A coercive combined field integral equation for electromagnetic scattering

SIAM Journal on Numerical Analysis

Volumn 42, Issue 2, 2004, Pages 621-640

  Buffa, A ^a   Hiptmair, R ^b  

^a CNR   (Italy)

^b ETH ZURICH   (Switzerland)

Author keywords

Boundary element methods; Coercivity; Combined field integral equations; Electromagnetic scattering; Galerkin scheme

Indexed keywords

APPROXIMATION THEORY; BOUNDARY ELEMENT METHOD; COERCIVE FORCE; DIFFERENTIAL EQUATIONS; ELECTROMAGNETIC WAVE SCATTERING; GALERKIN METHODS; MATHEMATICAL OPERATORS; PARAMETER ESTIMATION;

ANGULAR FREQUENCY; COMBINED FIELD INTEGRAL EQUATION; GALERKIN SCHEME; LIPSCHITZ-CONTINUOUS BOUNDARY;

INTEGRAL EQUATIONS;

EID: 15544364857     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142903423393     Document Type: Article

References (48)

1. R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. T. APEL, A.-M. SÄNDIG, AND J. WHITEMAN, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci., 19 (1996), pp. 63-85.
3. H. BRAKHAGE AND P. WERNER, Über das Dirichletsche Außenraumproblem für die Helm-holtzsche Schwingungsgleichung, Arch. Math., 16 (1965), pp. 325-329.
4. F. BREZZI, J. DOUGLAS, AND D. MARINI, Two families of mixed finite elements for 2nd order elliptic problems, Numer. Math., 47 (1985), pp. 217-235.
5. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.
6. A. BUFFA, Hodge decompositions on the boundary of a polyhedron: The multiconnected case, Math. Models Meth. Appl. Sci., 11 (2001), pp. 1491-1504.
7. A. BUFFA, Traces theorems on non-smmoth boundaries for functional spaces related to Maxwell equations: An overview, in Computational Electromagnetics, Lecture Notes Comput. Sci. Engrg. 28, C. Carstensen, S. Funken, W. Hackbusch, R. Hoppe, and P. Monk, eds., Springer, Berlin, 2003, pp. 23-34.
8. A. BUFFA AND S. CHRISTIANSEN, The electric field integral equation on Lipschitz screens: Definition and numerical approximation, Numer. Math., 94 (2003), pp. 229-267.
9. A. BUFFA AND P. CIARLET, On traces for functional spaces related to Maxwell's equations. Part I: An integration by parts formula in Lipschitz polyhedra., Math. Methods Appl. Sci., 24 (2001), pp. 9-30.
10. A. BUFFA AND P. CIARLET, On traces for functional spaces related to Maxwell's equations. Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications, Math. Methods Appl. Sci., 24 (2001), pp. 31-48.
11. A. BUFFA, M. COSTABEL, AND M. DAUGE, Algebraic Convergence for Edge Elements in Polyhedral Domains, Tech. report 28-PV, IMATI-CNR, Pavia, Italy, 2003.
12. A. BUFFA, M. COSTABEL, AND C. SCHWAB, Boundary element methods for Maxwell's equations on non-smooth domains, Numer. Math., 92 (2002), pp. 679-710.
13. A. BUFFA, M. COSTABEL, AND D. SHEEN, On traces for H(curl,Ω) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), pp. 845-867.
14. A. BUFFA AND R. HIPTMAIR, Regularized Combined Field Integral Equations, Tech. report 2003-06, SAM, ETH Zürich, Zürich, Switzerland, 2003.
15. A. BUFFA, R. HIPTMAIR, T. VON PETERSDORFF, AND C. SCHWAB, Boundary element methods for Maxwell equations on Lipschitz domains, Numer. Math., 95 (2003), pp. 459-485.
16. A. BUFFA AND S. SAUTER, Stabilization of the Acoustic Single Layer Potential on Non-smooth Domains, Tech. report 19-03, Institut für Mathematik, Universität Zürich, Zürich, Switzerland, 2003.
17. A. BURTON AND G. MILLER, The application of integral methods for the numerical solution of boundary value problems, Proc. Roy. Soc. London Ser. A, 323 (1971), pp. 201-210.
18. M. CESSENAT, Mathematical Methods in Electromagnetism, Ser. Adv. Math. Appl. Sci. 41, World Scientific, Singapore, 1996.
19. S. CHRISTIANSEN, Mixed Boundary Element Method for Eddy Current Problems, Research report 2002-16, SAM, ETH Zürich, Zürich, Switzerland, 2002.
20. S. CHRISTIANSEN, Discrete Fredholm properties and convergence estimates for the electric field integral equation, Math. Comp., 73 (2004), pp. 143-167.
21. S. H. CHRISTIANSEN AND J.-C. NÉDÉLEC, A preconditioner for the electric field integral equation based on Calderon formulas, SIAM J. Numer. Anal., 40 (2002), pp. 1100-1135.
22. P. CIARLET, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978.
23. D. COLTON AND R. KRESS, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Appl. Math. Sci. 93, Springer, Heidelberg, 1998.
24. M. COSTABEL, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), pp. 365-368.
25. M. COSTABEL AND M. DAUGE, Singularities of Maxwell's equations on polyhedral domains, in Analysis, Numerics and Applications of Differential and Integral Equations, Pitman Res. Notes Math. Ser. 379, M. Bach, ed., Longman, Harlow, 1998, pp. 69-76.
26. G. FAIRWEATHER AND A. KARAGEORGHIS, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69-95.
27. 3,), Ph.D. thesis, Fakultät für Mathematik, Universtität Karlsruhe, Karlsruhe, Germany, 1997.
28. V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986.
29. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
30. C. HAZARD AND M. LENOIR, On the solution of time-harmonic scattering problems for Maxwell's equations, SIAM J. Math. Anal., 27 (1996), pp. 1597-1630.
31. R. HIPTMAIR, Finite elements in computational electromagnetism, in Acta Numerica, Cambridge University Press, Cambridge, UK, 2002, pp. 237-339.
32. R. HIPTMAIR, Coupling of finite elements and boundary elements in electromagnetic scattering, SIAM J. Numer. Anal., 41 (2003), pp. 919-944.
33. R. HIFTMAIR AND C. SCHWAB, Natural boundary element methods for the electric field integral equation on polyhedra, SIAM J. Numer. Anal., 40 (2002), pp. 66-86.
34. G. HSIAO, Mathematical foundations for the boundary field equation methods in acoustic and electromagnetic scattering, in Analytical and Computational Methods in Scattering and Applied Mathematics. A Volume in the Memory of Ralph Ellis Kleinman, Chapman Hall/CRC Res. Notes Math. 417, F. Santosa and I. Stakgold, eds., Chapman and Hall/CRC, Boca Raton, PL, 2000, pp. 149-163.
35. D. JERISON AND C. KENIG, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), pp. 161-219.
36. R. KRESS, On the boundary operator in electromagnetic scattering, Proc. Roy. Soc. Edinburgh Sect. A, 103 (1986), pp. 91-98.
37. R. KRESS, Linear Integral Equations, Appl. Math. Sci. 82, Springer, Berlin, 1989.
38. R. KRESS AND W. T. SPASSOV, On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation, Numer. Math., 42 (1983), pp. 77-95.
39. J. LIONS AND F. MAGENES, Nonhomogeneous Boundary Value Problems and Applications, Springer, Berlin, 1972.
40. W. MCLEAN, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000.
41. J. NECǍS, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967.
42. J.-C. NÉDÉLEC, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Appl. Math. Sci. 144, Springer, Berlin, 2001.
43. O. PANICH, On the question of the solvability of the exterior boundary-value problems for the wave equation and maxwell's equations, Uspekhi Mat. Nauk., 20 (1965), pp. 221-226 (in Russian).
44. P. A. RAVIART AND J. M. THOMAS, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, Springer, New York, 1977, pp. 292-315.
45. M. REISSEL, On a transmission boundary-value problem for the time-harmonic Maxwell equations without displacement currents, SIAM J. Math. Anal., 24 (1993), pp. 1440-1457.
46. J. STRATTON AND L. CHU, Diffraction theory of electromagnetic waves, Phys. Rev., 56 (1939), pp. 99-107.
47. D. WILTON, Review of current status and trends in the use of integral equations in computational electromagnetics, Electromagnetics, 12 (1992), pp. 287-341.
48. J. XU AND L. ZIKATANOV, Some observations on Babuška and Brezzi theories, Numer. Math., 94 (2003), pp. 195-202.