Improvements for the ultra weak variational formulation

Luostari, Teemu; Huttunen, Tomi; Monk, Peter

doi:10.1002/nme.4469

Cited by 31 publications

(33 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Other basis functions can be utilized, such as Bessel functions [49,67,68] to improve the adaptivity to the curvature of the solution's wavefront and also reduce the linear dependence of the basis. Moreover, generalized plane waves [54][55][56] in the form e P(x) with an appropriate complex polynomial P(x) are developed to achieve high-order convergence for smooth heterogeneous media.…”

Section: Related Workmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

View full text Add to dashboard Cite

We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

show abstract

Section: Related Workmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

View full text Add to dashboard Cite

show abstract

“…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%

“…The use of evanescent plane waves in the basis is for instance examined to represent the transmission and reflection of sound at an interface between two media in [46] for the UWVF and in [45,29] for the DEM. However, this approach obviously requires a detailed knowledge of the solution behaviour before the numerical method is even formulated.…”

Section: Convergence Propertiesmentioning

confidence: 99%

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

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

View full text Add to dashboard Cite

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.

show abstract

“…We deliberately ignore the key challenge of ill-conditioning of linear systems arising from PWDG approaches, cf. [22,23]. We even acknowledge that an implementation of the method investigated below may severely be affected by numerical instability, see Remark 6.8.…”

Section: Introductionmentioning

confidence: 91%

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

2015

View full text Add to dashboard Cite

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftzdiscontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

show abstract

Improvements for the ultra weak variational formulation

Cited by 31 publications

References 35 publications

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

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

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

Contact Info

Product

Resources

About