Non-reflecting boundary conditions for waveguides

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.

[1]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[2]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[3]  C. Wilcox,et al.  Steady-state wave propagation in simple and compound acoustic waveguides , 1976 .

[4]  W. J. Fisher,et al.  Translated from the French , 1920 .

[5]  Reinhold Pregla,et al.  Analysis of rectangular waveguide junctions by the method of lines , 1995 .

[6]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[7]  P. Grisvard,et al.  Caractérisation de quelques espaces d'interpolation , 1967 .

[8]  Jacques Chazarain,et al.  Introduction to the theory of linear partial differential equations , 1982 .

[9]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[10]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[11]  Carretera de Valencia,et al.  The finite element method in electromagnetics , 2000 .

[12]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[13]  Raj Mittra,et al.  Computer techniques for electromagnetics , 1973 .

[14]  A. Taflove,et al.  Ultrawideband absorbing boundary condition for termination of waveguiding structures in FD-TD simulations , 1994, IEEE Microwave and Guided Wave Letters.

[15]  C. Wu.  CHAPTER 5 – Variational and Iterative Methods for Waveguides and Arrays , 1973 .

[16]  Chen Wu,et al.  A dispersive boundary condition for microstrip component analysis using the FD-TD method , 1992 .

[17]  A. Bendali,et al.  Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I - The continuous problem. II - The discrete problem , 1984 .

[18]  L. Galgani,et al.  Some rigorous results on the Pauli-Fierz model of classical electrodynamics , 1993 .

[19]  Jiayuan Fang,et al.  Numerical implementation and performance of perfectly matched layer boundary condition for waveguide structures , 1995 .

[20]  L. Trefethen,et al.  Well-Posedness of one-way wave equations and absorbing boundary conditions , 1986 .

[21]  C. Bernardi.  Optimal finite-element interpolation on curved domains , 1989 .

[22]  A. Bendali,et al.  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 , 1994 .

[23]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[24]  Marc Lenoir,et al.  The Localized Finite Element Method and Its Application to the Two-Dimensional Sea-Keeping Problem , 1988 .

[25]  Robert L. Higdon,et al.  Numerical absorbing boundary conditions for the wave equation , 1987 .

[26]  P. Glendinning.  Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations , 1994 .

[27]  H. Beckert,et al.  J. L. Lions and E. Magenes, Non‐Homogeneous Boundary Value Problems and Applications, II. (Die Grundlehren d. Math. Wissenschaften, Bd. 182). XI + 242 S. Berlin/Heidelberg/New York 1972. Springer‐Verlag. Preis geb. DM 58,— , 1973 .

[28]  Vidar Thomée,et al.  Discrete time Galerkin methods for a parabolic boundary value problem , 1974 .

[29]  M. J. Lighthill,et al.  The eigenvalues of ∇2u + λu=0 when the boundary conditions are given on semi-infinite domains , 1953, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[31]  E. Lindman.  “Free-space” boundary conditions for the time dependent wave equation , 1975 .

[32]  Philippe Guillaume,et al.  Solution to the Time-Harmonic Maxwell's Equations in a Waveguide; Use of Higher-Order Derivatives for Solving the Discrete Problem , 1997 .

[33]  Charles I. Goldstein,et al.  A Finite Element Method for Solving Heimholt/ Type Equations in Waveguides and Other Unbounded Domains* , 2010 .

[34]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .