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Convergence of infinite element methods for scalar waveguide problems

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We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the perfectly matched layer method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments.

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  1. Bécache, E., Bonnet-Ben Dhia, A.-S., Legendre, G.: Perfectly matched layers for the convected Helmholtz equation. SIAM J. Numer. Anal. 42, 409–433 (2004)

  2. Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bonnet-Ben Dhia, A.S., Jr Ciarlet, P., Zwölf, C.M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficient. J. Comput. Appl. Math. 234, 1912–1919 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Burger, S., Zschiedrich, L., Pomplun, J., Schmidt, F.: Finite element method for accurate 3d simulation of plasmonic waveguides. In Integrated optics: devices, materials, and technologies XIV, vol. 7604, Proc. SPIE, p 76040F (2010)

  5. Chen, Z., Wu, H.: An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41, 799–826 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Duren, P.L.: Theory of \(H^{p}\) Spaces, Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)

    Google Scholar 

  7. Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of linear operators. Vol. I, vol. 49 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1990)

    Book  Google Scholar 

  8. Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks]. In: B. G. Teubner, Stuttgart, second ed. (1996)

  9. Hein, S., Koch, W., Nannen, L.: Fano resonances in acoustics. J. Fluid Mech. 664, 238–264 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hein, S., Koch, W., Nannen, L.: Trapped modes and Fano resonances in two-dimensional acoustical ductcavity systems. J. Fluid Mech. 692, 257–287 (2012)

    Article  MATH  Google Scholar 

  11. Hislop, P.D., Sigal, I.M.: Introduction to spectral theory, vol. 113 of Applied Mathematical Sciences, Springer, New York. With applications to Schrödinger operators (1996)

  12. Hohage, T., Nannen, L.: Hardy space infinite elements for scattering and resonance problems. SIAM J. Numer. Anal. 47, 972–996 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. 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, 547–560 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kalvin, V.: Perfectly matched layers for diffraction gratings in inhomogeneous media, stability and error estimates. SIAM J. Appl. Math. 40, 309–330 (2011)

    MathSciNet  Google Scholar 

  15. Karma, O.: Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17, 365–387 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kim, S., Pasciak, J.E.: The computation of resonances in open systems using a perfectly matched layer. Math. Comp. 78, 1375–1398 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kress, R.: Linear integral equations. Applied Mathematical Sciences, vol. 82, 2nd edn. Springer, New York (1999)

  18. Lassas, M., Somersalo, E.: On the existence and the convergence of the solution of the PML equations. Computing 60, 229–241 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Levitin, M., Marletta, M.: A simple method of calculating eigenvalues and resonances in domains with infinite regular ends. Proc. Roy. Soc. Edinb. Sect. A 138, 1043–1065 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Moiseyev, N.: Quantum theory of resonances: calculating energies, width and cross-sections by complex scaling. Phys. Rep. 302, 211–293 (1998)

    Article  Google Scholar 

  21. Nannen, L., Hohage, T., Schädle, A., Schöberl, J.: Exact sequences of high order Hardy space infinite elements for exterior Maxwell problems. SIAM J. Sci. Comput. 35, A1024–A1048 (2013)

  22. Nannen, L., Schädle, A.: Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities. Wave Motion 48, 116–129 (2010)

    Article  Google Scholar 

  23. Nazarov, S.A., Plamenevsky, B.A.: Elliptic problems with domains with piecewise smooth boundaries. Walter de Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  24. Racec, P.N., Racec, E.R., Neidhardt, H.: Evanescent channels and scattering in cylindrical nanowire heterostructures. Phys. Rev. B 79, 155305 (2009)

    Article  Google Scholar 

  25. Rotter, S., Libisch, F., Burgdörfer, J., Kuhl, U., Stöckmann, H.-J.: Tunable Fano resonances in transport through microwave billiards. Phys. Rev. E 69, 046208 (2004)

    Article  Google Scholar 

  26. Schenk, O., Gärtner, K., Solving unsymmetric sparse systems of linear equations with PARDISO, in computational science–ICCS: part II (Amsterdam), vol. 2330 of Lecture Notes in Computer Science Springer, Berlin, 2002, 355–363 (2002)

  27. Schöberl, J.: Netgen—an advancing front 2d/3d-mesh generator based on abstract rules. Comput. Visual. Sci 1, 41–52 (1997)

    Article  MATH  Google Scholar 

  28. Steinbach, O., Unger, G.: Convergence analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem. SIAM J. Numer. Anal. 50, 710–728 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  29. Zschiedrich, L., Klose, R., Schädle, A., Schmidt, F.: A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in 2d. J. Comput. Appl. Math. 188, 12–32 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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The authors dedicate this work to Werner Koch for his inspiration, generosity and enthusiasm concerning the topic of resonances in waveguides. Unfortunately he passed away on August 28, 2012. Moreover, we would like to thank an anonymous referee for detailed and helpful suggestions and corrections. Financial support by the German Science Foundation through grant HO 2551/5 is gratefully acknowledged.

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Correspondence to Thorsten Hohage.

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Communicated by Ralf Hiptmair.

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Hohage, T., Nannen, L. Convergence of infinite element methods for scalar waveguide problems. Bit Numer Math 55, 215–254 (2015).

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