Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM

Ihlenburg, Frank; Babuška, Ivo

doi:10.1016/0898-1221(95)00144-n

Cited by 538 publications

(463 citation statements)

References 13 publications

Supporting

Mentioning

451

Contrasting

Unclassified

Order By: Relevance

“…With suitable boundary conditions the Helmholtz equation (5) can be solved using a variety of numerical methods [10][11][12][13][14] . The accuracy of the numerical solution from the Helmholtz equation depends significantly on the wavenumber, κ (κ=ω/c).…”

Section: Simulationsmentioning

confidence: 99%

“…Consequently, the discretisation stepsize h of a numerical method has to be sufficiently refined to resolve the oscillations. A natural rule for such adjustment is to force [11,15,16] constant ·h = κ (6) which implies the unchanged resolution, i.e. the same grid points (or elements) per wavelength used.…”

Section: Simulationsmentioning

confidence: 99%

“…the same grid points (or elements) per wavelength used. However, it is known [11] that, for κh=constant, the errors of the finite element solutions increase rapidly as the wavenumber κ increases. This non-robust behaviour…”

Section: Simulationsmentioning

confidence: 99%

See 2 more Smart Citations

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Klíma

Frías-Ferrer

González-Garcı́a

et al. 2007

Ultrasonics Sonochemistry

135

View full text Add to dashboard Cite

The intensity distribution of the ultrasonic energy is, after the frequency, the most significant parameter to characterize ultrasonic fields in any sonochemical experiment.Whereas in the case of low intensity ultrasound the measurement of intensity and its distribution is well solved, in the case of high intensity (when cavitation takes place) the measurement is much more complicated. That is why the predicting the acoustic pressure distribution within the cell is desirable.A numerical solution of the wave equation gave the distribution of intensity within the cell. The calculations together with experimental verification have shown that the whole reactor behaves like a resonator and the energy distribution depends strongly on its shape.The agreement between computational simulations and experiments allowed optimisation of the shape of the sonochemical reactor. The optimal geometry resulted in a 2 strong increase in intensity along a large part of the cell. The advantages of such optimised geometry are (i) the ultrasonic power necessary for obtaining cavitation is low, (ii) low power delivered to the system results in only weak heating; consequently no cooling is necessary and (iii) the "active volume" is large, i.e. the fraction of the reactor volume with high intensity is large and is not limited to a vicinity close to the horn tip.

show abstract

Section: Simulationsmentioning

confidence: 99%

Section: Simulationsmentioning

confidence: 99%

See 1 more Smart Citation

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Klíma

Frías-Ferrer

González-Garcı́a

et al. 2007

Ultrasonics Sonochemistry

135

View full text Add to dashboard Cite

show abstract

“…A hope (not realized in this study) is that the k dependence of the error could be derived by this theory. See Ihlenburg [13] for a discussion of k dependence of error estimates for the related one-dimensional Helmholtz equation.…”

Section: Theorem 23 For H Sufficiently Small (Or For H 0 Sufficientmentioning

confidence: 99%

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Monk

Demkowicz

2000

Math. Comp.

View full text Add to dashboard Cite

Abstract. We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

show abstract

“…The Galerkin finite element method (FEM) for (1.2) in the one-dimensional case was first carried out in [8], where the well-posedness and error estimates of the Galerkin FEM were established under the condition k 2 h 1 using the Green's function and an argument due to Schatz [21]. A refined analysis for (1.2) in the one-dimensional case was performed in [18] (resp., [19]) for the h version (resp., hp version) of FEM, where the well-posedness and error estimates were established under the condition kh 1 using the discrete Green's functions. The proofs in these works rely heavily on the use of explicit forms of continuous and/or discrete Green's functions.…”

Section: Introductionmentioning

confidence: 99%

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Shen¹,

Wang

2005

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract.A complete error analysis is performed for the spectral-Galerkin approximation of a model Helmholtz equation with high wave numbers. The analysis presented in this paper does not rely on the explicit knowledge of continuous/discrete Green's functions and does not require any mesh condition to be satisfied. Furthermore, new error estimates are also established for multidimensional radial and spherical symmetric domains. Illustrative numerical results in agreement with the theoretical analysis are presented.

show abstract

Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM

Cited by 538 publications

References 13 publications

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Optimisation of 20 kHz sonoreactor geometry on the basis of numerical simulation of local ultrasonic intensity and qualitative comparison with experimental results

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Spectral Approximation of the Helmholtz Equation with High Wave Numbers

Contact Info

Product

Resources

About