Abstract
We present an interface discontinuous Galerkin finite element method on non-fitted meshes for solving acoustic wave propagation problems in nonhomogeneous media. The proposed method uses the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one material while on interface elements containing multiple materials it uses a specially build piecewise polynomial shape functions that satisfy the interface jump conditions. We present several computational results that suggest that the proposed method has optimal convergence rates.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Hesthaven, J., Warburton, T.: Nodal high-order methods on unstructured grids-I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)
Minoli, C.A.A., Kopriva, D.A.: Discontinuous Galerkin spectral element approximations on moving meshes. J. Comput. Phys. 230, 1876–1902 (2011)
Wilcox, L.C., Stadler, G., Burstedde, C., Ghattas, O.: A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media. J. Comput. Phys. 229(24), 9373–9396 (2010)
Carpenter, M.H., Kennedy, C.A.: Fourth order 2N-storage Runge Kutta scheme. Techical Memorandum NASA-TM-109112, NASA Langley Research Center, Hampton (1994)
Manthey, J.L., Hu, F.Q., Hussaini, M.Y.: Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics. J. Comput. Phys. 124, 177–191 (1996)
Hu, F.Q., Hussaini, M.Y., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151, 921–946 (1999)
Williamson, J.H.: Low storage Runge Kutta schemes. J. Comput. Phys. 35, 48–56 (1980)
Falk, R., Richter, G.: Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36, 935–952 (1999)
Monk, P., Richter, G.R.: A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media. J. Sci. Comput. 22–23, 443–477 (2005)
Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)
Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundary and interfaces. Adv. Comput. Math. 6, 109–138 (1996)
Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79, 175–202 (1998)
Heinrich, B.: Finite Difference Methods on Irregular Networks. International Series of Numerical Mathematics, vol. 82. Birkhäuser, Boston (1987)
Li, Z., Ito, K.: The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. Frontiers in Applied Mathematics, vol. 33. Cambridge University Press, SIAM, Cambridge (2006)
Samarskiǐ, A.A., Andreev, V.B.: Méthodes aux Différences pour Équations Elliptiques. Mir, Moscow (1978)
Xu, J.: Estimate of the convergence rate of the finite element solutions to elliptic equation of second order with discontinuous coefficients. Nat. Sci. J. **angtan Univ. 1, 1–5 (1982)
Lombard, B., Piraux, J.: A new interface method for hyperbolic problems with discontinuous coefficients: one-dimensional acoustic example. J. Comput. Phys. 1168, 227–248 (2001)
Lombard, B., Piraux, J.: Numerical treatment of two dimensional interfaces for acoustic and elastic waves. J. Comput. Phys. 195, 90–116 (2004)
Adjerid, S., Ben-Romdhane, M., Lin, T.: Higher degree immersed finite element methods for elliptic interface problems. Int. J. Numer. Anal. Model. 11(3), 541–566 (2014)
Adjerid, S., Lin, T.: Higher-order immersed discontinuous Galerkin methods. Int. J. Inform. Syst. Sci. 3(4), 555–568 (2007)
Adjerid, S., Lin, T.: p-th degree immersed finite element for boundary value problems with discontinuous coefficients. Appl. Numer. Math. 59(6),1303–1321 (2009)
He, X., Lin, T., Lin Y.: Approximation capability of a bilinear immersed finite element space. Numer. Meth. Part. Differ. Equat. 24, 1265–1300 (2008)
He, X., Lin, T., Lin, Y.: Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions. Int. J. Numer. Anal. Model. 8(2), 284–301 (2011)
He, X., Lin, T., Lin, Y., Zhang, X.: Immersed finite element methods for parabolic equations with moving interface. Numer. Meth. Part. Differ. Equat. 29(2), 619–646 (2013)
Kafafy, R., Lin, T., Lin, Y., Wang, J.: Three-dimensional immersed finite element methods for electric field simulation in composite materials. Int. J. Numer. Meth. Eng. 64, 904–972 (2005)
Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198, 106–130 (2004)
Adjerid, S., Moon, K.: A high order immersed discontinuous Galerkin method for the two dimensional acoustic problem (2014, in preparation)
Acknowledgements
This research was partially supported by the National Science Foundation (Grant Number DMS 1016313).
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Adjerid, S., Moon, K. (2014). A Higher Order Immersed Discontinuous Galerkin Finite Element Method for the Acoustic Interface Problem. In: Ansari, A. (eds) Advances in Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-06923-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-06923-4_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06922-7
Online ISBN: 978-3-319-06923-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)