Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $$hp$$ h p -Version

Hiptmair, Ralf; Moiola, Andrea; Perugia, Ilaria

doi:10.1007/s10208-015-9260-1

Cited by 40 publications

(46 citation statements)

References 35 publications

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“…By combining an h-refinement near solution singularities with a p-refinement in the elements where the solution is sufficiently smooth, exponential convergence of the errors in terms of a proper root of the number of degrees of freedom is expected. In the framework of Trefftz methods for the Helmholtz equation, a full hp-analysis was investigated for the PWDG method [30], where exponential convergence in terms of the square root of the number of degrees of freedom was proven.…”

Section: Hp-versionmentioning

confidence: 99%

A nonconforming Trefftz virtual element method for the Helmholtz problem: Numerical aspects

Mascotto

Perugia

Pichler

2019

Computer Methods in Applied Mechanics and Engineering

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We discuss the implementation details and the numerical performance of the recently introduced nonconforming Trefftz virtual element method [37] for the 2D Helmholtz problem. In particular, we present a strategy to significantly reduce the ill-conditioning of the original method; such a recipe is based on an automatic filtering of the basis functions edge by edge, and therefore allows for a notable reduction of the number of degrees of freedom. A widespread set of numerical experiments, including an application to acoustic scattering, the h-, p-, and hp-versions of the method, is presented. Moreover, a comparison with other Trefftz-based methods for the Helmholtz problem shows that this novel approach results in robust and effective performance.combines the VE technology with the Trefftz setting in a nonconforming fashion (à la Crouzeix-Raviart) following the pioneering works on nonconforming VEM for elliptic problems [5,14] and their extension to other problems [3, 13-15, 25, 34, 36, 46]. This nonconforming Trefftz-VEM, which can be regarded as a generalization of the nonconforming harmonic VEM [36], is "morally" comparable to many other Trefftz methods for the Helmholtz equation such as the ultra weak variational formulation [16], the wave based method [22], discontinuous methods based on Lagrange multipliers [24] and on least square formulation [39], the plane wave discontinuous Galerkin method (PWDG) [27], and the variational theory of complex rays [41]; see [31] for an overview of such methods.It has to be mentioned that all of the above Trefftz methods are based on fully discontinuous approximation spaces. A peculiarity of the nonconforming Trefftz-VEM is that a "weak" notion (that is, via proper edge L 2 projections) of traces over the skeleton of the polytopal grid is, differently from discontinuous methods, available.The aim of the present paper is to continue the work begun in [37], where the nonconforming Trefftz-VEM was firstly introduced, an abstract error analysis was carried out, and h-version error estimates were derived. As already mentioned in [37], the original version of the method does not result in good numerical performance, mainly because of the strong ill-conditioning of the local plane wave basis functions.The scope of this contribution is manifold. After introducing the model problem and extending the original nonconforming Trefftz-VEM in Section 2, we discuss the implementation details of the method in Section 3. We will consider here a more general Helmholtz boundary value problem than originally done in [37], which will be reflected in the definition of the nonconforming Trefftz-VE spaces. Then, numerical results are presented in Section 4, in order to clarify that, rebus sic stantibus, the method severely suffers of ill-conditioning. A numerical recipe based on an edgewise orthonormalization procedure to mitigate this strong ill-conditioning is presented in Section 5. Additionally to the fact that the condition number of the resulting global matrix significantly improves, the number...

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Section: Hp-versionmentioning

confidence: 99%

A nonconforming Trefftz virtual element method for the Helmholtz problem: Numerical aspects

Mascotto

Perugia

Pichler

2019

Computer Methods in Applied Mechanics and Engineering

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“…Although the so-called pollution effect is not avoidable for the classical Helmholtz equation in dimension d ≥ 2 as shown in [4], much work in its reduction has been invested: Examples of the proposed methods are the hp-version of the finite element method [23,41], (hybridizable) discontinuous Galerkin methods [15,29], or plane wave Trefftz methods [33,34,47], just to name a few. Recently, it has been shown that the resolution condition can be relaxed to the natural assumption "kh sufficiently small" by applying a Localized Orthogonal Decomposition (LOD) to the Hemholtz equation, see [13,26,48].…”

Section: Quasi-optimality Of the Hmmmentioning

confidence: 99%

A New Heterogeneous Multiscale Method for the Helmholtz Equation with High Contrast

Ohlberger¹,

Verfürth²

2018

Multiscale Model. Simul.

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In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the Helmholtz equation with high contrast. The method is constructed for a setting as in Bouchitté and Felbacq (C.R. Math. Acad. Sci. Paris 339(5): [377][378][379][380][381][382] 2004), where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We revisit existing homogenization approaches for this special setting and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to the Helmholtz equation with discontinuous diffusion matrix. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and an a priori error estimate under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. Numerical experiments confirm our theoretical convergence results and examine the resolution condition. Moreover, the numerical simulation gives a good insight and explanation of the physical phenomenon of frequency band gaps.

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“…By now, the challenges for wave propagation problems on the one side and for high-contrast coefficients on the other side mostly have been studied separately. To reduce the pollution effect for the Helmholtz equation with constant or low-contrast coefficients, a number of approaches has been designed and analyzed, such as (hybridizable) discontinuous Galerkin methods [12,27], the hp-version of the finite element method [21,40,41,42], (plane wave) Trefftz methods [24,33,48], or the Localized Orthogonal Decomposition [10,23,50]. High-contrast coefficients mainly have been studied for elliptic problems using multiscale finite element methods [13,18,45] and the Localized Orthogonal Decomposition [29,51], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

Computational high frequency scattering from high-contrast heterogeneous media

Peterseim

Verfürth

2020

Math. Comp.

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This article considers the computational (acoustic) wave propagation in strongly heterogeneous structures beyond the assumption of periodicity. A high contrast between the constituents of microstructured multiphase materials can lead to unusual wave scattering and absorption, which are interesting and relevant from a physical viewpoint, for instance, in the case of crystals with defects. We present a computational multiscale method in the spirit of the Localized Orthogonal Decomposition and provide its rigorous a priori error analysis for two-phase diffusion coefficients that vary between 1 and very small values. Special attention is paid to the extreme regimes of high frequency, high contrast, and their previously unexplored coexistence. A series of numerical experiments confirms the theoretical results and demonstrates the ability of the multiscale approach to efficiently capture relevant physical phenomena.

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Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $$hp$$ h p -Version

Cited by 40 publications

References 35 publications

A nonconforming Trefftz virtual element method for the Helmholtz problem: Numerical aspects

A nonconforming Trefftz virtual element method for the Helmholtz problem: Numerical aspects

A New Heterogeneous Multiscale Method for the Helmholtz Equation with High Contrast

Computational high frequency scattering from high-contrast heterogeneous media

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