Dispersive properties of high order nedelec/edge element approximation of the time-harmonic Maxwell equations

Mark Ainsworth

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Abstract

The dispersive behaviour of high-order Næ#169;dæ#169;lec element approximation of the time harmonic Maxwell equations at a prescribed temporal frequency ω on tensor-product meshes of size h is analysed. A simple argument is presented, showing that the discrete dispersion relation may be expressed in terms of that for the approximation of the scalar Helmholtz equation in one dimension. An explicit form for the one-dimensional dispersion relation is given, valid for arbitrary order of approximation. Explicit expressions for the leading term in the error in the regimes where ωh is small, showing that the dispersion relation is accurate to order 2p for a pth-order method; and in the high-wavenumber limit where 1«ωh, showing that in this case the error reduces at a super-exponential rate once the order of approximation exceeds a certain threshold, which is given explicitly.
Original languageEnglish
Pages (from-to)471-493
Number of pages22
JournalPhilosophical Transactions A: Mathematical, Physical and Engineering Sciences
Volume362
Issue number1816
DOIs
Publication statusPublished - Mar 2004

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Edge Elements
Maxwell equations
Dispersion Relation
Maxwell's equations
Maxwell equation
Order of Approximation
Harmonic
Higher Order
harmonics
Approximation
approximation
Helmholtz equation
Helmholtz Equation
Tensor Product
One Dimension
Tensors
Helmholtz equations
Exceed
Scalar
Mesh

Keywords

  • edge finite element
  • numerical dispersion
  • discrete dispersion relation

Cite this

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AB - The dispersive behaviour of high-order Næ#169;dæ#169;lec element approximation of the time harmonic Maxwell equations at a prescribed temporal frequency ω on tensor-product meshes of size h is analysed. A simple argument is presented, showing that the discrete dispersion relation may be expressed in terms of that for the approximation of the scalar Helmholtz equation in one dimension. An explicit form for the one-dimensional dispersion relation is given, valid for arbitrary order of approximation. Explicit expressions for the leading term in the error in the regimes where ωh is small, showing that the dispersion relation is accurate to order 2p for a pth-order method; and in the high-wavenumber limit where 1«ωh, showing that in this case the error reduces at a super-exponential rate once the order of approximation exceeds a certain threshold, which is given explicitly.

KW - edge finite element

KW - numerical dispersion

KW - discrete dispersion relation

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JF - Proceedings A: Mathematical, Physical and Engineering Sciences

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