Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

Hoppe, Ronald H. W.; Sharma, Neeraj Kumar

doi:10.1093/imanum/drs028

Cited by 19 publications

(18 citation statements)

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“…General techniques of a posteriori error estimation for elliptic problems are described in [2], [39], [45] while the focus in [15] is on dG-methods. A posteriori error estimation for the conventional conforming discretization of the Helmholtz problem are described in [16] and for an IPDG method in [26]. For the derivation of an a posteriori error estimator for the dG-formulation of the Helmholtz problem the main challenges are a) the lower order term −k 2 (·, ·) in the sesquilinear forms a (·, ·) and a T (·, ·), which causes the problem to be highly indefinite and b) the integrals in (2.8b) containing the mean of the gradient on interior edges, which have the effect that a T (·, ·) + 2k 2 (·, ·) L 2 is not coercive on H…”

Section: A Posteriori Error Estimationmentioning

confidence: 99%

“…In contrast, we do not prove the convergence of the resulting adaptive method for our dG-formulation. On the other hand, our estimators are properly weighted with the polynomial degree and the estimates are explicit with respect to the wavenumber k, the mesh width h, and the polynomial degree p. In addition, the dependence of the constants in the estimates on the wavenumber k are milder in our approach compared to [26]; b) In [16], a residual a posteriori error estimator (cf. [5], [6], [2], [45]) has been developed for the conventional hp-finite element method.…”

Section: Introductionmentioning

confidence: 99%

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A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems

Sauter¹,

Zech²

2015

SIAM J. Numer. Anal.

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Abstract. In this paper, we will consider an hp-finite elements discretization of a highly indefinite Helmholtz problem by some dG formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés.We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters h and p. In contrast to the conventional conforming finite element method for indefinite problems, the dG formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh.Numerical experiments will illustrate the efficiency and robustness of the method.Key words. Helmholtz equation at high wavenumber, hp-finite elements, a posteriori error estimation, discontinuous Galerkin methods, ultra-weak variational formulation AMS subject classifications. 35J05, 65N12, 65N301. Introduction. High frequency scattering problems are ubiquitous in many fields of science and engineering and their reliable and efficient numerical simulation pervades numerous engineering applications such as detection (e.g., radar), communication (e.g., wireless), and medicine (e.g., sonic imaging) ([32], [1]). These phenomena are governed by systems of linear partial differential equations (PDEs); the wave equation for elastic waves and the Maxwell equations for electromagnetic scattering. We are here interested in time-harmonic problems where the equation can be reduced to purely spatial problems; for high frequencies these PDEs become highly indefinite and the development of accurate numerical solution methods is far from being in a mature state.In this paper we will consider the Helmholtz problem with high wavenumber as our model problem. Although the continuous problem with appropriate boundary conditions has a unique solution, conventional hp-finite element methods require a minimal resolution condition such that existence and uniqueness is guaranteed on the discrete level (see, e.g., [30], [29], [36], [37], [11]). However, this condition, typically, contains a generic constant C which is either unknown for specific problems or only very pessimistic estimates are available. This is one of the major motivations for the development of stabilized formulations such that the discrete system is always solvable -well-known examples include least square techniques [39,23,24,22] and discontinuous Galerkin (dG) methods [18,19,20,45,47]. These formulations lead to discrete systems which are unconditionally stable, i.e., no resolution condition is required. Although convergence starts for these methods only after a resolution condition is reached, the stability of the discrete system is considerably improved. The Ultra Weak Variational Formulation (UWVF) of Cessenat and Després [9,10,13] can be understood as a dG-method that permits the use of non-standard, discontinuous local discretization spaces such as plane waves (see [21,28,25,8]). In this paper we

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Section: A Posteriori Error Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems

Sauter¹,

Zech²

2015

SIAM J. Numer. Anal.

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“…where The exact solution u is not known explicitly. As a substitute for the exact solution we have used an approximate solution u s computed by the adaptive Interior Penalty Discontinuous Galerkin method from [12] with a sufficiently large number of refinement steps. For ω = 15, the approximate solution u s is displayed in Figure 6.6 (right).…”

Section: Projected Gradient Methodmentioning

confidence: 99%

“…Example 2: The computational domain Ω and the simplicial triangulation T h (Ω) for the PWDG method (left; the sound-soft scatterer is shown in blue). The substitute solution us computed by the adaptive Interior Penalty Discontinuous Galerkin method from[12] (right).…”

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confidence: 99%

Optimization of plane wave directions in plane wave discontinuous Galerkin methods for the Helmholtz equation

Agrawal¹,

Hoppe²

2017

Port. Math.

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Recently, the use of special local test functions other than polynomials in Discontinuous Galerkin (DG) approaches has attracted a lot of attention and became known as DG-Trefftz methods. In particular, for the 2D Helmholtz equation plane waves have been used in [10] to derive an Interior Penalty (IP) type Plane Wave DG (PWDG) method and to provide an a priori error analysis of its p-version with respect to equidistributed plane wave directions. However, the dependence on the distribution of the plane wave directions has not been studied. In this contribution, we study this dependence by formulating the choice of the directions as an optimal control problem with a tracking type objective functional and the variational formulation of the PWDG method as a constraint. The necessary optimality conditions are derived and numerically solved by a projected gradient method. Numerical results are given which illustrate the benefits of the approach.

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“…They have been in recent years among the primary candidates for solving problems involving convection/diffusion terms as well as reaction terms (see, e.g., [36,37] and the references cited therein). These methods received, more recently, a great deal of attention for wave problems, as attested by the various formulations that have been proposed for solving Helmholtz problems [10,[13][14][15][16][17][38][39][40][41][42][43]. This category of methods is very attractive because of several considerations, chief among them are the following:They offer cost-effective procedures for linking separate elements/domains in each of which finite elements, or plane waves, or any expansion series are used for approximation.…”

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confidence: 99%

Efficient DG‐like formulation equipped with curved boundary edges for solving elasto‐acoustic scattering problems

Barucq

Djellouli

Estecahandy

2014

Numerical Meth Engineering

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SUMMARYA Discontinuous Galerkin (DG)‐based approach is proposed for computing the scattered field from an elastic bounded object immersed in an infinite homogeneous fluid medium. The proposed method possesses two distinctive features. First, it employs higher‐order polynomial‐shape functions needed to address the high‐frequency propagation regime. Second, it is equipped with curved boundary edges to provide an accurate representation of the fluid–structure interface. The most salient benefits resulting from the latter feature, as demonstrated by the numerical investigation, are the following: (i) an improvement by—at least—two orders of magnitude on the relative error and (ii) the disappearance of spurious resonance frequencies in the surrounding fluid medium. In addition, the reported numerical results reveal that when using cubic polynomials with less than three elements per wavelength, the proposed DG method computes the scattered field with a relative error below 1% for an elastic scatterer of about 30 wavelengths. This observation highlights the potential of the proposed solution methodology for efficiently solving mid‐frequency to high‐frequency elasto‐acoustic scattering problems. Copyright © 2014 John Wiley & Sons, Ltd.

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Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

Cited by 19 publications

References 47 publications

A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems

A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems

Optimization of plane wave directions in plane wave discontinuous Galerkin methods for the Helmholtz equation

Efficient DG‐like formulation equipped with curved boundary edges for solving elasto‐acoustic scattering problems

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