A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems

Gamallo, Pablo; Astley, R.J.

doi:10.1002/nme.1948

Cited by 26 publications

(32 citation statements)

References 20 publications

(83 reference statements)

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“…This was noted in [15] where it was shown that the UWVF for Maxwell's equations can be derived using discontinuous Galerkin (DG) techniques and a special choice of degrees of freedom. This observation also holds for the Helmholtz equation (see also [9]). Using techniques of analysis appropriate for the discontinuous Galerkin method we can prove bounds on the jump of the error across element boundaries, and via duality techniques from [18] we obtain global convergence.…”

Section: Introductionsupporting

confidence: 56%

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Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Buffa¹,

Monk²

2008

ESAIM: M2AN

106

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Abstract. The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides avariational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.Mathematics Subject Classification. 65N15, 65N30, 35J05.

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Section: Introductionsupporting

confidence: 56%

“…First no estimates are available for the error in the presence of a boundary singularity. Computational results suggest that convergence should be provable in this case [9] but serious conditioning problems can arise. Secondly there are no estimates when both h and p are refined (an hp-method).…”

Section: Resultsmentioning

confidence: 98%

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Buffa¹,

Monk²

2008

ESAIM: M2AN

106

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“…Various choices of approximating functions are available: propagating plane waves [35,14,17,27], evanescent plane waves [29,45,46], Bessel functions [13], Green's functions [47]. Some elements of comparisons of these methods can be found in [48,36,49,19,37].…”

Section: Interpolation Basismentioning

confidence: 99%

“…It follows that the wavenumbers k n and vector amplitudes r n introduced in (26) have to be solution of the following eigenvalue problem:…”

Section: Interpolation Basismentioning

confidence: 99%

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A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.

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A comparison of wave‐based discontinuous Galerkin, ultra‐weak and least‐square methods for wave problems

Gabard

Gamallo

Huttunen

2010

Numerical Meth Engineering

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SUMMARYSeveral numerical methods using non-polynomial interpolation have been proposed for wave propagation problems at high frequencies. The common feature of these methods is that in each element, the solution is approximated by a set of local solutions. They can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation. There are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated. The similarities and discrepancies between several non-polynomial numerical methods are discussed in the context of the Helmholtz equation. The present comparison is concerned with the ultraweak variational formulation (UWVF), the least-squares method (LSM) and the discontinuous Galerkin method with numerical flux (DGM). An analysis in terms of Trefftz methods provides an interesting insight into the properties of these methods. Second, the UWVF and the LSM are reformulated in a similar fashion to that of the DGM. This offers a unified framework to understand the properties of several non-polynomial methods. Numerical results are also presented to put in perspective the relative accuracy of the methods. The numerical accuracies of the methods are compared with the interpolation errors of the wave bases.

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A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems

Cited by 26 publications

References 20 publications

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A comparison of wave‐based discontinuous Galerkin, ultra‐weak and least‐square methods for wave problems

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