Multigrid method: Difference between revisions

Revision as of 23:40, 7 December 2020

Revision as of 15:41, 23 December 2020

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==Generalized multigrid methods==

Multigrid methods can be generalized in many different ways. They can be applied naturally in a time-stepping solution of [[parabolic partial differential equation]]s, or they can be applied directly to time-dependent [[partial differential equation]]s.<ref>{{cite book |chapter-url=https://books.google.com/books?id=GKDQUXzLTkIC&pg=PA165 |editor1=Are Magnus Bruaset |editor2=Aslak Tveito |title=Numerical solution of partial differential equations on parallel computers |page=165 |chapter=Parallel geometric multigrid |author1=F. Hülsemann |author2=M. Kowarschik |author3=M. Mohr |author4=U. Rüde |publisher=Birkhäuser |year=2006 |isbn=978-3-540-29076-6}}</ref> Research on multilevel techniques for [[hyperbolic partial differential equation]]s is underway.<ref>For example, {{cite book |title=Computational fluid dynamics: principles and applications |page=305 |url=https://books.google.com/books?id=asWGy362QFIC&q=%22The+goal+of+the+current+research+is+the+significant+improvement+of+the+efficiency+of+multigrid+for+hyperbolic+flow+problems%22&pg=PA305 |author= J. Blaz̆ek |year=2001 |isbn=978-0-08-043009-6 |publisher=Elsevier}} and {{cite book |chapter-url=https://books.google.com/books?id=TapltAX3ry8C&pg=PA369 |author=Achi Brandt and Rima Gandlin |chapter=Multigrid for Atmospheric Data Assimilation: Analysis |page=369 |editor1=Thomas Y. Hou |editor2=[[Eitan Tadmor]] |editor2-link=Eitan Tadmor |title=Hyperbolic problems: theory, numerics, applications: proceedings of the Ninth International Conference on Hyperbolic Problems of 2002 |year=2003 |isbn=978-3-540-44333-9 |publisher=Springer}}</ref> Multigrid methods can also be applied to [[integral equation]]s, or for problems in [[statistical physics]].<ref>{{cite book |title=Multiscale and multiresolution methods: theory and applications |author=Achi Brandt |chapter-url=https://books.google.com/books?id=mtsy6Ci2TRoC&pg=PA53 |editor1=Timothy J. Barth |editor2=Tony Chan |editor3=Robert Haimes |page=53 |chapter=Multiscale scientific computation: review |isbn=978-3-540-42420-8 |year=2002 |publisher=Springer}}</ref>

Another set of multiresolution methods is based upon [[wavelets]]. These wavelet methods can be combined with multigrid methods.<ref>{{cite book |chapter-url=https://books.google.com/books?id=mtsy6Ci2TRoC&pg=PA140 |author1=Björn Engquist |author2=Olof Runborg |editor1=Timothy J. Barth |editor2=Tony Chan |editor3=Robert Haimes |chapter=Wavelet-based numerical homogenization with applications |title=Multiscale and Multiresolution Methods |isbn=978-3-540-42420-8 |volume=Vol. 20 of Lecture Notes in Computational Science and Engineering |publisher=Springer |year=2002 |page=140 ''ff''}}</ref><ref>{{cite book |url=https://books.google.com/books?id=-og1wD-Nx_wC&q=wavelet+ |author1=U. Trottenberg |author2=C. W. Oosterlee |author3=A. Schüller |title=Multigrid |isbn=978-0-12-701070-0|year=2001 }}</ref> For example, one use of wavelets is to reformulate the finite element approach in terms of a multilevel method.<ref>{{cite book |title=Numerical Analysis of Wavelet Methods |author=Albert Cohen |url=https://books.google.com/books?id=Dz9RnDItrAYC&pg=PA44 |page=44 |publisher=Elsevier |year=2003 |isbn=978-0-444-51124-9}}</ref>

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==Notes==

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== References ==