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Complete radiation boundary conditions for the Helmholtz equation II: domains with corners

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This paper continues Part I (Hagstrom and Kim in Numer Math 141(4):917–966, 2019) of the investigation on the complete radiation boundary condition (CRBC) in waveguides. In this paper, we propose corner compatibility conditions for CRBC applied to the Helmholtz equation posed in \(\mathbb {R}^2\). Since CRBC is developed as a high-order absorbing boundary condition approximating the radiation condition by using rational functions via the cross-sectional Fourier analysis, it is well-studied and its accurate performance is validated on a straight/planar fictitious boundary in waveguides. However in the presence of corners on artificial absorbing boundaries such as boundaries of rectangular domains, a special treatment for corner conditions is required. We design and validate the accurate CRBC with the corner compatibility conditions on rectangular domains. We also analyze the existence and uniqueness of solutions to the Helmholtz equation coupled with CRBC with the corner compatibility conditions. Finally, numerical experiments illustrating the accuracy of CRBC will be presented.

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References

  1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2(2), 151–218 (1975)

    MathSciNet  MATH  Google Scholar 

  2. Akhiezer, N.I.: Elements of the theory of elliptic functions. American Mathematical Society, Providence, RI (1990)

    MATH  Google Scholar 

  3. Antoine, X., Barucq, H., Bendali, A.: Bayliss–Turkel-like radiation conditions on surfaces of arbitrary shape. J. Math. Anal. Appl. 229(1), 184–211 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Antoine, X., Darbas, M., Lu, Y.: An improved surface radiation condition for high-frequency acoustic scattering problems. Comput. Methods Appl. Mech. Engrg. 195(33–36), 4060–4074 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Bamberger, A., Engquist, B., Halpern, L., Joly, P.: Higher order paraxial wave equation approximations in heterogeneous media. SIAM J. Appl. Math. 48(1), 129–154 (1988)

    MathSciNet  MATH  Google Scholar 

  6. Bamberger, A., Joly, P., Roberts, J.E.: Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM J. Numer. Anal. 27(2), 323–352 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general-purpose object-oriented finite element library. ACM Trans. Math. Software 33(4), 24 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Bayliss, A., Gunzburger, M., Turkel, E.: Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math. 42(2), 430–451 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Bramble, J.H., Pasciak, J.E.: Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comput. 76(258), 597–614 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Chandler-Wilde, S.N., Elschner, J.: Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces. SIAM J. Math. Anal. 42(6), 2554–2580 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Chandler-Wilde, S.N., Monk, P.: The PML for rough surface scattering. Appl. Numer. Math. 59(9), 2131–2154 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Chen, Z., Liu, X.: An adaptive perfectly matched layer technique for time-harmonic scattering problems. SIAM J. Numer. Anal. 43(2), 645–671 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Chen, Z., Zheng, W.: Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media. SIAM J. Numer. Anal. 48(6), 2158–2185 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Collino, F.: High order absorbing boundary conditions for wave propagation medels. In: 2nd International Conference on Mathematical and Numerical Aspects of Wave Propagation, pp. 161–171. Philadelphia, PA (1993)

  15. Druskin, V., Güttel, S., Knizhnerman, L.: Near-optimal perfectly matched layers for indefinite Helmholtz problems. SIAM Rev. 58(1), 90–116 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Nat. Acad. Sci. USA 74(5), 1765–1766 (1977)

    MathSciNet  MATH  Google Scholar 

  17. Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31(139), 629–651 (1977)

    MathSciNet  MATH  Google Scholar 

  18. Ghorpade, S.R., Limaye, B.V.: A Course in Multivariable Calculus and Analysis. Springer, New York (2010)

    MATH  Google Scholar 

  19. Grote, M.J., Keller, J.B.: On nonreflecting boundary conditions. J. Comput. Phys. 122(2), 231–243 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Guddati, M.N., Lim, K.-W.: Continued fraction absorbing boundary conditions for convex polygonal domains. Int. J. Numer. Methods Eng. 66(6), 949–977 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Hagstrom, T., Kim, S.: Complete radiation boundary conditions for the Helmholtz equation I: waveguides. Numer. Math. 141(4), 917–966 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Hagstrom, T., Warburton, T.: A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems. Wave Mot. 39(4), 327–338 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Hagstrom, T., Warburton, T.: Complete radiation boundary conditions: minimizing the long time error growth of local methods. SIAM J. Numer. Anal. 47(5), 3678–3704 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Hohage, T., Schmidt, F., Zschiedrich, L.: Solving time-harmonic scattering problems based on the pole condition. I. Theory. SIAM J. Math. Anal. 35(1), 183–210 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Hohage, T., Schmidt, F., Zschiedrich, L.: Solving time-harmonic scattering problems based on the pole condition. II. Convergence of the PML method. SIAM J. Math. Anal. 35(3), 547–560 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Hsiao, G.C., Nigam, N., Pasciak, J.E., Xu, L.: Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis. J. Comput. Appl. Math. 235(17), 4949–4965 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Kechroud, R., Antoine, X., Soulaïmani, A.: Numerical accuracy of a Padé-type non-reflecting boundary condition for the finite element solution of acoustic scattering problems at high-frequency. Int. J. Numer. Methods Eng. 64(10), 1275–1302 (2005)

    MATH  Google Scholar 

  28. Keller, J.B., Givoli, D.: Exact nonreflecting boundary conditions. J. Comput. Phys. 82(1), 172–192 (1989)

    MathSciNet  MATH  Google Scholar 

  29. Kim, S.: Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides. ESAIM Math. Model. Numer. Anal. 53(4), 1191–1222 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Kim, S.: Analysis of complete radiation boundary conditions for the Helmholtz equation in perturbed waveguides. J. Comput. Appl. Math. 367, 112458 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Kim, S.: Hybrid absorbing boundary conditions of PML and CRBC. J. Comput. Appl. Math. 399, 113713 (2022)

    MathSciNet  MATH  Google Scholar 

  32. Kim, S., Pasciak, J.E.: Analysis of a Cartesian PML approximation to acoustic scattering problems in \(\mathbb{R} ^2\). J. Math. Anal. Appl. 370(1), 168–186 (2010)

    MathSciNet  MATH  Google Scholar 

  33. Kirby, R.C., Klöckner, A., Sepanski, B.: Finite elements for Helmholtz equations with a nonlocal boundary condition. SIAM J. Sci. Comput. 43(3), A1671–A1691 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Kriegsmann, G.A., Taflove, A., Umashankar, K.R.: A new formulation of electromagnetic wave scattering using an on-surface radiation boundary condition approach. IEEE Trans. Antennas Propag. 35(2), 153–161 (1987)

    MathSciNet  MATH  Google Scholar 

  35. Li, P., Wu, H., Zheng, W.: Electromagnetic scattering by unbounded rough surfaces. SIAM J. Math. Anal. 43(3), 1205–1231 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Medovikov, A.A., Lebedev, V.I.: Variable time steps optimization of \(L_\omega \)-stable Crank–Nicolson method. Russ. J. Numer. Anal. Math. Model. 20(3), 283–303 (2005)

    MathSciNet  MATH  Google Scholar 

  37. Mennicken, R., Möller, M.: Non-self-adjoint boundary eigenvalue problems. North-Holland Mathematics Studies, vol. 192. North-Holland Publishing Co., Amsterdam (2003)

  38. Modave, A., Geuzaine, C., Antoine, X.: Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering. J. Comput. Phys. 401, 109029 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Petrushev, P., Popov, V.: Rational Approximation of Real Functions. Encyclopedia of Mathematics, vol. 28. Cambridge University Press, Cambridge (1987)

  40. Rabinovich, D., Givoli, D., Bécache, E.: Comparison of high-order absorbing boundary conditions and perfectly matched layers in the frequency domain. Int. J. Numer. Methods Biomed. Eng. 26(10), 1351–1369 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Schmidt, F., Hohage, T., Klose, R., Schädle, A., Zschiedrich, L.: Pole condition: a numerical method for Helmholtz-type scattering problems with inhomogeneous exterior domain. J. Comput. Appl. Math. 218(1), 61–69 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Singer, I., Turkel, E.: A perfectly matched layer for the Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 201(2), 439–465 (2004)

    MathSciNet  MATH  Google Scholar 

  43. Vacus, O.: Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem. Math. Comput. 74(249), 177–200 (2005)

    MathSciNet  MATH  Google Scholar 

  44. Vaĭnberg, B.R.: Asymptotic Methods in Equations of Mathematical Physics. Gordon & Breach Science Publishers, New York (1989). Translated from the Russian by E. Primrose

  45. Zhang, G.Q.: High order approximation of one-way wave equations. J. Comput. Math. 3(1), 90–97 (1985)

    MathSciNet  MATH  Google Scholar 

  46. Zolotarev, E.I.: Applications of elliptic functions to problems on functions deviating least or most from zero (Russian). Zap. Imper. Akad. Nauk St. Petersburg 30(5) (1877); reprinted in his Collected works Vol 2. Izadt. Akad. Nauk SSSR, Moscow. Ibuch. Fortschritte Math., 9(343):1–59, 1932

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  1. Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX, 75275-0156, USA

    Thomas Hagstrom

  2. Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul, 02447, Korea

    Seungil Kim

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  1. Thomas Hagstrom
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The research of the first author was supported in part by NSF Grant DMS-2012296. The research of the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2018R1D1A1B07047416) funded by the Ministry of Education, Science and Technology. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Hagstrom, T., Kim, S. Complete radiation boundary conditions for the Helmholtz equation II: domains with corners. Numer. Math. 153, 775–825 (2023). https://doi.org/10.1007/s00211-023-01352-0

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