An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations

Marcella Bonazzoli, Victorita Dolean, Frédéric Hecht, Francesca Rapetti

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell's equations in waveguide configurations.The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell's equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.
LanguageEnglish
Pages1498-1514
Number of pages17
JournalComputers and Mathematics with Applications
Volume75
Early online date6 Dec 2017
DOIs
Publication statusPublished - 1 Mar 2018

Fingerprint

Edge Finite Elements
Vandermonde Matrix
High-order Finite Elements
Maxwell equations
Preconditioning
Maxwell's equations
Waveguide
Harmonic
Higher Order
Finite Element
Schwarz Methods
Higher Order Approximation
Configuration
Curl
Triangular pyramid
Waveguides
Finite Element Approximation
Preconditioner
Open Source
Overlapping

Keywords

  • high order finite elements
  • edge elements
  • Schwarz preconditioners
  • time-harmonic Maxwell’s equations
  • FreeFem++

Cite this

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title = "An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations",
abstract = "In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell's equations in waveguide configurations.The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell's equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.",
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An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations. / Bonazzoli, Marcella; Dolean, Victorita; Hecht, Frédéric; Rapetti, Francesca.

In: Computers and Mathematics with Applications, Vol. 75, 01.03.2018, p. 1498-1514.

Research output: Contribution to journalArticle

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