Summary
We present hybrid finite element methods for the Helmholtz equation and the time harmonic Maxwell equations, which allow us to reduce the unknowns to degrees of freedom supported only on the element facets and to use efficient iterative solvers for the resulting system of equations. For solving this system, additive and multiplicative Schwarz preconditioners with local smoothers and a domain decomposition preconditioner with an exact subdomain solver are presented. Good convergence properties of these preconditioners are shown by numerical experiments.
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Keywords
- Domain Decomposition
- Helmholtz Equation
- Discontinuous Galerkin Method
- Domain Decomposition Method
- Iterative Solver
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Bibliography
D.N. Arnold and F. Brezzi. Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer., 19(1):7–32, 1985.
O. Cessenat and B. Despres. Application of an Ultra Weak Variational Formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal., 35(1):255–299, 1998.
B. Engquist and L. Ying. Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation. Comm. Pure Appl. Math., 64(5):697–735, 2011.
Y.A. Erlangga, C. Vuik, and C.W. Oosterlee. On a class of preconditioners for solving the Helmholtz equation. Appl. Numer. Math., 50(3-4):409–425, 2004.
C. Farhat, I. Harari, and U. Hetmaniuk. A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Engrg., 192 (11-12):1389–1419, 2003.
X. Feng and H. Wu. Discontinuous Galerkin methods for the Helmholtz equation with large wave number. SIAM J. Numer. Anal., 47(4): 2872–2896, 2009.
I. Harari. A survey of finite element methods for time harmonic acoustics. Comput. Methods Appl. Mech. Engrg., 195(13-16):1594–1607, 1997.
F. Ihlenburg and I. Babuska. Finite element solution of the Helmholtz equation with high wave number part ii: \(hp\)-version of the FEM. SIAM J. Numer. Anal., 34(1):315–358, 1997.
J.M. Melenk. On Generalized Finite Element Methods. Phd thesis, University of Maryland, 1995.
P. Monk. Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford, 2003.
P. Monk, A. Sinwel, and J. Schöberl. Hybridizing Raviart-Thomas elements for the Helmholtz equation. Electromagnetics, 30(1):149–176, 2010.
K. Zhao, V. Rawat, S.C. Lee, and J.F. Lee. A domain decomposition method with non-conformal meshes of finite periodic and semi-periodic structures. IEEE Trans. Antennas and Propagation, 55(9):2559–2570, 2007.
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Huber, M., Pechstein, A., Schöberl, J. (2013). Hybrid Domain Decomposition Solvers for the Helmholtz and the Time Harmonic Maxwell’s Equation. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_32
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DOI: https://doi.org/10.1007/978-3-642-35275-1_32
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