March  2017, 37(3): 1159-1181. doi: 10.3934/dcds.2017048

Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers

1. 

CMLA, ENS Cachan, CNRS, Universit, Paris-Saclay, 94235 Cachan, France

2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago 851 S Morgan St, Chicago, IL 60607, USA

3. 

Department of Mathematical Sciences, School of Natural Science Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan 689-798, Republic of Korea

* Corresponding author

Received  November 2015 Revised  October 2016 Published  December 2016

We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on k.

Citation: Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
References:
[1]

M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D. C. 1964.

[2]

I. Andronov, D. Bouche and F. Molinet, Asymptotic and Hybrid Methods in Electromagnetics volume 48 of IEE Electromagnetic Waves Series, Institution of Electrical Engineers (IEE), London, 2005. doi: 10.1049/PBEW051E.

[3]

V. M. Babič and V. S. Buldyrev, Short-wavelength Diffraction Theory volume 4 of Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1991. Asymptotic methods, Translated from the 1972 Russian original by E. F. Kuester. doi: 10.1007/978-3-642-83459-2.

[4]

A. H. Barnett and T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227 (2008), 7003-7026. doi: 10.1016/j.jcp.2008.04.008.

[5]

A. H. Barnett and T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput., 32 (2010), 1417-1441. doi: 10.1137/090768667.

[6]

A. Barnett and T. Betcke, Mpspack, url: https://code.google.com/p/mpspack/.

[7]

A. BaylissM. Gunzburger and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (1982), 430-451. doi: 10.1137/0142032.

[8]

D. Bouche, F. Molinet and R. Mittra, Asymptotic Methods in Electromagnetics Springer-Verlag, Berlin, 1997. Translated from the 1994 French original by Patricia and Daniel Gogny and revised by the authors. doi: 10.1007/978-3-642-60517-8.

[9]

R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), 85-114. doi: 10.1002/cpa.3160130109.

[10]

S. N. Chandler-WildeI. G. GrahamS. Langdon and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), 89-305. doi: 10.1017/S0962492912000037.

[11]

S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618. doi: 10.1137/040615523.

[12]

S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), 1774-1790(electronic). doi: 10.1137/S0036139996309722.

[13]

D. Colton, Partial Differential Equations Dover Publications, Inc. , Mineola, NY, 2004. An introduction, Corrected reprint of the 1988 original.

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory volume 93 of Applied Mathematical Sciences, Springer, New York, third edition, 2013. doi: 10.1007/978-1-4614-4942-3.

[15]

P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci., 16 (2006), 139-160. doi: 10.1142/S021820250600108X.

[16]

J. Douglas JrJ. E. SantosD. Sheen and L. Schreyer Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci., 3 (1993), 171-194. doi: 10.1142/S0218202593000102.

[17]

W. Eckhaus and E. M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal., 23 (1966), 26-86. doi: 10.1007/BF00281135.

[18]

K. Feng, Finite element method and natural boundary reduction, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) pages 1439-1453. PWN, Warsaw, 1984.

[19]

X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872-2896. doi: 10.1137/080737538.

[20]

X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), 1997-2024. doi: 10.1090/S0025-5718-2011-02475-0.

[21]

E. Giladi, Asymptotically derived boundary elements for the Helmholtz equation in high frequencies, J. Comput. Appl. Math., 198 (2007), 52-74. doi: 10.1016/j.cam.2005.11.024.

[22]

E. Giladi and J. B. Keller, A hybrid numerical asymptotic method for scattering problems, J. Comput. Phys., 174 (2001), 226-247. doi: 10.1006/jcph.2001.6903.

[23]

M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243. doi: 10.1006/jcph.1995.1210.

[24]

G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008.

[25]

G.-M. Gie and C.-Y. Jung, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition, Asymptot. Anal., 84 (2013), 17-33.

[26]

Y. HongC.-Y. Jung and R. Temam, On the numerical approximations of stiff convection-diffusion equations in a circle, Numerische Mathematik, 127 (2014), 291-313. doi: 10.1007/s00211-013-0585-x.

[27]

F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37. doi: 10.1016/0898-1221(95)00144-N.

[28]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering volume 132 of Applied Mathematical Sciences Springer-Verlag, New York, 1998. doi: 10.1007/b98828.

[29]

C.-Y. JungM. Petcu and R. Temam, Singular perturbation analysis on a homogeneous ocean circulation model, Anal. Appl. (Singap.), 9 (2011), 275-313. doi: 10.1142/S0219530511001832.

[30]

J. B. Keller, Diffraction by a convex cylinder, Div. Electromag. Res., Inst. Math. Sci., New York Univ., Res. Rep. No. EM-94 (1956), 10 pp. Also: Trans. I.R.E., AP-4 (1956), 312-321.

[31]

J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer., 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116.

[32]

J. B. Keller and D. Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys., 82 (1989), 172-192. doi: 10.1016/0021-9991(89)90041-7.

[33]

J. B. Keller, R. M. Lewis and B. D. Seckler, Asymptotic Solution of Some Diffraction Problems Res. Rep. No. EM-81. Div. Electromag. Res. , Inst. Math. Sci. , New York Univ. , 1955.

[34]

L. J. Landau, Bessel functions: Monotonicity and bounds, J. London Math. Soc. (2), 61 (2000), 197-215. doi: 10.1112/S0024610799008352.

[35]

N. N. Lebedev, Special Functions and Their Applications Revised English edition. Translated and edited by Richard A. Silverman. Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1965.

[36]

A. Lechleiter and S. Ritterbusch, A variational method for wave scattering from penetrable rough layers, IMA J. Appl. Math., 75 (2010), 366-391. doi: 10.1093/imamat/hxp040.

[37]

J. M. Melenk, On Generalized Finite-Element Methods, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph. D. )-University of Maryland, College Park.

[38]

R. B. Melrose and M. E. Taylor, The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672. doi: 10.1080/03605308608820439.

[39]

J. R. Ockendon and R. H. Tew, Thin-layer solutions of the Helmholtz and related equations, SIAM Rev., 54 (2012), 3-51. doi: 10.1137/090761641.

[40]

F. W. J. Olver, Asymptotics and Special Functions AKP Classics. A K Peters Ltd. , Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)].

[41]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors, NIST Handbook Of Mathematical Functions U. S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.

[42]

L. Prandtl, Verber fl¨ussigkeiten bei sehr kleiner reibung, Verk. Ⅲ Intem. Math. Kongr. Heidelberg, pages 484-491,1905.

[43]

M. LebentalJ. S. LauretJ. ZyssC. Schmit and E. Bogomolny, Directional emission of stadium-shaped microlasers, Physical Review A, 75 (2007), 033806. doi: 10.1103/PhysRevA.75.033806.

[44]

L. Prandtl, Gesammelte Abhandlungen zur angewandten Mechanik, Hydro-und Aerodynamik Herausgegeben von Walter Tollmien, Hermann Schlichting, Henry Görtler. Schriftleitung: F. W. Riegels. In 3 Teilen. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[45]

J. Shen and L.-L. Wang, Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), 623-644. doi: 10.1137/040607332.

[46]

J. Shen and L.-L. Wang, Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954-1978. doi: 10.1137/060665737.

[47]

A. Sommerfeld, Partial Differential Equations in Physics Academic Press, Inc. , New York, N. Y. , 1949. Translated by Ernst G. Straus.

[48]

A. Sommerfeld, Optics. Lectures on Theoretical Physics, Vol. Ⅳ Academic Press Inc. , New York, 1954. Translated by O. Laporte and P. A. Moldauer.

[49]

W. A. Strauss, Partial Differential Equations John Wiley & Sons, Inc. , New York, 1992. An introduction.

[50]

R. H. TewS. J. ChapmanJ. R. KingJ. R. Ockendon and I. Zafarullah, Scalar wave diffraction by tangent rays, Wave Motion, 32 (2000), 363-380. doi: 10.1016/S0165-2125(00)00051-2.

[51]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D. C. 1964.

[2]

I. Andronov, D. Bouche and F. Molinet, Asymptotic and Hybrid Methods in Electromagnetics volume 48 of IEE Electromagnetic Waves Series, Institution of Electrical Engineers (IEE), London, 2005. doi: 10.1049/PBEW051E.

[3]

V. M. Babič and V. S. Buldyrev, Short-wavelength Diffraction Theory volume 4 of Springer Series on Wave Phenomena, Springer-Verlag, Berlin, 1991. Asymptotic methods, Translated from the 1972 Russian original by E. F. Kuester. doi: 10.1007/978-3-642-83459-2.

[4]

A. H. Barnett and T. Betcke, Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains, J. Comput. Phys., 227 (2008), 7003-7026. doi: 10.1016/j.jcp.2008.04.008.

[5]

A. H. Barnett and T. Betcke, An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput., 32 (2010), 1417-1441. doi: 10.1137/090768667.

[6]

A. Barnett and T. Betcke, Mpspack, url: https://code.google.com/p/mpspack/.

[7]

A. BaylissM. Gunzburger and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (1982), 430-451. doi: 10.1137/0142032.

[8]

D. Bouche, F. Molinet and R. Mittra, Asymptotic Methods in Electromagnetics Springer-Verlag, Berlin, 1997. Translated from the 1994 French original by Patricia and Daniel Gogny and revised by the authors. doi: 10.1007/978-3-642-60517-8.

[9]

R. N. Buchal and J. B. Keller, Boundary layer problems in diffraction theory, Comm. Pure Appl. Math., 13 (1960), 85-114. doi: 10.1002/cpa.3160130109.

[10]

S. N. Chandler-WildeI. G. GrahamS. Langdon and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), 89-305. doi: 10.1017/S0962492912000037.

[11]

S. N. Chandler-Wilde and P. Monk, Existence, uniqueness, and variational methods for scattering by unbounded rough surfaces, SIAM J. Math. Anal., 37 (2005), 598-618. doi: 10.1137/040615523.

[12]

S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), 1774-1790(electronic). doi: 10.1137/S0036139996309722.

[13]

D. Colton, Partial Differential Equations Dover Publications, Inc. , Mineola, NY, 2004. An introduction, Corrected reprint of the 1988 original.

[14]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory volume 93 of Applied Mathematical Sciences, Springer, New York, third edition, 2013. doi: 10.1007/978-1-4614-4942-3.

[15]

P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci., 16 (2006), 139-160. doi: 10.1142/S021820250600108X.

[16]

J. Douglas JrJ. E. SantosD. Sheen and L. Schreyer Bennethum, Frequency domain treatment of one-dimensional scalar waves, Math. Models Methods Appl. Sci., 3 (1993), 171-194. doi: 10.1142/S0218202593000102.

[17]

W. Eckhaus and E. M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal., 23 (1966), 26-86. doi: 10.1007/BF00281135.

[18]

K. Feng, Finite element method and natural boundary reduction, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) pages 1439-1453. PWN, Warsaw, 1984.

[19]

X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872-2896. doi: 10.1137/080737538.

[20]

X. Feng and H. Wu, hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 80 (2011), 1997-2024. doi: 10.1090/S0025-5718-2011-02475-0.

[21]

E. Giladi, Asymptotically derived boundary elements for the Helmholtz equation in high frequencies, J. Comput. Appl. Math., 198 (2007), 52-74. doi: 10.1016/j.cam.2005.11.024.

[22]

E. Giladi and J. B. Keller, A hybrid numerical asymptotic method for scattering problems, J. Comput. Phys., 174 (2001), 226-247. doi: 10.1006/jcph.2001.6903.

[23]

M. J. Grote and J. B. Keller, On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243. doi: 10.1006/jcph.1995.1210.

[24]

G.-M. Gie and J. P. Kelliher, Boundary layer analysis of the Navier-Stokes equations with generalized Navier boundary conditions, J. Differential Equations, 253 (2012), 1862-1892. doi: 10.1016/j.jde.2012.06.008.

[25]

G.-M. Gie and C.-Y. Jung, Vorticity layers of the 2D Navier-Stokes equations with a slip type boundary condition, Asymptot. Anal., 84 (2013), 17-33.

[26]

Y. HongC.-Y. Jung and R. Temam, On the numerical approximations of stiff convection-diffusion equations in a circle, Numerische Mathematik, 127 (2014), 291-313. doi: 10.1007/s00211-013-0585-x.

[27]

F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37. doi: 10.1016/0898-1221(95)00144-N.

[28]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering volume 132 of Applied Mathematical Sciences Springer-Verlag, New York, 1998. doi: 10.1007/b98828.

[29]

C.-Y. JungM. Petcu and R. Temam, Singular perturbation analysis on a homogeneous ocean circulation model, Anal. Appl. (Singap.), 9 (2011), 275-313. doi: 10.1142/S0219530511001832.

[30]

J. B. Keller, Diffraction by a convex cylinder, Div. Electromag. Res., Inst. Math. Sci., New York Univ., Res. Rep. No. EM-94 (1956), 10 pp. Also: Trans. I.R.E., AP-4 (1956), 312-321.

[31]

J. B. Keller, Geometrical theory of diffraction, J. Opt. Soc. Amer., 52 (1962), 116-130. doi: 10.1364/JOSA.52.000116.

[32]

J. B. Keller and D. Givoli, Exact nonreflecting boundary conditions, J. Comput. Phys., 82 (1989), 172-192. doi: 10.1016/0021-9991(89)90041-7.

[33]

J. B. Keller, R. M. Lewis and B. D. Seckler, Asymptotic Solution of Some Diffraction Problems Res. Rep. No. EM-81. Div. Electromag. Res. , Inst. Math. Sci. , New York Univ. , 1955.

[34]

L. J. Landau, Bessel functions: Monotonicity and bounds, J. London Math. Soc. (2), 61 (2000), 197-215. doi: 10.1112/S0024610799008352.

[35]

N. N. Lebedev, Special Functions and Their Applications Revised English edition. Translated and edited by Richard A. Silverman. Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1965.

[36]

A. Lechleiter and S. Ritterbusch, A variational method for wave scattering from penetrable rough layers, IMA J. Appl. Math., 75 (2010), 366-391. doi: 10.1093/imamat/hxp040.

[37]

J. M. Melenk, On Generalized Finite-Element Methods, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph. D. )-University of Maryland, College Park.

[38]

R. B. Melrose and M. E. Taylor, The radiation pattern of a diffracted wave near the shadow boundary, Comm. Partial Differential Equations, 11 (1986), 599-672. doi: 10.1080/03605308608820439.

[39]

J. R. Ockendon and R. H. Tew, Thin-layer solutions of the Helmholtz and related equations, SIAM Rev., 54 (2012), 3-51. doi: 10.1137/090761641.

[40]

F. W. J. Olver, Asymptotics and Special Functions AKP Classics. A K Peters Ltd. , Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)].

[41]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors, NIST Handbook Of Mathematical Functions U. S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.

[42]

L. Prandtl, Verber fl¨ussigkeiten bei sehr kleiner reibung, Verk. Ⅲ Intem. Math. Kongr. Heidelberg, pages 484-491,1905.

[43]

M. LebentalJ. S. LauretJ. ZyssC. Schmit and E. Bogomolny, Directional emission of stadium-shaped microlasers, Physical Review A, 75 (2007), 033806. doi: 10.1103/PhysRevA.75.033806.

[44]

L. Prandtl, Gesammelte Abhandlungen zur angewandten Mechanik, Hydro-und Aerodynamik Herausgegeben von Walter Tollmien, Hermann Schlichting, Henry Görtler. Schriftleitung: F. W. Riegels. In 3 Teilen. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[45]

J. Shen and L.-L. Wang, Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), 623-644. doi: 10.1137/040607332.

[46]

J. Shen and L.-L. Wang, Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954-1978. doi: 10.1137/060665737.

[47]

A. Sommerfeld, Partial Differential Equations in Physics Academic Press, Inc. , New York, N. Y. , 1949. Translated by Ernst G. Straus.

[48]

A. Sommerfeld, Optics. Lectures on Theoretical Physics, Vol. Ⅳ Academic Press Inc. , New York, 1954. Translated by O. Laporte and P. A. Moldauer.

[49]

W. A. Strauss, Partial Differential Equations John Wiley & Sons, Inc. , New York, 1992. An introduction.

[50]

R. H. TewS. J. ChapmanJ. R. KingJ. R. Ockendon and I. Zafarullah, Scalar wave diffraction by tangent rays, Wave Motion, 32 (2000), 363-380. doi: 10.1016/S0165-2125(00)00051-2.

[51]

G. N. Watson, A Treatise on the Theory of Bessel Functions Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition.

Figure 1.  Scattering by a circular object
Figure 2.  Sound-soft scattering from a circular object as in Figure 1 at (Top) $k=50$, (Bottom) $k=500$: (a) $\Re (u_{inc}^{(k)}) = \cos(ikx)$: incident wave traveling to the right, (b) $\Re u_s^{(k)}$: scattered wave, (c) $\Re u^{(k)} = \Re u^{(k)}_{inc} + \Re u^{(k)}_s$: total solution. This simulation is generated using the MPSPACK (see [6])
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