The Costabel-Stephan system of boundary integral equations in the time domain

Qiu, Tianyu; Sayas, Francisco-Javier

doi:10.1090/mcom3053

The Costabel-Stephan system of boundary integral equations in the time domain

Authors: Tianyu Qiu and Francisco-Javier Sayas
Journal: Math. Comp. 85 (2016), 2341-2364
MSC (2010): Primary 65N30, 65N38, 65N12, 65N15
DOI: https://doi.org/10.1090/mcom3053
Published electronically: November 18, 2015
MathSciNet review: 3511284
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Abstract: In this paper we formulate a transmission problem for the transient acoustic wave equation as a system of retarded boundary integral equations. We then analyse a fully discrete method using a general Galerkin semidiscretization-in-space and convolution quadrature (CQ) in time. All proofs are developed using recent techniques based on the theory of evolution equations. Some numerical experiments are provided.

[1] Toufic Abboud, Patrick Joly, Jerónimo Rodríguez, and Isabelle Terrasse, Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains, J. Comput. Phys. 230 (2011), no. 15, 5877–5907. MR 2804957, https://doi.org/10.1016/j.jcp.2011.03.062
[2] A. Bamberger and T. Ha Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique. I, Math. Methods Appl. Sci. 8 (1986), no. 3, 405–435 (French, with English summary). MR 859833
[3] A. Bamberger and T. Ha Duong, Formulation variationnelle pour le calcul de la diffraction d’une onde acoustique par une surface rigide, Math. Methods Appl. Sci. 8 (1986), no. 4, 598–608 (French, with English summary). MR 870995
[4] Lehel Banjai, Antonio R. Laliena, and Francisco-Javier Sayas, Fully discrete Kirchhoff formulas with CQ-BEM, IMA J. Numer. Anal. 35 (2015), no. 2, 859–884. MR 3335227, https://doi.org/10.1093/imanum/dru017
[5] Lehel Banjai, Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments, SIAM J. Sci. Comput. 32 (2010), no. 5, 2964–2994. MR 2729447, https://doi.org/10.1137/090775981
[6] Lehel Banjai, Christian Lubich, and Francisco-Javier Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math. 129 (2015), no. 4, 611–646. MR 3317813, https://doi.org/10.1007/s00211-014-0650-0
[7] Lehel Banjai, Matthias Messner, and Martin Schanz, Runge-Kutta convolution quadrature for the boundary element method, Comput. Methods Appl. Mech. Engrg. 245/246 (2012), 90–101. MR 2969188, https://doi.org/10.1016/j.cma.2012.07.007
[8] Lehel Banjai and Martin Schanz, Wave propagation problems treated with convolution quadrature and BEM, Fast boundary element methods in engineering and industrial applications, Lect. Notes Appl. Comput. Mech., vol. 63, Springer, Heidelberg, 2012, pp. 145–184. MR 3059731, https://doi.org/10.1007/978-3-642-25670-7_5
[9] Yassine Boubendir, Victor Dominguez, David Levadoux, and Catalin Turc, Regularized combined field integral equations for acoustic transmission problems, 2013, arXiv:1312.6598.
[10] John Fun-Choi Chan and Peter Monk, Time dependent electromagnetic scattering by a penetrable obstacle, BIT 55 (2015), no. 1, 5–31. MR 3313600, https://doi.org/10.1007/s10543-014-0500-6
[11] Qiang Chen and Peter Monk, Discretization of the time domain CFIE for acoustic scattering problems using convolution quadrature, SIAM J. Math. Anal. 46 (2014), no. 5, 3107–3130. MR 3257633, https://doi.org/10.1137/110833555
[12] Xavier Claeys and Ralf Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures, Comm. Pure Appl. Math. 66 (2013), no. 8, 1163–1201. MR 3069956, https://doi.org/10.1002/cpa.21462
[13] Martin Costabel and Ernst Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985), no. 2, 367–413. MR 782799, https://doi.org/10.1016/0022-247X(85)90118-0
[14] Víctor Domínguez, Sijiang L. Lu, and Francisco-Javier Sayas, A Nyström flavored Calderón calculus of order three for two dimensional waves, time-harmonic and transient, Comput. Math. Appl. 67 (2014), no. 1, 217–236. MR 3141718, https://doi.org/10.1016/j.camwa.2013.11.005
[15] Víctor Domínguez and Francisco-Javier Sayas, Some properties of layer potentials and boundary integral operators for the wave equation, J. Integral Equations Appl. 25 (2013), no. 2, 253–294. MR 3161614, https://doi.org/10.1216/JIE-2013-25-2-253
[16] Silvia Falletta and Giovanni Monegato, An exact non reflecting boundary condition for 2D time-dependent wave equation problems, Wave Motion 51 (2014), no. 1, 168–192. MR 3127695, https://doi.org/10.1016/j.wavemoti.2013.06.001
[17] S. P. Groth, D. P. Hewett, and S. Langdon, Hybrid numerical-asymptotic approximation for high-frequency scattering by penetrable convex polygons, IMA Journal of Applied Mathematics (2013).
[18] Matthew Hassell and Francisco-Javier Sayas, Convolution quadrature for wave simulations, 2014, arXiv:1407.0345.
[19] R. Hiptmair and C. Jerez-Hanckes, Multiple traces boundary integral formulation for Helmholtz transmission problems, Adv. Comput. Math. 37 (2012), no. 1, 39–91. MR 2927645, https://doi.org/10.1007/s10444-011-9194-3
[20] R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math. 48 (1988), no. 2, 307–325. MR 933037, https://doi.org/10.1137/0148016
[21] R. Kreß and G. F. Roach, Transmission problems for the Helmholtz equation, J. Mathematical Phys. 19 (1978), no. 6, 1433–1437. MR 495653, https://doi.org/10.1063/1.523808
[22] Antonio R. Laliena and Francisco-Javier Sayas, Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numer. Math. 112 (2009), no. 4, 637–678. MR 2507621, https://doi.org/10.1007/s00211-009-0220-z
[23] C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. MR 923707, https://doi.org/10.1007/BF01398686
[24] Ch. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math. 67 (1994), no. 3, 365–389. MR 1269502, https://doi.org/10.1007/s002110050033
[25] F.-J. Sayas, Retarded potentials and time domain integral equations: a roadmap, to appear in Springer series in computational mathematics, 2015.
[26] Francisco-Javier Sayas, Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations, Numer. Math. 124 (2013), no. 1, 121–149. MR 3041732, https://doi.org/10.1007/s00211-012-0506-4
[27] Rodolfo H. Torres and Grant V. Welland, The Helmholtz equation and transmission problems with Lipschitz interfaces, Indiana Univ. Math. J. 42 (1993), no. 4, 1457–1485. MR 1266102, https://doi.org/10.1512/iumj.1993.42.42067
[28] T. von Petersdorff, Boundary integral equations for mixed Dirichlet, Neumann and transmission problems, Math. Methods Appl. Sci. 11 (1989), no. 2, 185–213. MR 984053, https://doi.org/10.1002/mma.1670110203
[29] A. Zinn, A numerical method for transmission problems for the Helmholtz equation, Computing 41 (1989), no. 3, 267–274 (English, with German summary). MR 988240, https://doi.org/10.1007/BF02259097