Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-12T23:02:31.415Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly elliptic boundary integral equations for electromagnetic transmission problems

Published online by Cambridge University Press:  14 November 2011

Martin Costabel  and
Ernst P. Stephan
Martin Costabel
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, 6100 Darmstadt, Germany
Ernst P. Stephan
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 330332, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study a boundary integral equation method for transmission problems for strongly elliptic differential operators, which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with Galerkin's procedure. The method is shown to work for the vector Helmholtz equation in ℝ3 with electromagnetic transmission conditions. We propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over H1. We analyse the boundary integral equations using the calculus of pseudodifferential operators. Here the concept of the principal symbol is used to derive existence and regularity results for the solution.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 3-4 , 1988 , pp. 271 - 296
Copyright
Copyright © Royal Society of Edinburgh 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Chazarain, J. and Piriou, A.. Introduction à la Thórie des Equations aux Dérivées Partielles Linéaires (Paris: Gauthier–Villars, 1981).Google Scholar
2Colton, P. and Kress, R.. Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics (New York: John Wiley, 1983).Google Scholar
3Costabel, M.. Starke Elliptizitàt von Randintegraloperatoren erster Art (Habilitationsschrift, Technische Hochschule Darmstadt, 1984).Google Scholar
4Costabel, M. and Stephan, E. P.. A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985), 367–413.CrossRefGoogle Scholar
5Costabel, M. and Wendland, W. L.. Strong ellipticity of boundary integral operators. J. Reine Angew. Math. 372 (1986), 39–63.Google Scholar
6Dieudonné, J.. Eléments d'analyse, Vol. 8 (Paris: Gauthier-Villars, 1978).Google Scholar
7Eskin, G. I.. Boundary Problems for Elliptic Pseudo-Differential Operators.Translations of Mathematical Monographs 52 (Providence, R.I.: American Mathematical Society, 1981).Google Scholar
8Girault, V. and Raviart, D. A.. Finite Element Approximation of the Navier–Stokes Equations. Lecture Notes in Mathematics 149 (Berlin: Springer, 1979).Google Scholar
9Giroire, J. and Nedelec, J. C.. Numerical solution of an exterior Neumann problem using a double layer potential. Math. Comp. 32 (1978), 973–990.CrossRefGoogle Scholar
10Giinther, N. M.. Potential Theory and its Applications to Basic Problems of Mathematical Physics (New York: Ungar, 1967).Google Scholar
11Hariharan, S. L. and Stephan, E.. A boundary element method for a two-dimensional interface problem in electromagnetics. Numer. Math. 42 (1983), 311–322.CrossRefGoogle Scholar
12Hildebrandt, S. and Wienholtz, E.. Constructive proofs of representation theorems in separable Hubert space. Comm. Pure. Appl. Math. 17 (1964), 369–373.CrossRefGoogle Scholar
13Hsiao, G. C. and Wendland, W. L.. A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58 (1977), 449–481.CrossRefGoogle Scholar
14Knauff, W. and Kress, R.. On the exterior boundary value problem for the time-harmonic Maxwell equations. J. Math. Anal. Appl. 72 (1979), 215–235.CrossRefGoogle Scholar
15Kress, R. and Roach, G. F.. Transmission problems for the Helmholtz equation. J. Math. Phys. 19 (1978), 1433–1437.CrossRefGoogle Scholar
16Kupradze, W. D.. Randwertaufgaben der Schwingungstheorie und Integralgleichungen (Berlin: VEB DVW, 1956).Google Scholar
17Lions, J. L. and Magenes, E.. Non-Homogeneous Boundary Value Problems and Applications, Vol. I (Berlin: Springer, 1972).Google Scholar
18MacCamy, R. C. and Stephan, E. P.. A boundary element method for an exterior problem for three-dimensional Maxwell's equations. Appl. Anal. 16 (1983), 141–163.CrossRefGoogle Scholar
19MacCamy, R. C. and Stephan, E. P.. Solution procedures for three-dimensional eddy current problems. J. Math. Anal. Appl. 101 (1984), 348–379.CrossRefGoogle Scholar
20Müller, C., Foundations of the Mathematical Theory of Electromagnetic Waves (Berlin: Springer, 1969).CrossRefGoogle Scholar
21TPetersdorff, . von. Randintegralgleichungen für kombinierte Dirichlet-, Neumann und Transmissionsprobleme (Diploma Thesis, Technische Hochschule Darmstadt, 1987).Google Scholar
22Seeley, R. T.. Singular integrals and boundary value problems. Amer. J. Math. 88 (1966), 781–809.CrossRefGoogle Scholar
23Stephan, E. and Wendland, W. L.. Remarks to Galerkin and least squares methods with finite elements for general elliptic problems. Manuscripta Geodaetica 1 (1976), 93–123.Google Scholar
24Stephan, E. P.. Boundary integral equations for magnetic screens in ℝ3. Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), 189–210.CrossRefGoogle Scholar
25Stephan, E. P.. Boundary Integral Equations for Mixed Boundary Value Problems, Screen and Transmission Problems in ℝ3 (Habilitationsschrift, Technische Hochschule Darmstadt, 1984).Google Scholar
26Stephan, E. P.. Solution procedures for interface problems in acoustics and electromagnetics. In Theoretical Acoustics and Numerical Techniques, CISM Courses 277, ed. Filippi, P., pp. 291–348 (Wien–New York: Springer, 1983).CrossRefGoogle Scholar
27Taylor, M.. Pseudodifferential Operators (Princeton: University Press, 1981).CrossRefGoogle Scholar
28Vainikko, G.. On the question of convergence of Galerkin's method. Tartu Rükl. Ul. Toimetised 177 (1965), 148–152.Google Scholar
29Wendland, W. L.. Asymptotic convergence of boundary element methods. In Lectures on the Numerical Solution of Partial Differential Equations, eds. Babuška, I., Liu, T. P. and Osborn, J., pp. 435–528, Lecture Notes #20 (College Park, MD: University of Maryland, 1981).Google Scholar
30Wendland, W. L.. Boundary element methods and their asymptotic convergence. In Theoretical Acoustics and Numerical Techniques, CISM Courses 277, ed. Filippi, P., pp. 135–216 (Wien: Springer, 1983).CrossRefGoogle Scholar
31Wendland, W. L.. On some mathematical aspects of boundary element methods for elliptic problems. In Mathematics of Finite Elements and Applications V, ed. Whiteman, J., pp. 193–227 (London: Academic Press, 1985).CrossRefGoogle Scholar