Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems
-
Gabriel N. Gatica
und
Salim Meddahi
Abstract
This paper extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. The well-posedness of the continuous and discrete formulations are established, and then Cea-type estimates and consequent rates of convergence are derived.
Funding statement: This research was partially supported by ANID-Chile through the project Centro de Modelamiento Matemático (AFB170001) of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal; by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; and by Spain’s Ministry of Economy Project MTM2017-87162-P.
References
[1] B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013), No. 3, 376–391.10.1016/j.camwa.2013.05.015Suche in Google Scholar
[2] L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016), No. 4, 729–750.10.1142/S0218202516500160Suche in Google Scholar
[3] S. C. Brenner, Q. Guan, and L. Y. Sung, Some estimates for virtual element methods. Comput. Methods Appl. Math. 17 (2017), No. 4, 553–574.10.1515/cmam-2017-0008Suche in Google Scholar
[4] D. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory Second edition. Applied Mathematical Sciences, Vol. 93. Springer-Verlag, Berlin, 1998.10.1007/978-3-662-03537-5Suche in Google Scholar
[5] M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements. In: Boundary Elements IX (Eds. C. A. Brebbia, G. Kuhn, and W. L. Wendland), Springer, Berlin, pp. 411–420, (1987).10.1007/978-3-662-21908-9_26Suche in Google Scholar
[6] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), No. 150, 441–463.10.1090/S0025-5718-1980-0559195-7Suche in Google Scholar
[7] G. N. Gatica, A. Márquez, and S. Meddahi, A virtual marriage a la mode: some recent results on the coupling of VEM and BEM. In: The Virtual Element Method and its Applications (Eds. P. Antonietti, L. Beirao da Veiga, and G. Manzini). SEMA-SIMA Springer series, to appear.Suche in Google Scholar
[8] G. N. Gatica and S. Meddahi, On the coupling of VEM and BEM in two and three dimensions. SIAM J. Numer. Anal. 57 (2019), No. 6, 2493–2518.10.1137/18M1202487Suche in Google Scholar
[9] G. N. Gatica and S. Meddahi, Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems. Preprint 2019-22 Centro de Investigación en Ingeniería Matemática, Universidad de Concepción, Concepción, Chile, 2019.Suche in Google Scholar
[10] H. Han, A new class of variational formulations for the coupling of finite and boundary element methods. J. Comput. Math. 8 (1990), No. 3, 223–232.Suche in Google Scholar
[11] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems Springer-Verlag, New York, 1996.10.1007/978-1-4612-5338-9Suche in Google Scholar
[12] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations Cambridge University Press, Cambridge, 2000.Suche in Google Scholar
[13] S. Meddahi, A. Márquez, and V. Selgas, Computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements. SIAM J. Numer. Anal. 41 (2003), No. 5, 1729–1750.10.1137/S0036142902406624Suche in Google Scholar
[14] P. Monk, Finite Element Methods for Maxwell’s Equations Oxford University Press, New York, 2003.10.1093/acprof:oso/9780198508885.001.0001Suche in Google Scholar
[15] S. A. Sauter and C. Schwab, Boundary Element Methods Springer Series in Computational Mathematics, Vol. 39. Springer-Verlag, Berlin, 2011.10.1007/978-3-540-68093-2Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
- Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems
- P1-nonconforming divergence-free finite element method on square meshes for Stokes equations
Artikel in diesem Heft
- Frontmatter
- Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
- Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems
- P1-nonconforming divergence-free finite element method on square meshes for Stokes equations
