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Semi-discrete Galerkin approximation method applied to initial boundary value problems for Maxwell's equations in anisotropic, inhomogeneous media

Published online by Cambridge University Press:  14 November 2011

Pekka Neittaanmäki  and
Jukka Saranen
Pekka Neittaanmäki
Affiliation:
Department of Mathematics, University of Jyväskylä, Sammonkatu 6, SF-40100 Jyväskylä 10, Finland
Jukka Saranen
Affiliation:
Department of Mathematics, University of Jyväskylä, Sammonkatu 6, SF-40100 Jyväskylä 10, Finland
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Synopsis

In this paper the semi-discrete Galerkin approximation of initial boundary value problems for Maxwell's equations is analysed. For the electric field a hyperbolic system of equations is first derived. The standard Galerkin method is applied to this system and a priori error estimates are established for the approximation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 1-2 , 1981 , pp. 125 - 133
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Baker, G. A.. Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal. 13 (1976), 564–576.CrossRefGoogle Scholar
2Dupont, T.. L 2-estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10 (1973), 880–889.CrossRefGoogle Scholar
3Duvaut, G. and Lions, J. L.. Inequalities in mechanics and physics. Grundlehren der mathematischen Wissenschaften 219 (Berlin: Springer, 1976).Google Scholar
4Fairweather, G.. The finite element Galerkin methods for differential equations (New York: Marcel Dekker, 1978).Google Scholar
5Martensen, E.. The convergence of the horizontal line method for Maxwell's equations. Math. Methods Appl. Sci. 1 (1979), 101–113.CrossRefGoogle Scholar
6Mehra, L. M.. Zur asymptotischen Verteilung der Eigenwerte des Maxwellschen Randwertproblems. Bonn. Math. Schr. 109 (1978).Google Scholar
7Neittaanmäki, P. and Picard, R.. Error estimates for the finite element approximation to a Maxwell-type boundary value problem. Numer. Funct. Anal. Optim. 2/4 (1980), 267–285.CrossRefGoogle Scholar
8Picard, R.. On convergence and error estimates for the horizontal line method applied to Maxwell's boundary value problem in anisotropic, inhomogeneous media. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 53–59.CrossRefGoogle Scholar
9Saranen, J.. Über die Approximation der Lösungen der Maxwellschen Randwertaufgabe mit der Methode der finiten Elemente. Applicable Anal. 10 (1980), 15–30.CrossRefGoogle Scholar
10Saranen, J.. Some remarks on the convergence of the horizontal line method for Maxwell's equations. Ber. Univ. Jyväskylä Math. Inst. 23 (1980).Google Scholar
11Wheeler, M. F.. A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973), 723–759.CrossRefGoogle Scholar
12Wilcox, C. H.. The mathematical foundations for diffraction theory. In Langer, R. E. (ed.) Electromagnetic Waves (Madison, Wisc: The Univ. of Wisconsin Press, 1962), 65–97.Google Scholar
13Adam, J. C., Gourdin-Serveniere, A., Nedelec, J. C. and Raviart, P. A.. Study of an implicit scheme for integrating Maxwell's equations. Comput. Methods Appl. Mech. Engrg 22 (1980), 327–346.CrossRefGoogle Scholar
14Raviart, P. A. and Thomas, J. M.. A mixed finite element for 2nd order elliptic problems. Lecture Notes in Mathematics 606 (Berlin: Springer, 1977), 292–315.Google Scholar