Old and new convergence proofs for multigrid methods

Yserentant, Harry

doi:10.1017/s0962492900002385

Cited by 186 publications

(166 citation statements)

References 62 publications

Supporting

Mentioning

164

Contrasting

Order By: Relevance

“…Hence, it follows that McCormick's analysis cannot prove optimal bounds for the V-cycle if the problem does not possess full regularity. Considering the results in [26], the same conclusions hold for Hackbusch's analysis [22, Section 7.2], and the SSC theory with a-orthogonal decomposition [16,17]. Thus, for the case when A k and A k M (2 ) k behave similarly with respect to the problem size, we show here that another type of analysis, as developed in, e.g.…”

Section: 2]) Thatsupporting

confidence: 75%

“…Thus, for the case when A k and A k M (2 ) k behave similarly with respect to the problem size, we show here that another type of analysis, as developed in, e.g. [13][14][15][16][17][18], is really needed to get uniform results for the V-cycle for problems with less than full regularity. …”

Section: 2]) Thatmentioning

confidence: 89%

“…To analyze the V-cycle, one possibility consists of defining an appropriate subspace decomposition and then applying successive subspace correction (SSC) theory [13][14][15][16][17][18]. Another possibility consists in checking the so-called smoothing and approximation properties [19][20][21][22][23][24][25].…”

mentioning

confidence: 99%

“…On the other hand, we also consider in [26] the classical formulation of the SSC theory (as stated in [16,17]), and discuss how to obtain a bound that could also be assessed considering only two levels at a time. It turns out that this requires the use of the so-called a-orthogonal decomposition, which corresponds to the choice most frequently made when applying the SSC theory to multigrid methods for H 2 -regular problems.…”

mentioning

confidence: 99%

See 3 more Smart Citations

When does two‐grid optimality carry over to the V‐cycle?

Napov¹,

Notay²

2009

Numerical Linear Algebra App

View full text Add to dashboard Cite

SUMMARYWe investigate additional condition(s) that confirm that a V-cycle multigrid method is satisfactory (say, optimal) when it is based on a two-grid cycle with satisfactory (say, level-independent) convergence properties. The main tool is McCormick's bound on the convergence factor (SIAM J. Numer. Anal. 1985; 22:634-643), which we showed in previous work to be the best bound for V-cycle multigrid among those that are characterized by a constant that is the maximum (or minimum) over all levels of an expression involving only two consecutive levels; that is, that can be assessed considering only two levels at a time. We show that, given a satisfactorily converging two-grid method, McCormick's bound allows us to prove satisfactory convergence for the V-cycle if and only if the norm of a given projector is bounded at each level. Moreover, this projector norm is simple to estimate within the framework of Fourier analysis, making it easy to supplement a standard two-grid analysis with an assessment of the V-cycle potentialities. The theory is illustrated with a few examples that also show that the provided bounds may give a satisfactory sharp prediction of the actual multigrid convergence.

show abstract

Section: 2]) Thatsupporting

confidence: 75%

Section: 2]) Thatmentioning

confidence: 89%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

When does two‐grid optimality carry over to the V‐cycle?

Napov¹,

Notay²

2009

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…For selfadjoint and coercive linear elliptic boundary value problems the convergence theory has reached a mature, if not its final state, cf. [14]. In other areas the state of the art is (far) less advanced.…”

Section: Introductionmentioning

confidence: 99%

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

Reusken

1995

Notes on Numerical Fluid Mechanics (NNFM)

View full text Add to dashboard Cite

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.

show abstract

Multilevel Minimal Residual Methods for Nonsymmetric Elliptic Problems

Starke

1996

Numer. Linear Algebra Appl.

View full text Add to dashboard Cite

Old and new convergence proofs for multigrid methods

Cited by 186 publications

References 62 publications

When does two‐grid optimality carry over to the V‐cycle?

When does two‐grid optimality carry over to the V‐cycle?

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

Multilevel Minimal Residual Methods for Nonsymmetric Elliptic Problems

Contact Info

Product

Resources

About