# Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation

D. Sarmany, Mikhail A. Bochev, Jacobus J.W. van der Vegt, J.G. Verwer

Research output: Book/ReportReport

### Abstract

This article compares the discontinuous Galerkin finite element method (DG-FEM) with the $H(\mathrm{curl})$-conforming FEM in the discretisation of the second-order time-domain Maxwell equations with possibly nonzero conductivity term. While DG-FEM suffers from an increased number of degrees of freedom compared with $H(\mathrm{curl})$-conforming FEM, it has the advantage of a purely block-diagonal mass matrix. This means that, as long as an explicit time-integration scheme is used, it is no longer necessary to solve a linear system at each time step -- a clear advantage over $H(\mathrm{curl})$-conforming FEM. It is known that DG-FEM generally favours high-order methods whereas $H(\mathrm{curl})$-conforming FEM is more suitable for low-order ones. The novelty we provide in this work is a direct comparison of the performance of the two methods when hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$. The motivation behind this choice of basis functions is its growing importance in the development of $p$- and $hp$-adaptive FEMs. The fact that we allow for nonzero conductivity requires special attention with regards to the time-integration methods applied to the semi-discrete systems. High-order polynomial basis warrants the use of high-order time-integration schemes, but existing high-order schemes may suffer from a too severe time-step stability restriction as result of the conductivity term. We investigate several alternatives from the point of view of accuracy, stability and computational work. Finally, we carry out a numerical Fourier analysis to study the dispersion and issipation properties of the semi-discrete DG-FEM scheme and several of the time-integration methods. It is instructive in our approach that the dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.
Language Undefined Enschede Numerical Analysis and Computational Mechanics (NACM) 23 Published - 1 Dec 2009

### Publication series

Name Memorandum / Department of Applied Mathematics Department of Applied Mathematics, University of Twente 1912 1874-4850 1874-4850

### Keywords

• second-order Maxwell wave equation
• numerical time integration
• Discontinuous Galerkin finite element method
• MSC-65L06
• METIS-264211
• MSC-65M60
• IR-68865
• H(curl) conforming finite element methods
• MSC-65M20
• EWI-16951

### Cite this

Sarmany, D., Bochev, M. A., van der Vegt, J. J. W., & Verwer, J. G. (2009). Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation. (Memorandum / Department of Applied Mathematics; No. 1912). Enschede: Numerical Analysis and Computational Mechanics (NACM).
Sarmany, D. ; Bochev, Mikhail A. ; van der Vegt, Jacobus J.W. ; Verwer, J.G./ Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation. Enschede : Numerical Analysis and Computational Mechanics (NACM), 2009. 23 p. (Memorandum / Department of Applied Mathematics; 1912).
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abstract = "This article compares the discontinuous Galerkin finite element method (DG-FEM) with the $H(\mathrm{curl})$-conforming FEM in the discretisation of the second-order time-domain Maxwell equations with possibly nonzero conductivity term. While DG-FEM suffers from an increased number of degrees of freedom compared with $H(\mathrm{curl})$-conforming FEM, it has the advantage of a purely block-diagonal mass matrix. This means that, as long as an explicit time-integration scheme is used, it is no longer necessary to solve a linear system at each time step -- a clear advantage over $H(\mathrm{curl})$-conforming FEM. It is known that DG-FEM generally favours high-order methods whereas $H(\mathrm{curl})$-conforming FEM is more suitable for low-order ones. The novelty we provide in this work is a direct comparison of the performance of the two methods when hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$. The motivation behind this choice of basis functions is its growing importance in the development of $p$- and $hp$-adaptive FEMs. The fact that we allow for nonzero conductivity requires special attention with regards to the time-integration methods applied to the semi-discrete systems. High-order polynomial basis warrants the use of high-order time-integration schemes, but existing high-order schemes may suffer from a too severe time-step stability restriction as result of the conductivity term. We investigate several alternatives from the point of view of accuracy, stability and computational work. Finally, we carry out a numerical Fourier analysis to study the dispersion and issipation properties of the semi-discrete DG-FEM scheme and several of the time-integration methods. It is instructive in our approach that the dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.",
keywords = "second-order Maxwell wave equation, numerical time integration, Discontinuous Galerkin finite element method, MSC-65L06, METIS-264211, MSC-65M60, IR-68865, H(curl) conforming finite element methods, MSC-65M20, EWI-16951",
author = "D. Sarmany and Bochev, {Mikhail A.} and {van der Vegt}, {Jacobus J.W.} and J.G. Verwer",
note = "Please note an alternative spelling of the name of the 2nd author: Botchev or Bochev.",
year = "2009",
month = "12",
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series = "Memorandum / Department of Applied Mathematics",
publisher = "Numerical Analysis and Computational Mechanics (NACM)",
number = "1912",

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Sarmany, D, Bochev, MA, van der Vegt, JJW & Verwer, JG 2009, Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation. Memorandum / Department of Applied Mathematics, no. 1912, Numerical Analysis and Computational Mechanics (NACM), Enschede.

Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation. / Sarmany, D.; Bochev, Mikhail A.; van der Vegt, Jacobus J.W.; Verwer, J.G.

Enschede : Numerical Analysis and Computational Mechanics (NACM), 2009. 23 p. (Memorandum / Department of Applied Mathematics; No. 1912).

Research output: Book/ReportReport

TY - BOOK

T1 - Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation

AU - Sarmany,D.

AU - Bochev,Mikhail A.

AU - van der Vegt,Jacobus J.W.

AU - Verwer,J.G.

N1 - Please note an alternative spelling of the name of the 2nd author: Botchev or Bochev.

PY - 2009/12/1

Y1 - 2009/12/1

N2 - This article compares the discontinuous Galerkin finite element method (DG-FEM) with the $H(\mathrm{curl})$-conforming FEM in the discretisation of the second-order time-domain Maxwell equations with possibly nonzero conductivity term. While DG-FEM suffers from an increased number of degrees of freedom compared with $H(\mathrm{curl})$-conforming FEM, it has the advantage of a purely block-diagonal mass matrix. This means that, as long as an explicit time-integration scheme is used, it is no longer necessary to solve a linear system at each time step -- a clear advantage over $H(\mathrm{curl})$-conforming FEM. It is known that DG-FEM generally favours high-order methods whereas $H(\mathrm{curl})$-conforming FEM is more suitable for low-order ones. The novelty we provide in this work is a direct comparison of the performance of the two methods when hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$. The motivation behind this choice of basis functions is its growing importance in the development of $p$- and $hp$-adaptive FEMs. The fact that we allow for nonzero conductivity requires special attention with regards to the time-integration methods applied to the semi-discrete systems. High-order polynomial basis warrants the use of high-order time-integration schemes, but existing high-order schemes may suffer from a too severe time-step stability restriction as result of the conductivity term. We investigate several alternatives from the point of view of accuracy, stability and computational work. Finally, we carry out a numerical Fourier analysis to study the dispersion and issipation properties of the semi-discrete DG-FEM scheme and several of the time-integration methods. It is instructive in our approach that the dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.

AB - This article compares the discontinuous Galerkin finite element method (DG-FEM) with the $H(\mathrm{curl})$-conforming FEM in the discretisation of the second-order time-domain Maxwell equations with possibly nonzero conductivity term. While DG-FEM suffers from an increased number of degrees of freedom compared with $H(\mathrm{curl})$-conforming FEM, it has the advantage of a purely block-diagonal mass matrix. This means that, as long as an explicit time-integration scheme is used, it is no longer necessary to solve a linear system at each time step -- a clear advantage over $H(\mathrm{curl})$-conforming FEM. It is known that DG-FEM generally favours high-order methods whereas $H(\mathrm{curl})$-conforming FEM is more suitable for low-order ones. The novelty we provide in this work is a direct comparison of the performance of the two methods when hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$. The motivation behind this choice of basis functions is its growing importance in the development of $p$- and $hp$-adaptive FEMs. The fact that we allow for nonzero conductivity requires special attention with regards to the time-integration methods applied to the semi-discrete systems. High-order polynomial basis warrants the use of high-order time-integration schemes, but existing high-order schemes may suffer from a too severe time-step stability restriction as result of the conductivity term. We investigate several alternatives from the point of view of accuracy, stability and computational work. Finally, we carry out a numerical Fourier analysis to study the dispersion and issipation properties of the semi-discrete DG-FEM scheme and several of the time-integration methods. It is instructive in our approach that the dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.

KW - second-order Maxwell wave equation

KW - numerical time integration

KW - Discontinuous Galerkin finite element method

KW - MSC-65L06

KW - METIS-264211

KW - MSC-65M60

KW - IR-68865

KW - H(curl) conforming finite element methods

KW - MSC-65M20

KW - EWI-16951

M3 - Report

T3 - Memorandum / Department of Applied Mathematics

BT - Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation

PB - Numerical Analysis and Computational Mechanics (NACM)

CY - Enschede

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Sarmany D, Bochev MA, van der Vegt JJW, Verwer JG. Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation. Enschede: Numerical Analysis and Computational Mechanics (NACM), 2009. 23 p. (Memorandum / Department of Applied Mathematics; 1912).