On the Convergence of Cascadic Iterations for Elliptic Problems.
Please always quote using this URN: urn:nbn:de:0297-zib-1389
- We consider nested iterations, in which the multigrid method is replaced by some simple basic iteration procedure, and call them {\em cascadic iterations}. They were introduced by Deuflhard, who used the conjugate gradient method as basic iteration (CCG method). He demonstrated by numerical experiments that the CCG method works within a few iterations if the linear systems on coarser triangulations are solved accurately enough. Shaidurov subsequently proved multigrid complexity for the CCG method in the case of $H^2$-regular two-dimensional problems with quasi-uniform triangulations. We show that his result still holds true for a large class of smoothing iterations as basic iteration procedure in the case of two- and three-dimensional $H^{1+\alpha}$-regular problems. Moreover we show how to use cascadic iterations in adaptive codes and give in particular a new termination criterion for the CCG method.
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| Author: | Folkmar A. Bornemann |
|---|---|
| Document Type: | ZIB-Report |
| Date of first Publication: | 1994/03/30 |
| Series (Serial Number): | ZIB-Report (SC-94-08) |
| ZIB-Reportnumber: | SC-94-08 |
| Published in: | Appeared as: F. A. Bornemann, P. Deuflhard: The cascadic multigrid method for elliptic problems. in: Numer. Math. 75 (1996) pp. 135-152 |
