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

Jinchao Xu; Long Chen; Ricardo H. Nochetto

doi:10.1007/978-3-642-03413-8_14

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

Jinchao Xu, Long Chen, Ricardo H. Nochetto

Research output: Chapter in Book/Report/Conference proceeding › Chapter

45 Scopus citations

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. © Springer-Verlag Berlin Heidelberg 2009.

Original language	English (US)
Title of host publication	Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday
Publisher	Springer Berlin Heidelberg
Pages	599-659
Number of pages	61
ISBN (Print)	9783642034121
DOIs	https://doi.org/10.1007/978-3-642-03413-8_14
State	Published - Dec 1 2009
Externally published	Yes

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Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids. / Xu, Jinchao; Chen, Long; Nochetto, Ricardo H.
Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday. Springer Berlin Heidelberg, 2009. p. 599-659.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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AU - Xu, Jinchao

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AU - Nochetto, Ricardo H.

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PY - 2009/12/1

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AB - 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. © Springer-Verlag Berlin Heidelberg 2009.

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

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