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

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

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 [1] 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. [1] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999.
Original languageUndefined
Place of PublicationEnschede
PublisherDepartment of Applied Mathematics, University of Twente
Number of pages23
StatePublished - Jul 2006

Publication series

NameApplied Mathematics Memoranda
PublisherDepartment of Applied Mathematics, University of Twente
No.1806
ISSN (Print)0169-2690

Fingerprint

Maxwell's equations
Finite element solution
Frequency domain
kernel
Edge finite elements
Iterative solver
Iterative solvers
Krylov subspace
Multigrid method
Domain decomposition
Iterative scheme
Preconditioner
Rotor
Test problems
Conductivity
Jump
Projection
Finite element
Operator
Demonstrate

Keywords

  • METIS-237884
  • Nédélec vector finite elements
  • MSC-65N30
  • MSC-65N22
  • EWI-8960
  • hierarchical preconditioners
  • kernel of the rotor operator
  • MSC-65N55
  • IR-66839
  • Domain decomposition
  • multilevel iterative solvers

Cite this

Nechaev, O. V., Shurina, E. P., & Bochev, M. A. (2006). Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation. (Applied Mathematics Memoranda; No. 1806). Enschede: Department of Applied Mathematics, University of Twente.

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

Enschede : Department of Applied Mathematics, University of Twente, 2006. 23 p. (Applied Mathematics Memoranda; No. 1806).

Research output: ProfessionalReport

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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 [1] 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. [1] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999.",
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Nechaev, OV, Shurina, EP & Bochev, MA 2006, Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation. Applied Mathematics Memoranda, no. 1806, Department of Applied Mathematics, University of Twente, Enschede.

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

Enschede : Department of Applied Mathematics, University of Twente, 2006. 23 p. (Applied Mathematics Memoranda; No. 1806).

Research output: ProfessionalReport

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N2 - 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 [1] 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. [1] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999.

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Nechaev OV, Shurina EP, Bochev MA. Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation. Enschede: Department of Applied Mathematics, University of Twente, 2006. 23 p. (Applied Mathematics Memoranda; 1806).