The hierarchical basis multigrid method

Bank, Randolph E.; Dupont, Todd; Yserentant, Harry

doi:10.1007/bf01462238

Cited by 303 publications

(237 citation statements)

References 18 publications

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“…In this paper we are able to prove that a sufficiently damped version of the V-cycle is almost robust in the sense that the rate of convergence depends at most on the number of levels. Our result is of the same quality as typical estimates known for the convergence rates of multigrid solvers without regularity assumptions on the continuous problem, see [1], [3] and [17]. Indeed, the situation without regularity assumptions and the situation for the FDMGM are comparable in their lack of the approximation property.…”

Section: Introductionsupporting

confidence: 81%

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On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

Rieder

Zhou

1994

Computing

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--ZusammenfassungOn the Robustness of the Damped V-Cycle of the Wavelet Frequency Decomposition Multigrid Method. The damped V-cycle of the wavelet variation of the "Frequency decomposition multigrid method" of Hackbusch [Numer. Math. 56, pp. 229-245 (1989)] is considered. It is shown that the convergence speed under sufficient damping is not affected by the presence of anisotropy but still depends on the number of levels. Our analysis is based on properties of wavelet packets which are supplied and proved. Numerical approximations to the speed of convergence illustrate the theoretical results. AMS Subject Classifications: 65F10, 65N30Key words: Wavelets, wavelet packets, robust multilevel methods, V-cycle.Zur Robustheit des ged~impften V-Zyklus bei der FDMGM mit Wavelets. Wir betrachten den ged~impften V-Zyklus ffir die Wavelet-Variante der "Frequenzzerlegungs-Multigridmethode" yon Hackbusch [Numer. Math. 56, 229-245 (1989)]. Es wird gezeigt, dab die Konvergenzgeschwindigkeit bei hinreichender D/impfung durch Anisotropie nicht beeinflul3t wird, aber noch yon der Anzahl des Niveaus abh~ingt. Unsere Analyse beruht auf Eigenschaften von Wavelet-Paketen, die formuliert und bewiesen werden. Numerische Schgtzungen der Konvergenzgeschwindigkeit erlfiutern die theoretischen Ergebnisse.

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Section: Introductionsupporting

confidence: 81%

“…The iteration matrix of the basic iterative method (BIM) (also called smoothing iteration [1][2][3][4][5][6][7][8][9][10]) with respect to Al,,, will be denoted by Sz,,,, I -~ = -W~,,,A,,m.…”

Section: A~_lzm = (H | H)atm(h' | H T) and At_~zm+~ = (G | H)at Mmentioning

confidence: 99%

On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

Rieder

Zhou

1994

Computing

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“…The solve operation corresponds to the smoother in multigrid. Note, however, that in ULsolve every unknown is involved in the solve operations of precisely one level (as in hierarchical basis multigrid, d. [2]). The implementation of cyclic reduction presented in this section clearly has a modular structure.…”

Section: Cyclic Reductionmentioning

confidence: 99%

Approximate Cyclic Reduction Preconditioning

Reusken

1998

Lecture Notes in Computational Science and Engineering

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“…vVe assume two nested grids ("coarse" and "fine") and on the finest grid the new mesh points are ordered first and then the old ones. Now, as in the hierarchical basis multigrid method (HBMG; [4]), we use a transformation from the original matrix to another matrix that has a significantly lower generalized condition number. To this transformed matrix we apply a block Jacobi type of method.…”

Section: Introductionmentioning

confidence: 99%

A new robust multigrid method for 2D convection-diffusion problems

Reusken

1995

Notes on Numerical Fluid Mechanics (NNFM)

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Citation for published version (APA): Reusken, A. A. (1994). A new robust multigrid method for 2D convection-diffusion problems. (RANA : reports on applied and numerical analysis; Vol. 9404). Eindhoven: Eindhoven University of Technology. Document status and date:Published: 01/01/1994 Document Version:Publisher's PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publicationGeneral rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nl providing details and we will investigate your claim. SummaryIn this paper we introduce a new multigrid approach for solving 2D convection-diffusion problems. As in the hierarchical basis multigrid method we use a transformed matrix with a relatively low generalized condition number. The transformation we use is not based on the nesting of the coarse and fine grid only, but uses more information from the matrix. After transforming back to the original matrix this results in a robust multigrid algorithm with matrix-dependent prolongations and restrictions.

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The hierarchical basis multigrid method

Cited by 303 publications

References 18 publications

On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

On the robustness of the dampedV-cycle of the wavelet frequency decomposition multigrid method

Approximate Cyclic Reduction Preconditioning

A new robust multigrid method for 2D convection-diffusion problems

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