Application of a complete radiation boundary condition for the Helmholtz equation in locally perturbed waveguides. (English) Zbl 1428.78021

Summary: This paper deals with an application of a complete radiation boundary condition (CRBC) for the Helmholtz equation in locally perturbed waveguides. The CRBC, one of efficient high-order absorbing boundary conditions, has been analyzed in straight waveguides in [T. Hagstrom and S. Kim, Numer. Math. 141, No. 4, 917–966 (2019; Zbl 1412.65187)]. In this paper, we apply CRBC to the Helmholtz equation posed in locally perturbed waveguides and establish the well-posedness of the problem and convergence of CRBC approximate solutions. The new CRBC proposed in this paper improves the one studied in [loc. cit.] in two aspects. The first one is that the new CRBC involves more damping parameters with the same computational cost as that of CRBC in [loc. cit.], which results in 50% smaller reflection errors. The second one is that the new CRBC takes a Neumann terminal condition of three term recurrence relations of auxiliary variables instead of a Dirichlet terminal condition used in [loc. cit.] so that it can treat cutoff modes effectively. Finally, we present numerical experiments illustrating the convergence theory.


78A50 Antennas, waveguides in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs


Zbl 1412.65187


Full Text: DOI


[1] Bécache, E.; Bonnet-Ben Dhia, A.-S.; Legendre, G., Perfectly matched layers for the convected Helmholtz equation, SIAM J. Numer. Anal., 42, 1, 409-433 (2004) · Zbl 1089.76045
[2] Kim, S., Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides, ESAIM Math. Model. Numer. Anal., 53, 4, 1191-1222 (2019) · Zbl 1455.78014
[3] Bendali, A.; Guillaume, P., Non-reflecting boundary conditions for waveguides, Math. Comp., 68, 225, 123-144 (1999) · Zbl 0907.35127
[4] Goldstein, C. I., A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains, Math. Comp., 39, 160, 309-324 (1982) · Zbl 0493.65046
[5] Harari, I.; Patlashenko, I.; Givoli, D., Dirichlet-to-Neumann maps for unbounded wave guides, J. Comput. Phys., 143, 1, 200-223 (1998) · Zbl 0928.65140
[6] Mitsoudis, D. A.; Makridakis, C.; Plexousakis, M., Helmholtz Equation with artificial boundary conditions in a two-dimensional waveguide, SIAM J. Math. Anal., 44, 6, 4320-4344 (2012) · Zbl 1312.76057
[7] Druskin, V.; Güttel, S.; Knizhnerman, L., Near-optimal perfectly matched layers for indefinite Helmholtz problems, SIAM Rev., 58, 1, 90-116 (2016) · Zbl 1344.35009
[8] Druskin, V.; Moskow, S., Three-point finite-difference schemes, Padé and the spectral Galerkin method. I One-sided impedance approximation, Math. Comp., 71, 239, 995-1019 (2002), (electronic) · Zbl 0997.65117
[9] Ingerman, D.; Druskin, V.; Knizherman, L., Optimal finite difference grids and rational approximations of the square root. I. elliptic functions, Comm. Pure Appl. Math., 53, 1039-1066 (2000) · Zbl 1021.65051
[10] Hagstrom, T.; Kim, S., Complete radiation boundary conditions for the Helmholtz equation I: waveguides, Numer. Math., 141, 4, 917-966 (2019) · Zbl 1412.65187
[11] Kim, S., Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions, J. Math. Anal. Appl., 410, 1, 275-291 (2014) · Zbl 1316.35094
[12] Higdon, R. L., Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp., 47, 176, 437-459 (1986) · Zbl 0609.35052
[13] Higdon, R. L., Numerical absorbing boundary conditions for the wave equation, Math. Comp., 49, 179, 65-90 (1987) · Zbl 0654.65083
[14] Druskin, V.; Knizhnerman, L., Gaussian spectral rules for the three-point second differences. I. a two-point positive definite problem in a semi-infinite domain, SIAM J. Numer. Anal., 37, 2, 403-422 (2000), (electronic) · Zbl 0947.65127
[15] Davies, E. B.; Parnovski, L., Trapped modes in acoustic waveguides, Quart. J. Mech. Appl. Math., 51, 3, 477-492 (1998) · Zbl 0908.76083
[16] Evans, D. V.; Levitin, M.; Vassiliev, D., Existence theorems for trapped modes, J. Fluid Mech., 261, 21-31 (1994) · Zbl 0804.76075
[17] Jones, D. S., The eigenvalues of \(\nabla^2 u + \lambda u = 0\) when the boundary conditions are given on semi-infinite domains, Proc. Camb. Phil. Soc., 49, 668-684 (1953) · Zbl 0051.07704
[18] Linton, C. M.; McIver, P., Embedded trapped modes in water waves and acoustics, Wave Motion, 45, 1-2, 16-29 (2007) · Zbl 1231.76046
[19] Krejčiřík, D.; Kříz, J., On the spectrum of curved planar waveguides, Publ. Res. Inst. Math. Sci., 41, 3, 757-791 (2005) · Zbl 1113.35143
[20] Borisov, D.; Exner, P.; Gadyl’shin, R.; Krejčiřík, D., Bound states in weakly deformed strips and layers, Ann. Henri Poincaré, 2, 3, 553-572 (2001) · Zbl 1043.35046
[21] Exner, P.; Vugalter, S. A., Bound states in a locally deformed waveguide: the critical case, Lett. Math. Phys., 39, 1, 59-68 (1997) · Zbl 0871.35067
[22] Exner, P.; Šeba, P.; Tater, M.; Vaněk, D., Bound states and scattering in quantum waveguides coupled laterally through a boundary window, J. Math. Phys., 37, 10, 4867-4887 (1996) · Zbl 0883.35085
[23] Courant, R.; Hilbert, D., Methods of Mathematical Physics, vol. 1 (1953), Wiley-Interscience: Wiley-Interscience New York
[24] Petrushev, P.; Popov, V., Rational Approximation of Real Functions, 28 of Encyclopedia of Mathematics (1987), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0644.41010
[25] Akhiezer, N. I., Elements of the Theory of Elliptic Functions (1990), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0694.33001
[26] Medovikov, A. A.; Lebedev, V. I., Variable time steps optimization of \(L_\omega \)-stable Crank-Nicolson method, Russian J. Numer. Anal. Math. Modelling, 20, 3, 283-303 (2005) · Zbl 1080.65077
[27] Beckermann, B.; Townsend, A., On the singular values of matrices with displacement structure, SIAM J. Matrix Anal. Appl., 38, 4, 1227-1248 (2017) · Zbl 1386.15024
[28] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 (1964), Dover: Dover New York · Zbl 0171.38503
[29] Bangerth, W.; Hartmann, R.; Kanschat, G., Deal.II—a general-purpose object-oriented finite element library, ACM Trans. Math. Software, 33, 4, 24 (2007) · Zbl 1365.65248
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