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Boundary integral equations for magnetic screens in ℝ3*

Published online by Cambridge University Press:  14 November 2011

Ernst P. Stephan
Ernst P. Stephan
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A.
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Synopsis

A boundary integral method is developed for the scattering of electromagnetic waves at thin obstacles. The exterior boundary value problem for the vector Helmholtz equation with given Neumann data on an open surface piece (screen S) is converted into a system of integral equations for the jumps of the tangential component of the field and its divergence across the screen. A slight modification of the Cauchy data yields a strongly elliptic system of pseudodifferential equations on S which can therefore be used for numerical computations using Galerkin's procedure. The resulting boundary integral equations are analysed using pseudodifferential operator calculus. The principal symbol concept, together with the Wiener–Hopf technique, are used to derive existence and regularity results for the solutions to the boundary integral equations. Quasi-optimal error estimates in the energy norm are given for the numerical scheme.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 189 - 210
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Agmon, S., Douglis, A. and Nirenberg, L.. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Comm. Pure Appl. Math. 12 (1959), 623–727; II. Comm. Pure Appl. Math. 17 (1964), 35–92.CrossRefGoogle Scholar
2Colton, D. and Kress, R.. Integral Equation Methods in Scattering Theory (New York: John Wiley and Sons, 1983).Google Scholar
3Costabel, M. and Stephan, E.. A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985), 367–413.CrossRefGoogle Scholar
4Costabel, M. and Stephan, E.. Strongly elliptic boundary integral equations for electromagnetic transmission problems. In preparation.Google Scholar
5Durand, M.. Layer potentials and boundary value problems for the Helmholtz equation in the complement of a thin obstacle. Math. Methods Appl. Sci. 5 (1983), 389–421.CrossRefGoogle Scholar
6Duvaut, G. and Lions, J. L.. Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften 219 (Berlin: Springer, 1976).CrossRefGoogle Scholar
7Eskin, G. I.. Boundary Problems for Elliptic Pseudo–Differential Operators. Transl. of Math. Mon. 52. (Providence, R.I.: American Mathematical Society, 1981).Google Scholar
8Hildebrandt, St. and Wienholtz, E.. Constructive proofs of representation theorems in separable Hilbert space. Comm. Pure Appl. Math. 17 (1964), 369–373.CrossRefGoogle Scholar
9Höormander, L.. Linear Partial Differential Operators (Berlin: Springer, 1969).CrossRefGoogle Scholar
10Knauff, 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
11Lions, J. L. and Magenes, E.. Non–Homogeneous Boundary Value Problems and Applications I. (Berlin: Springer, 1972).Google Scholar
12MacCamy, R. C. and Stephan, E.. Solution procedures for three–dimensional eddy current problems. J. Math. Anal. Appl. 101 (1984), 348–379.CrossRefGoogle Scholar
13MacCamy, R. C. and Stephan, E.. A boundary element method for an exterior problem for three–dimensional Maxwell's equations. Applicable Anal. 16 (1983), 141–163.CrossRefGoogle Scholar
14Müller, C.. Foundations of the Mathematical Theory of Electromagnetic Waves. (Berlin: Springer, 1969).CrossRefGoogle Scholar
15Stephan, E.. Solution procedures for interface problems in acoustics and electromagnetics. In CISM, Courses and Lectures 277, 291–348. (Vienna: Springer, 1983).Google Scholar
16Stephan, E.. Boundary integral equations for mixed boundary value problems, screen and transmission problems in ℝ3. Habilitationsschrift. THD–Preprint 848, Darmstadt, 1984.Google Scholar
17Stephan, E. and Wendland, W. L.. Remarks to Galerkin and least squares methods with finite elements for general elliptic problems. Lecture Notes in Mathematics 564, 461–471 (Berlin: Springer, 1976). Manuscripta Geodaetica 1 (1976), 93–123.Google Scholar
18Stephan, E. and Wendland, W. L.. An augmented Galerkin procedure for the boundary integral method applied to two–dimensional screen and crack problems. Applicable Anal. 18 (1984), 183–219.CrossRefGoogle Scholar
19Taylor, M.. Pseudodifferential Operators. (Princeton: University Press, 1981).CrossRefGoogle Scholar
20Wendland, W. L.. On the asymptotic convergence of some boundary element methods. Mafelap IV, 281–312 (ed. Whiteman, J..) (London: Academic Press, 1982).Google Scholar
21Wendland, W. L.. Boundary element methods and their asymptotic convergence. In CISM, Courses and Lectures 277, 135–216 (Vienna: Springer, 1983).Google Scholar