Skip to main content
Erschienen in: Calcolo 3/2018

01.09.2018

A discontinuous Galerkin method for a time-harmonic eddy current problem

verfasst von: Ana Alonso Rodríguez, Antonio Márquez, Salim Meddahi, Alberto Valli

Erschienen in: Calcolo | Ausgabe 3/2018

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

We introduce and analyze a discontinuous Galerkin method for a time-harmonic eddy current problem formulated in terms of the magnetic field. The scheme is obtained by putting together a DG method for the approximation of the vector field variable representing the magnetic field in the conductor and a DG method for the Laplace equation whose solution is a scalar magnetic potential in the insulator. The transmission conditions linking the two problems are taken into account weakly in the global discontinuous Galerkin scheme. We prove that the numerical method is uniformly stable and obtain quasi-optimal error estimates in the DG-energy norm.
Literatur
1.
Zurück zum Zitat Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRef Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRef
2.
Zurück zum Zitat Außerhofer, S., Bíró, O., Preis, K.: Discontinuous Galerkin formulation for eddy-current problems. COMPEL 28, 1081–1090 (2009)MathSciNetCrossRef Außerhofer, S., Bíró, O., Preis, K.: Discontinuous Galerkin formulation for eddy-current problems. COMPEL 28, 1081–1090 (2009)MathSciNetCrossRef
3.
Zurück zum Zitat Bermúdez, A., Rodríguez, R., Salgado, P.: A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Numer. Anal. 40, 1823–1849 (2002)MathSciNetCrossRef Bermúdez, A., Rodríguez, R., Salgado, P.: A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations. SIAM J. Numer. Anal. 40, 1823–1849 (2002)MathSciNetCrossRef
4.
Zurück zum Zitat Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Heidelberg (2013)CrossRef Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Heidelberg (2013)CrossRef
5.
Zurück zum Zitat Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Heidelberg (2012)CrossRef Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Heidelberg (2012)CrossRef
7.
Zurück zum Zitat Hiptmair, R.: Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J. Numer. Anal. 41, 919–944 (2003)MathSciNetCrossRef Hiptmair, R.: Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM J. Numer. Anal. 41, 919–944 (2003)MathSciNetCrossRef
8.
Zurück zum Zitat Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100, 485–518 (2005)MathSciNetCrossRef Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100, 485–518 (2005)MathSciNetCrossRef
9.
Zurück zum Zitat McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)MATH McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)MATH
10.
Zurück zum Zitat Meddahi, S., Selgas, V.: A mixed-FEM and BEM coupling for a three-dimensional eddy current problem, M2AN Math. Model. Numer. Anal. 37, 291–318 (2003)MathSciNetCrossRef Meddahi, S., Selgas, V.: A mixed-FEM and BEM coupling for a three-dimensional eddy current problem, M2AN Math. Model. Numer. Anal. 37, 291–318 (2003)MathSciNetCrossRef
11.
Zurück zum Zitat Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRef Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRef
12.
13.
Zurück zum Zitat Perugia, I., Schötzau, D.: The \(hp\)-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comp. 72, 1179–1214 (2003)MathSciNetCrossRef Perugia, I., Schötzau, D.: The \(hp\)-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comp. 72, 1179–1214 (2003)MathSciNetCrossRef
14.
Zurück zum Zitat Rodríguez, A.A., Valli, A.: A FEM-BEM approach for electro-magnetostatics and time-harmonic eddy-current problems. Appl. Numer. Math. 59, 2036–2049 (2009)MathSciNetCrossRef Rodríguez, A.A., Valli, A.: A FEM-BEM approach for electro-magnetostatics and time-harmonic eddy-current problems. Appl. Numer. Math. 59, 2036–2049 (2009)MathSciNetCrossRef
15.
Zurück zum Zitat Rodríguez, A.A., Valli, A.: Eddy Current Approximation of Maxwell Equations. Springer, Milan (2010)CrossRef Rodríguez, A.A., Valli, A.: Eddy Current Approximation of Maxwell Equations. Springer, Milan (2010)CrossRef
16.
Zurück zum Zitat Rodríguez, A.A., Bertolazzi, E., Ghiloni, R., Valli, A.: Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems. SIAM J. Numer. Anal. 51, 2380–2402 (2013)MathSciNetCrossRef Rodríguez, A.A., Bertolazzi, E., Ghiloni, R., Valli, A.: Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3D magnetostatic problems. SIAM J. Numer. Anal. 51, 2380–2402 (2013)MathSciNetCrossRef
17.
Zurück zum Zitat Rodríguez, A.A., Meddahi, S., Vall, A.: Coupling DG-FEM and BEM for a time harmonic eddy current problem. In: Bittencourt, M.L., Dumont, N.A., Hesthaven, J.S. (eds.) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016, pp. 147–160. Springer, Cham (2017)CrossRef Rodríguez, A.A., Meddahi, S., Vall, A.: Coupling DG-FEM and BEM for a time harmonic eddy current problem. In: Bittencourt, M.L., Dumont, N.A., Hesthaven, J.S. (eds.) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016, pp. 147–160. Springer, Cham (2017)CrossRef
Metadaten
Titel
A discontinuous Galerkin method for a time-harmonic eddy current problem
verfasst von
Ana Alonso Rodríguez
Antonio Márquez
Salim Meddahi
Alberto Valli
Publikationsdatum
01.09.2018
Verlag
Springer International Publishing
Erschienen in
Calcolo / Ausgabe 3/2018
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-018-0267-2

Weitere Artikel der Ausgabe 3/2018

Calcolo 3/2018 Zur Ausgabe