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High Order Transparent Boundary Conditions for the Helmholtz Equation

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Modern Solvers for Helmholtz Problems

Part of the book series: Geosystems Mathematics ((GSMA))

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Abstract

We consider finite element simulations of the Helmholtz equation in unbounded domains. For computational purposes, these domains are truncated to bounded domains using transparent boundary conditions at the artificial boundaries. We present here two numerical realizations of transparent boundary conditions: the complex scaling or perfectly matched layer method and the Hardy space infinite element method. Both methods are Galerkin methods, but their variational framework differs. Proofs of convergence of the methods are given in detail for one dimensional problems. In higher dimensions radial as well as Cartesian constructions are introduced with references to the known theory.

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Notes

  1. 1.

    H loc r(Ω) denotes the space of functions, which belong to \(H^{r}(\hat{\varOmega })\) for each compact \(\hat{\varOmega }\subset \varOmega\).

  2. 2.

    \(\mathbb{N}\) denotes the set of all positive natural numbers and \(\mathbb{N}_{0}:=\{ 0\} \cup \mathbb{N}\).

  3. 3.

    H +(S 1) ⊂ L 2(S 1) consists of functions of the form ∑ j = 0 ∞ α j z j, z ∈ S 1, with a square summable series (α j ). These functions are boundary values of some functions, which are holomorphic in the complex unit disk. Equipped with the L 2(S 1) scalar product, H +(S 1) is a Hilbert space. For more details to Hardy spaces we refer to [14].

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Acknowledgements

Support from the Austrian Science Fund (FWF) through grant P26252 is gratefully acknowledged.

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Authors and Affiliations

  1. Technische Universität Wien, Wien, Austria

    Lothar Nannen

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  1. Lothar Nannen
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Correspondence to Lothar Nannen .

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Editors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Domenico Lahaye

  2. Delft Institute of Applied Mathematics, Delft University of Technology VORtech B.V., Delft, The Netherlands

    Jok Tang

  3. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Kees Vuik

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Nannen, L. (2017). High Order Transparent Boundary Conditions for the Helmholtz Equation. In: Lahaye, D., Tang, J., Vuik, K. (eds) Modern Solvers for Helmholtz Problems. Geosystems Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28832-1_2

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