Non-reflecting boundary conditions for waveguides

Bendali, A.; Guillaume, Ph.

doi:10.1090/S0025-5718-99-01016-9

Non-reflecting boundary conditions for waveguides
HTML articles powered by AMS MathViewer

by A. Bendali and Ph. Guillaume PDF

Math. Comp. 68 (1999), 123-144 Request permission

Abstract:

New non-reflecting boundary conditions are introduced for the solution of the Helmholtz equation in a waveguide. These boundary conditions are perfectly transparent for all propagating modes. They do not require the determination of these propagating modes but only their propagation constants. A quasi-local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes. Related convergence properties to the exact solution and optimal error estimates are established.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
Alvin Bayliss and Eli Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. 33 (1980), no. 6, 707–725. MR 596431, DOI 10.1002/cpa.3160330603
A. Bendali, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I. The continuous problem, Math. Comp. 43 (1984), no. 167, 29–46. MR 744923, DOI 10.1090/S0025-5718-1984-0744923-1
A. Bendali and M. Souilah, Consistency estimates for a double-layer potential and application to the numerical analysis of the boundary-element approximation of acoustic scattering by a penetrable object, Math. Comp. 62 (1994), no. 205, 65–91. MR 1201067, DOI 10.1090/S0025-5718-1994-1201067-9
Dario Bambusi and Luigi Galgani, Some rigorous results on the Pauli-Fierz model of classical electrodynamics, Ann. Inst. H. Poincaré Phys. Théor. 58 (1993), no. 2, 155–171 (English, with English and French summaries). MR 1217117
Christine Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26 (1989), no. 5, 1212–1240 (English, with French summary). MR 1014883, DOI 10.1137/0726068
James H. Bramble and Vidar Thomée, Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl. (4) 101 (1974), 115–152. MR 388805, DOI 10.1007/BF02417101
Z. Bi, K. Wu, C. Wu and J. Litva, A Dispersive Boundary Condition for Microstrip Component Analysis Using The FD-TD Method, IEEE Transactions on Microwave Theory and Techniques, 40 (1992), 774–777.
Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
P. G. Ciarlet and J.-L. Lions (eds.), Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR 1115235
Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 436612, DOI 10.1090/S0025-5718-1977-0436612-4
Björn Engquist and Andrew Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math. 32 (1979), no. 3, 314–358. MR 517938, DOI 10.1002/cpa.3160320303
Charles I. Goldstein, A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains, Math. Comp. 39 (1982), no. 160, 309–324. MR 669632, DOI 10.1090/S0025-5718-1982-0669632-7
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
P. Grisvard, Caractérisation de quelques espaces d’interpolation, Arch. Rational Mech. Anal. 25 (1967), 40–63 (French). MR 213864, DOI 10.1007/BF00281421
Jean-Claude Guillot and Calvin H. Wilcox, Steady-state wave propagation in simple and compound acoustic waveguides, Math. Z. 160 (1978), no. 1, 89–102. MR 496589, DOI 10.1007/BF01182333
Ph. Guillaume and M. Masmoudi, Solution to the time-harmonic Maxwell’s equations in a waveguide, use of higher order derivatives for solving the discrete problem, SIAM J. Numer. Anal., 34 (1997), 1306–1330.
Lloyd N. Trefethen and Laurence Halpern, Well-posedness of one-way wave equations and absorbing boundary conditions, Math. Comp. 47 (1986), no. 176, 421–435. MR 856695, DOI 10.1090/S0025-5718-1986-0856695-2
Robert L. Higdon, Numerical absorbing boundary conditions for the wave equation, Math. Comp. 49 (1987), no. 179, 65–90. MR 890254, DOI 10.1090/S0025-5718-1987-0890254-1
J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, New-York, 1993.
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
M. Lenoir and A. Tounsi, The localized finite element method and its application to the two-dimensional sea-keeping problem, SIAM J. Numer. Anal. 25 (1988), no. 4, 729–752. MR 954784, DOI 10.1137/0725044
E. L. Lindman, “Free-space” boundary conditions for the time-dependent wave equation, J. Comp. Phys. 18 (1975), 66–78.
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
W. Pascher and R. Pregla, Analysis of rectangular waveguide junctions by the method of lines, IEEE Trans. on Microwaves Theory and Techniques 43 (1995), 2649–2653.
C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz and A. Taflove, Ultrawideband Absorbing Boundary Condition for Termination of Waveguiding Structures in FD-TD Simulations, IEEE Microwave and guided wave letters 4 (1994), 344–346.
Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
Allen Taflove, Computational electrodynamics, Artech House, Inc., Boston, MA, 1995. The finite-difference time-domain method; With contributions by Stephen D. Gedney, Faiza S. Lansing, Thomas G. Jurgens, Gregory W. Saewert, Melinda J. Piket-May, Eric T. Thiele and Stephen T. Barnard. MR 1338377
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463, DOI 10.1515/9781400886104
Raj. Mittra (ed.), Computer techniques for electromagnetics, Computer Techniques for Electromagnetics, Vol. 7, Pergamon Press, Oxford-New York-Toronto, Ont., 1973. MR 0462236
Z. Wu and J. Fang, Numerical implementation and performance of perfectly matched layer boundary condition for waveguide structure, IEEE Trans. on Microwaves Theory and Techniques, 43 (1995), 2676–2683.