Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Conference paper

Abstract

We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids.

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.LMAM, The School of Mathematical SciencesPeking UniversityBeijingChina
  3. 3.Department of MathematicsUniversity of California at IrvineIrvineUSA
  4. 4.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

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