The Complex-Scaled Half-Space Matching Method
Abstract
The half-space matching (HSM) method has recently been developed as a new method for the solution of two-dimensional scattering problems with complex backgrounds, providing an alternative to perfectly matched layers or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretization localized around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.
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References
1.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1964.
2.
J. Aguilar and J.-M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys., 22 (1971), pp. 269--279.
3.
H. Amann and J. Escher, Analysis III, Birkhäuser, Basel, 2009.
4.
T. Arens and T. Hohage, On radiation conditions for rough surface scattering problems, IMA J. Appl. Math., 70 (2005), pp. 839--847.
5.
J.-P. Aubin, Applied Functional Analysis, 2nd ed., John Wiley & Sons, New York, 2000.
6.
E. Balslev and J.-M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Comm. Math. Phys., 22 (1971), pp. 280--294.
7.
V. Baronian, A.-S. Bonnet-BenDhia, S. Fliss, and Y. Tjandrawidjaja, The half-space matching method for Lamb wave scattering in anisotropic plates, in Proceedings of the 2nd International Conference on Advanced Modelling of Wave Propagation in Solids, Prague, 2018.
8.
E. Bécache, S. Fauqueux, and P. Joly, Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys., 188 (2003), pp. 399--433.
9.
A.-S. Bonnet-Ben Dhia, S. N. Chandler-Wilde, and S. Fliss, On the Half-Space Matching Method for Real Wavenumber, preprint, arXiv:2111.06793.
10.
A.-S. Bonnet-Ben Dhia, G. Dakhia, C. Hazard, and L. Chorfi, Diffraction by a defect in an open waveguide: A mathematical analysis based on a modal radiation condition, SIAM J. Appl. Math., 70 (2009), pp. 677--693.
11.
A.-S. Bonnet-BenDhia, S. Fliss, and Y. Tjandrawidjaja, Numerical analysis of the half-space matching method with Robin traces on a convex polygonal scatterer, in Maxwell's Equations: Analysis and Numerics, U. Langer, D. Pauly, and S. Repin, eds., De Gruyter, Berlin, 2019, pp. 105--144.
12.
A.-S. Bonnet-BenDhia, S. Fliss, and A. Tonnoir, The halfspace matching method: A new method to solve scattering problem in infinite media, J. Comput. Appl. Math., 338 (2018), pp. 44--68.
13.
A.-S. Bonnet-BenDhia, B. Goursaud, and C. Hazard, Mathematical analysis of the junction of two acoustic open waveguides, SIAM J. Appl. Math., 71 (2011), pp. 2048--2071.
14.
A.-S. Bonnet-BenDhia and A. Tillequin, A generalized mode matching method for the junction of open waveguides, in Proceedings of the 5th International Conference on Mathematical and Numerical Aspects of Wave Propagation, A. Bermúdez, D. Gómez, C. Hazard, P. Joly, and J. E. Roberts, eds., SIAM, Philadelphia, 2000, pp. 399--403.
15.
A.-S. Bonnet-BenDhia and A. Tillequin, A limiting absorption principle for scattering problems with unbounded obstacles, Math. Methods Appl. Sci., 24 (2001), pp. 1089--1111.
16.
H. Brakhage and P. Werner, Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung, Arch. Math., 16 (1965), pp. 325--329.
17.
G. A. Chandler, Galerkin's method for boundary integral equations on polygonal domains, ANZIAM J., 26 (1984), pp. 1--13.
18.
S. Chandler-Wilde and B. Zhang, Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate, Proc. A, 454 (1998), pp. 519--542.
19.
S. N. Chandler-Wilde, The impedance boundary value problem for the Helmholtz equation in a half-plane, Math. Methods Appl. Sci., 20 (1997), pp. 813--840.
20.
S. N. Chandler-Wilde and J. Elschner, Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal., 42 (2010), pp. 2554--2580.
21.
S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence, Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering, Acta Numer., 21 (2012), pp. 89--305.
22.
S. N. Chandler-Wilde, D. P. Hewett, S. Langdon, and A. Twigger, A high frequency boundary element method for scattering by a class of nonconvex obstacles, Numer. Math., 129 (2015), pp. 647--689.
23.
S. N. Chandler-Wilde and S. Langdon, A Galerkin boundary element method for high frequency scattering by convex polygons, SIAM J. Numer. Anal., 45 (2007), pp. 610--640.
24.
F. Collino and P. Monk, The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), pp. 2061--2090.
25.
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Springer, New York, 1998.
26.
D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
27.
J. B. Conway, Functions of One Complex Variable I, Springer, New York, 1978.
28.
J.-P. Croisille and G. Lebeau, Diffraction by an Immersed Elastic Wedge, Lecture Notes in Math. 1723, Springer-Verlag, Berlin, 1999.
29.
P.-M. Cutzach and C. Hazard, Existence, uniqueness and analyticity properties for electromagnetic scattering in a two-layered medium, Math. Methods Appl. Sci., 21 (1998), pp. 433--461.
30.
NIST Digital Library of Mathematical Functions, Release 1.0.28, F. W. J. Olver et al., eds., 2015, http://dlmf.nist.gov/.
31.
S. Dyatlov and M. Zworski, Mathematical Theory of Scattering Resonances, Grad. Stud. Math. 200, AMS, Providence, RI, 2019.
32.
S. Fliss and P. Joly, Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media, Appl. Numer. Math., 59 (2009), pp. 2155--2178.
33.
S. Fliss and P. Joly, Solutions of the time-harmonic wave equation in periodic waveguides: Asymptotic behaviour and radiation condition, Arch. Ration. Mech. Anal., 219 (2016), pp. 349--386.
34.
S. Fliss, P. Joly, and J. Li, Exact boundary conditions for wave propagation in periodic media containing a local perturbation, in Wave Propagation in Periodic Media: Analysis, Numerical Techniques and Practical Applications, Progr. Comput. Phys. 1, Bentham, Sharjah, 2010, pp. 108--134.
35.
D. P. Hewett, S. L. Langdon, and J. M. Melenk, A high frequency $hp$ boundary element method for scattering by convex polygons, SIAM J. Numer. Anal., 51 (2013), pp. 629--653.
36.
C. Jerez-Hanckes and J.-C. Nédélec, Asymptotics for Helmholtz and Maxwell solutions in 3-D open waveguides, Commun. Comput. Phys., 11 (2012), pp. 629--646.
37.
V. V. Kamotski\u\i, An application of the method of spectral functions to the problem of scattering by two wedges, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 324 (2005), pp. 61--76 (in Russian); J. Math. Sci., 138 (2006), pp. 5514--5523, (in English).
38.
N. Kielbasiewicz and E. Lunéville, User Documentation for XLiFE++, 2019, https://uma.ensta-paris.fr/soft/XLiFE++/var/files/docs/usr/user_documentation.pdf.
39.
W. Lu, Y. Y. Lu, and J. Qian, Perfectly-matched-layer boundary integral equation method for wave scattering in a layered medium, SIAM J. Appl. Math., 78 (2018), pp. 246--265.
40.
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000.
41.
I. Mitrea, On the spectra of elastostatic and hydrostatic layer potentials on curvilinear polygons, J. Fourier Anal. Appl., 8 (2002), pp. 443--487.
42.
J. Ott, Halfspace Matching: A Domain Decomposition Method for Scattering by 2D Open Waveguides, Ph.D. thesis, Karlsruhe Institute of Technology, 2017, https://doi.org/10.5445/IR/1000070898.
43.
K.-M. Perfekt and M. Putinar, The essential spectrum of the Neumann-Poincaré operator on a domain with corners, Arch. Ration. Mech. Anal., 223 (2017), pp. 1019--1033.
44.
S. A. Sauter and C. Schwab, Boundary Element Methods, Springer, New York, 2011.
45.
Y. Tjandrawidjaja, Some Contributions to the Analysis of the Half-Space Matching Method for Scattering Problems and Extension to 3D Elastic Plates, Ph.D. thesis, Université Paris Saclay, 2019.
46.
A. Tonnoir, Conditions transparentes pour la diffraction d'ondes en milieu élastique anisotrope, Ph.D. thesis, École Polytechnique, 2015.
Information & Authors
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Published In
SIAM Journal on Mathematical Analysis
Pages: 512 - 557
DOI: 10.1137/20M1387122
ISSN (online): 1095-7154
Copyright
© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 21 December 2020
Accepted: 7 September 2021
Published online: 18 January 2022
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Funding Information
Commissariat à l'Énergie Atomique et aux Énergies Alternatives https://doi.org/10.13039/501100006489
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/S029486/1
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