Conference on Domain Decomposition Methods 1996, Ullensvang, Norway. Bergen: Domain Decomposition Press 1998. Pp. 64-71
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