Top

Journal of Scientific Computing

Published in:

01-10-2015

A Weak Galerkin Finite Element Method for the Maxwell Equations

Authors: Lin Mu, Junping Wang, Xiu Ye, Shangyou Zhang

Published in: Journal of Scientific Computing | Issue 1/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either \(H^1\)-like or \(L^2\) and \(L^2\)-like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.

previous article The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes

next article -matrix Accelerated Second Moment Analysis for Potentials with Rough Correlation

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Bossavit, A.: Computational Electromagnetism. Academic Press, San Diego (1998)MATH

Brenner, S., Li, F., Sung, L.: A locally divergence-free interior penalty method for two-dimensional curl–curl problems. SIAM J. Numer. Anal. 42, 1190–1211 (2008)MathSciNetCrossRef

Brenner, S., Cui, J., Nan, Z., Sung, L.: Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations. Math. Comput. 81, 643–659 (2012)MATHMathSciNetCrossRef

Cai, W.: Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport. Cambridge University Press, New York (2013)

Cai, W.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)

Houston, P., Perugia, I., Schotzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator. Technical Report 02-16, University of Basel, Department of Mathematics, Basel, Switzerland (2002)

Houston, P., Perugia, I., Schotzau, D.: hp-DGFEM for Maxwell’s equations. In: Brezzi, F., Buffa, A., Corsaro, S., Murli, A. (eds.) Numerical Mathematics and Advanced Applications: ENUMATH 2001, pp. 785–794. Springer, Berlin (2003)CrossRef

Houston, P., Perugia, I., Schotzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42, 434–459 (2004)MATHMathSciNetCrossRef

Jin, J.: The Finite Element Method in Electromagnetics, 2nd edn. Wiley, New York (2002)MATH

10.

Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)MATHCrossRef

11.

Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods Partial Differ. Equ. 30, 1003–1029 (2014)

12.

Nedelec, J.: Mixed finite elements in R3. Numer. Math. 35, 315–341 (1980)MATHMathSciNetCrossRef

13.

Nedelec, J.: A new family of mixed finite elements in R3. Numer. Math. 50, 57–81 (1986)MATHMathSciNetCrossRef

14.

Perugia, I., Schotzau, D.: The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72, 1179–1214 (2003)MATHMathSciNetCrossRef

15.

Perugia, I., Schotzau, D., Monk, P.: Stabilized interior penalty methods for the time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Eng. 191, 4675–4697 (2002)MATHMathSciNetCrossRef

16.

Vardapetyan, L., Demkowicz, L.: hp-adaptive nite elements in electromagnetics. Comput. Methods Appl. Mech. Eng. 169, 331–344 (1999)MATHMathSciNetCrossRef

17.

Wang, C., Wang, J.: An efficient numerical scheme for the Biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. Comput. Math. Appl. 68, 2314–2330 (2014)

18.

Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)MATHMathSciNetCrossRef

19.

Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83, 2101–2126 (2014)

20.

Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. arXiv:1302.2707v1

Title: A Weak Galerkin Finite Element Method for the Maxwell Equations
Authors: Lin Mu
Junping Wang
Xiu Ye
Shangyou Zhang
Publication date: 01-10-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-014-9964-4

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2015

A Maximum-Principle-Satisfying High-Order Finite Volume Compact WENO Scheme for Scalar Conservation Laws with Applications in Incompressible Flows

Second-Order Stable Finite Difference Schemes for the Time-Fractional Diffusion-Wave Equation

Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation

Towards Smooth Particle Methods Without Smoothing

Strong Stability Preserving General Linear Methods

Numerical Approximation of the Fractional Laplacian via $$hp$$ h p -finite Elements, with an Application to Image Denoising

Premium Partner