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Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous media. (English) Zbl 1496.65229

The authors are concerned with an adaptive coupling FEM-BEM in order to solve a Helmholtz acoustic/electromagnetic 2D wave propagation problem with a bounded heterogeneous region. First they present the Helmholtz model and an equivalent decomposition formulation. Then they accomplish a numerical analysis of the FEM-BEM algorithm establishing optimal order convergence of the hybridized numerical solution. On the bounded part of the domain they approximate the solution by a FEM with classical continuous piecewise polynomials on triangular meshes. On the other hand a high-order Nyström BEM is used to compute the scattered wave in the unbounded part of the domain. The solutions are coupled by requiring the coinciding in the two artificial boundaries that ensures the matching of FEM and BEM solutions in the common region of the partition of domain. Three distinct sets of experiments are carried out, the most challenging being the algorithm for multiple-particle Janus-type configurations with non-smooth solutions.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
45P05 Integral operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Software:

Gmsh; TMATROM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adams, R. A.; Fournier, J. J.F., Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), vol. 140 (2003), Elsevier/Academic Press: Elsevier/Academic Press Amsterdam · Zbl 1098.46001
[2] Bertoluzza, S., The discrete commutator property of approximation spaces, C. R. Acad. Sci., Sér. 1 Math., 329, 12, 1097-1102 (1999) · Zbl 0940.65053
[3] Brakhage, H.; Werner, P., Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung, Arch. Math., 16, 325-329 (1965) · Zbl 0132.33601
[4] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15 (2008), Springer: Springer New York · Zbl 1135.65042
[5] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics (2002), Society for Industrial and Applied Mathematics · Zbl 0999.65129
[6] Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (2019), Springer · Zbl 1425.35001
[7] Costabel, M., Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal., 19, 3, 613-626 (1988) · Zbl 0644.35037
[8] Coyle, J.; Monk, P., Scattering of time-harmonic electromagnetic waves by anisotropic in homogeneous scatterers or impenetrable obstacles, SIAM J. Math. Anal., 37, 1590-1617 (2004) · Zbl 0979.78015
[9] de Gennes, P. G., Soft matter (Nobel lecture), Angew. Chem., Int. Ed. Engl., 31, 842-845 (1992)
[10] Domínguez, V.; Ganesh, M.; Sayas, F. J., An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: formulation, analysis, algorithm, and simulation, J. Comput. Phys., 403, Article 109052 pp. (2020) · Zbl 1453.65405
[11] Domínguez, V.; Sayas, F.-J., Stability of discrete liftings, C. R. Math. Acad. Sci. Paris, 337, 12, 805-808 (2003) · Zbl 1036.65089
[12] Domínguez, V.; Turc, C., High order Nyström methods for transmission problems for Helmholtz equations, (Trends in Differential Equations and Applications. Trends in Differential Equations and Applications, SEMA SIMAI Springer Ser., vol. 8 (2016), Springer: Springer Cham), 261-285 · Zbl 06981860
[13] Ganesh, M.; Hawkins, S. C., Algorithm 975: TMATROM—a T-matrix reduced order model software, ACM Trans. Math. Softw., 44, 9 (2017) · Zbl 1484.78006
[14] Ganesh, M.; Hawkins, S. C.; Hiptmair, R., Convergence analysis with parameter estimates for a reduced basis acoustic scattering T-matrix method, IMA J. Numer. Anal., 32, 1348-1374 (2012) · Zbl 1275.65074
[15] Ganesh, M.; Morgenstern, C., High-order FEM-BEM computer models for wave propagation in unbounded and heterogeneous media: application to time-harmonic acoustic horn problem, J. Comput. Appl. Math., 307, 183-203 (2016) · Zbl 1382.76163
[16] Ganesh, M.; Morgenstern, C., High-order FEM domain decomposition models for high-frequency wave propagation in heterogeneous media, Comput. Math. Appl., 75, 1961-1972 (2018) · Zbl 1409.78007
[17] Ganesh, M.; Morgenstern, C., A coercive heterogeneous media Helmholtz model: formulation, wavenumber-explicit analysis, and preconditioned high-order FEM, Numer. Algorithms, 83, 1441-1487 (2020) · Zbl 1436.35079
[18] Geuzaine, C.; Remacle, J.-F., Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 1309-1331 (2009) · Zbl 1176.74181
[19] Gillman, A.; Barnett, A. H.; Martinsson, P.-G., A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media, BIT Numer. Math., 55, 1, 141-170 (2015) · Zbl 1312.65201
[20] Grisvard, P., Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, vol. 69 (2011), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 1231.35002
[21] Hawkins, S. C.; Rother, T.; Wauer, J., Numerical study of acoustic scattering by Janus spheres, J. Acoust. Soc. Am., 147, 4097 (2020)
[22] Hazard, C.; Lenoir, M., On the solutions of time-harmonic scattering problems for maxwell’s equations, SIAM J. Math. Anal., 27, 1597-1630 (1996) · Zbl 0860.35129
[23] Hsiao, G. C.; Wendland, W. L., Boundary Integral Equations, Applied Mathematical Sciences, vol. 164 (2008), Springer-Verlag: Springer-Verlag Berlin · Zbl 1157.65066
[24] Ihlenburg, F., Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, vol. 132 (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0908.65091
[25] Jami, A.; Lenoir, M., A variational formulation for exterior problems in linear hydrodynamics, Comput. Methods Appl. Mech. Eng., 16, 341-359 (1978) · Zbl 0392.76020
[26] Kirsch, A.; Monk, P., Convergence analysis of a coupled finite element and spectral method in acoustic scattering, IMA J. Numer. Anal., 10, 3, 425-447 (1990) · Zbl 0712.65095
[27] Kirsch, A.; Monk, P., An analysis of the coupling of finite-element and Nyström methods in acoustic scattering, IMA J. Numer. Anal., 14, 4, 523-544 (1994) · Zbl 0816.65104
[28] Kress, R., Linear Integral Equations, Applied Mathematical Sciences, vol. 82 (2014), Springer: Springer New York · Zbl 1328.45001
[29] Lattuada, M.; Hatton, A., Synthesis, properties and applications of Janus nanoparticles, Nano Today, 6, 286-308 (2011)
[30] McLean, W., Strongly Elliptic Systems and Boundary Integral Equations (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0948.35001
[31] Nédélec, J.-C., Acoustic and Electromagnetic EquationsIntegral Representations for Harmonic Problems, Applied Mathematical Sciences, vol. 144 (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0981.35002
[32] Nitsche, J. A.; Schatz, A. H., Interior estimates for Ritz-Galerkin methods, Math. Comput., 28, 937-958 (1974) · Zbl 0298.65071
[33] Rother, T., Sound Scattering on Spherical Objects (2020), Springer: Springer New York · Zbl 1503.76087
[34] Ruhland, T. M.; Gröschel, A. H.; Ballard, N.; Skelhon, T. S.; Walther, A.; Müller, A. H.E.; Bon, S. A.F., Influence of Janus particle shape on their interfacial behavior at liquid-liquid interfaces, Langmuir, 29, 1388-1394 (2013)
[35] Saranen, J.; Vainikko, G., Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monographs in Mathematics (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 0991.65125
[36] Sayas, F. J., The validity of Johnson-Nédélec’s BEM-FEM coupling on polygonal interfaces, SIAM Rev., 55, 131-146 (2013) · Zbl 1270.65070
[37] Scott, L. R.; Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54, 190, 483-493 (1990) · Zbl 0696.65007
[38] Vafaeezadeh, M.; Thiel, W. R., Janus interphase catalysts for interfacial organic reactions, J. Mol. Liq., 315, Article 113735 pp. (2020)
[39] Zhang, J.; Gryzbowski, A.; Granick, S., Janus particle synthesis, assembly and application, Langmuir, 33, 6964-6977 (2017)
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