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Volume 27, Issue 5
Local Multigrid in H(Curl)

Ralf Hiptmair & Weiying Zheng

DOI: 10.4208/jcm.2009.27.5.012

J. Comp. Math., 27 (2009), pp. 573-603.

Published online: 2009-10

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  • Abstract

We consider $\boldsymbol{H}$(curl, $Ω$)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a $H^1(Ω)$-context along with local discrete Helmholtz-type decompositions of the edge element space.

  • Keywords

Edge elements, Local multigrid, Stable multilevel splittings, Subspace correction theory, Regular decompositions of $\boldsymbol{H}(curl, Ω)$, Helmholtz-type decompositions, Local mesh refinement.

  • AMS Subject Headings

65N30, 65N55, 78A25.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-573, author = {Ralf Hiptmair and Weiying Zheng}, title = {Local Multigrid in H(Curl)}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {5}, pages = {573--603}, abstract = {

We consider $\boldsymbol{H}$(curl, $Ω$)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a $H^1(Ω)$-context along with local discrete Helmholtz-type decompositions of the edge element space.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.27.5.012}, url = {http://global-sci.org/intro/article_detail/jcm/8591.html} }
TY - JOUR T1 - Local Multigrid in H(Curl) AU - Ralf Hiptmair & Weiying Zheng JO - Journal of Computational Mathematics VL - 5 SP - 573 EP - 603 PY - 2009 DA - 2009/10 SN - 27 DO - http://doi.org/10.4208/jcm.2009.27.5.012 UR - https://global-sci.org/intro/article_detail/jcm/8591.html KW - Edge elements, Local multigrid, Stable multilevel splittings, Subspace correction theory, Regular decompositions of $\boldsymbol{H}(curl, Ω)$, Helmholtz-type decompositions, Local mesh refinement. AB -

We consider $\boldsymbol{H}$(curl, $Ω$)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a $H^1(Ω)$-context along with local discrete Helmholtz-type decompositions of the edge element space.

Ralf Hiptmair and Weiying Zheng. (2009). Local Multigrid in H(Curl). Journal of Computational Mathematics. 27 (5). 573-603. doi:10.4208/jcm.2009.27.5.012
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