Hardy Space Infinite Elements for Scattering and Resonance Problems

Hohage, Thorsten; Nannen, Lothar

doi:10.1137/070708044

Cited by 52 publications

(77 citation statements)

References 14 publications

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“…This permits an easy implementation of the expansion coefficients on the same grid as the solution, as illustrated in Figure 3, and is the reason why it is so easy to use the pole condition truncation. Note that this is the same system of equations for the a n as obtained using a Galerkin ansatz in the Hardyspace of the unit disc by Hohage and Nannen in [10]. …”

Section: The Pole Conditionmentioning

confidence: 85%

“…The pole condition leads to a numerical method for domain truncation which is easy to implement and has shown great promise in numerical experiments for a variety of problems, see [12,13,10]. We show in this paper for a model problem of diffusive nature an error estimate for the numerical method based on the pole condition: the domain truncation achieved is a Padé approximation of the transparent boundary condition.…”

Section: Introductionmentioning

confidence: 93%

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The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator

Gander

Schädle

2010

Lecture Notes in Computational Science and Engineering

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Section: The Pole Conditionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 93%

The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator

Gander

Schädle

2010

Lecture Notes in Computational Science and Engineering

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“…The original Hardy space infinite element method [12,17,18] employs the Hardy space H + (S 1 ) of the complex unit sphere. Since F ∈ H + (S 1 ) is the boundary value of functions, which are holomorphic in the complex unit disk, it can be written as …”

Section: Two Pole Basis Functionsmentioning

confidence: 99%

“…Since the wavenumbers and the waveguide modes depend non-linearly on the frequency, these approaches result for resonance problems into the solution of a non-standard eigenvalue problem. Frequency independent methods like the perfectly matched layer method [1,16,18] or the Hardy space infinite element method [12,17,18] lead to linear eigenvalue problems. But the performance of these methods deteriorates in the vicinity of cut-off frequencies, where an evanescent mode becomes a propagating mode.…”

Section: Introductionmentioning

confidence: 99%

Two scale Hardy space infinite elements for scalar waveguide problems

Halla

Nannen

2017

Adv Comput Math

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We consider the numerical solution of the Helmholtz equation in domains with one infinite cylindrical waveguide. Such problems exhibit wavenumbers on different scales in the vicinity of cut-off frequencies. This leads to performance issues for non-modal methods like the perfectly matched layer or the Hardy space infinite element method. To improve the latter, we propose a two scale Hardy space infinite element method which can be optimized for wavenumbers on two different scales. It is a tensor product Galerkin method and fits into existing analysis. Up to arbitrary small thresholds it converges exponentially with respect to the number of longitudinal unknowns in the waveguide. Numerical experiments support the theoretical error bounds.

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“…It was further explored in [6,11,16,21]. An alternative formulation of the pole condition is presented in [17], which provides a noticeably simplified implementation and is also used in the present paper.…”

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confidence: 99%

Transparent Boundary Conditions for Time-Dependent Problems

Ruprecht¹,

Schädle²,

Schmidt³

et al. 2008

SIAM J. Sci. Comput.

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Abstract. The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some problem dependent complex half-plane, these modes can be suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schrödinger and the drift-diffusion equation but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.

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Hardy Space Infinite Elements for Scattering and Resonance Problems

Cited by 52 publications

References 14 publications

The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator

The Pole Condition: A Padé Approximation of the Dirichlet to Neumann Operator

Two scale Hardy space infinite elements for scalar waveguide problems

Transparent Boundary Conditions for Time-Dependent Problems

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