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High Order Whitney Forms on Simplices and the Question of Potentials Read chapter Read first chapter

2021 | OriginalPaper | Chapter

On the Dirichlet-to-Neumann Coarse Space for Solving the Helmholtz Problem Using Domain Decomposition

Authors : Niall Bootland, Victorita Dolean

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2019

Publisher: Springer International Publishing

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Abstract

We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

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previous chapter Multipreconditioning with Application to Two-Phase Incompressible Navier–Stokes Flow
next chapter A Comparison of Boundary Element and Spectral Collocation Approaches to the Thermally Coupled MHD Problem
Footnotes
1
Note that if ΓR = ∅ then the problem will be ill-posed for certain choices of k corresponding to Dirichlet eigenvalues of the corresponding Laplace problem.
 
Literature
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Babuška, I.M., Sauter, S.A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal. 34(6), 2392–2423 (1997)MathSciNetCrossRef
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Conen, L., Dolean, V., Krause, R., Nataf, F.: A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator. J. Comput. Appl. Math. 271, 83–99 (2014)MathSciNetCrossRef
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Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation, vol. 144. SIAM (2015)
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Ernst, O.G., Gander, M.J.: Why it is difficult to solve Helmholtz problems with classical iterative methods. In: I.G. Graham, T.Y. Hou, O. Lakkis, R. Scheichl (eds.) Numerical Analysis of Multiscale Problems, pp. 325–363. Springer (2012)
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Nataf, F., Xiang, H., Dolean, V., Spillane, N.: A coarse space construction based on local Dirichlet-to-Neumann maps. SIAM J. Sci. Comput. 33(4), 1623–1642 (2011)MathSciNetCrossRef
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Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014)MathSciNetCrossRef
Metadata
Title
On the Dirichlet-to-Neumann Coarse Space for Solving the Helmholtz Problem Using Domain Decomposition
Authors
Niall Bootland
Victorita Dolean
Copyright Year
2021
Book
Numerical Mathematics and Advanced Applications ENUMATH 2019

Print ISBN: 978-3-030-55873-4

Electronic ISBN: 978-3-030-55874-1

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-55874-1_16

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