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

In: Multiscale, Nonlinear and Adaptive Approximation

Author

Listed:
  • Jinchao Xu

    (Pennsylvania State University, Department of Mathematics
    Peking University, LMAM, The School of Mathematical Sciences)

  • Long Chen

    (University of California at Irvine, Department of Mathematics)

  • Ricardo H. Nochetto

    (University of Maryland, Department of Mathematics and Institute for Physical Science and Technology)

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.

Suggested Citation

  • Jinchao Xu & Long Chen & Ricardo H. Nochetto, 2009. "Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids," Springer Books, in: Ronald DeVore & Angela Kunoth (ed.), Multiscale, Nonlinear and Adaptive Approximation, pages 599-659, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-03413-8_14
    DOI: 10.1007/978-3-642-03413-8_14
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