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In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.
[3] A. Asheim and D. Huybrechs, Local solutions to high-frequency 2D scattering problems, J. Comput. Phys., 229 (2010), pp. 5357-5372.
[4] I. Babuˇska, B. Q. Guo, and E. P. Stephan, On the exponential convergence of the h-p version for boundary element Galerkin methods on polygons, Math. Methods Appl. Sci., 12 (1990), pp. 413-427.
[5] I. Babuˇska and M. Suri, The p and h-p versions of the finite element method, basic principles and properties, SIAM Rev., 36 (1994), pp. 578-632.
[6] T. Betcke, S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustics and their boundary element discretisation, Numer. Methods Partial Differential Equations, 27 (2011), pp. 31-69.
[8] O. P. Bruno, C. A. Geuzaine, J. A. Monro, Jr., and F. Reitich, Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: The convex case, Philos. Trans. R. Soc. Lond. Ser. A, 362 (2004), pp. 629-645.
[9] O. P. Bruno and F. Reitich, High order methods for high-frequency scattering applications, in Modeling and Computations in Electromagnetics, H. Ammari, ed., Lect. Notes Comput. Sci. Eng. 59, Springer, Berlin, 2007, pp. 129-164.
[10] S. N. Chandler-Wilde and I. G. Graham, Boundary integral methods in high frequency scattering, in Highly Oscillatory Problems, B. Engquist, T. Fokas, E. Hairer, and A. Iserles, eds., London Math. Soc. Lecture Note Ser. 366, Cambridge University Press, Cambridge, UK, 2009, pp. 154-193.
[11] S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering, J. Integral Equations Appl., 21 (2009), pp. 229-279.
[12] S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence, Numericalasymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), pp. 89-305.
[15] S. N. Chandler-Wilde, S. Langdon, and M. Mokgolele, A high frequency boundary element method for scattering by convex polygons with impedance boundary conditions, Commun. Comput. Phys., 11 (2012), pp. 575-593.
[16] S. N. Chandler-Wilde and P. Monk, Wave-number-explicit bounds in time-harmonic scattering, SIAM J. Math. Anal., 39 (2008), pp. 1428-1455.
[17] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, SpringerVerlag, Berlin, 1992.
[18] D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.
[19] C. P. Davis and W. C. Chew, Frequency-independent scattering from a flat strip with T Ezpolarized fields, IEEE Trans. Antennas and Propagation, 56 (2008), pp. 1008-1016.
[20] V. Dominguez, I. G. Graham, and V. P. Smyshlyaev, A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering, Numer. Math., 106 (2007), pp. 471-510.
[21] M. Ganesh and S. C. Hawkins, A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions, J. Comput. Phys., 230 (2011), pp. 104-125.
[22] W. Gui and I. Babuˇska, The h-, p- and hp-versions of the finite element method in 1 dimension, parts I, II, III, Numer. Math., 49 (1986), pp. 577-683.
[23] B. Q. Guo and I. Babuˇska, The hp-version of the finite element method. Part 1: The basic approximation results, Comp. Mech., 1 (1986), pp. 21-41.
[24] B. Q. Guo and I. Babuˇska, The hp-version of the finite element method. Part 2: General results and applications, Comp. Mech., 1 (1986), pp. 203-226.
[25] N. Heuer, M. Maischak, and E. P. Stephan, Exponential convergence of the hp-version for the boundary element method on open surfaces, Numer. Math., 83 (1999), pp. 641-666.
[26] D. P. Hewett, S. Langdon, and J. M. Melenk, A High Frequency hp Boundary Element Method for Scattering by Convex Polygons, Technical report MPS-2011-18, Department of Mathematics and Statistics, preprint, University of Reading, 2011.
[31] M. Lo¨hndorf and J. M. Melenk, Wavenumber-explicit hp-BEM for high frequency scattering, SIAM J. Numer. Anal., 49 (2011), pp. 2340-2363.
[32] T. Luostari, T. Huttunen, and P. Monk, Plane wave methods for approximating the time harmonic wave equation, in Highly Oscillatory Problems, B. Engquist, T. Fokas, E. Hairer, and A. Iserles, eds., London Math. Soc. Lecture Note Ser. 366, Cambridge University Press, Cambridge, UK, 2009, pp. 127-153.
[33] M. Maischak and E. P. Stephan, The hp-version of the boundary element method in R3: The basic approximation results, Math. Methods Appl. Sci., 20 (1997), pp. 461-476.
[34] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000.
[35] J. M. Melenk, hp-Finite Element Methods for Singular Perturbations, Springer, Berlin, 2003.
[36] J. M. Melenk and S. Langdon, A fully discrete hp boundary element method for high frequency scattering by convex polygons, in preparation.
[41] S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Ser. Comput. Math. 39, Springer-Verlag, Berlin, 2011.
[42] K. Scherer, On optimal global error bounds obtained by scaled local error estimates, Numer. Math., 36 (1981), pp. 151-176.
[43] C. Schwab, p- and hp-Finite Element Methods, Clarendon Press, Oxford, UK, 1998.
[44] E. A. Spence, S. N. Chandler-Wilde, I. G. Graham, and V. P. Smyshlyaev, A new frequency-uniform coercive boundary integral equation for acoustic scattering, Comm. Pure Appl. Math., 64 (2011), pp. 1384-1415.
[45] E. P. Stephan, The h-p boundary element method for solving 2- and 3-dimensional problems, Comput. Methods Appl. Mech. Engrg., 133 (1996), pp. 183-208.
[46] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, UK, 1944.