Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation

O.V. Nechaev, E.P. Shurina, Mikhail A. Bochev

  • 8 Citations

Abstract

In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1) (1999) 204–225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements. The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.
Original languageEnglish
Pages (from-to)2346-2362
Number of pages23
JournalComputers and mathematics with applications
Volume55
Issue number274/10
DOIs
StatePublished - 2008

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Maxwell equations
Rotors
Decomposition

Keywords

  • EWI-12880
  • MSC-65N22
  • MSC-65N30
  • MSC-65N55
  • Domain decomposition
  • IR-62360
  • Nédélec vector finite elements
  • kernel of the rotor operator
  • hierarchical preconditioners
  • METIS-251009
  • multilevel iterative solvers

Cite this

Nechaev, O.V.; Shurina, E.P.; Bochev, Mikhail A. / Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation.

In: Computers and mathematics with applications, Vol. 55, No. 274/10, 2008, p. 2346-2362.

Research output: Scientific - peer-reviewArticle

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abstract = "In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1) (1999) 204–225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements. The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.",
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Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation. / Nechaev, O.V.; Shurina, E.P.; Bochev, Mikhail A.

In: Computers and mathematics with applications, Vol. 55, No. 274/10, 2008, p. 2346-2362.

Research output: Scientific - peer-reviewArticle

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T1 - Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation

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