Numerische Mathematik

, Volume 134, Issue 3, pp 637–666 | Cite as

A nearly optimal multigrid method for general unstructured grids

  • Lars Grasedyck
  • Lu WangEmail author
  • Jinchao Xu


In this paper, we develop a multigrid method on unstructured shape-regular grids. For a general shape-regular unstructured grid of \({\mathcal O}(N)\) elements, we present a construction of an auxiliary coarse grid hierarchy on which a geometric multigrid method can be applied together with a smoothing on the original grid by using the auxiliary space preconditioning technique. Such a construction is realized by a cluster tree which can be obtained in \({\mathcal O}(N\log N)\) operations for a grid of N elements. This tree structure in turn is used for the definition of the grid hierarchy from coarse to fine. For the constructed grid hierarchy we prove that the convergence rate of the multigrid preconditioned CG for an elliptic PDE is \(1 - {\mathcal O}({1}/{\log N})\). Numerical experiments confirm the theoretical bounds and show that the total complexity is in \({\mathcal O}(N\log N)\).


Clustering Multigrid Auxiliary space Finite elements 

Mathematics Subject Classification

65N22 65F10 65N30 65N55 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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