Dispersion analysis of a nonconforming finite element method for the three-dimensional scalar and elastic wave equations

Zyserman, Fabio; Gauzellino, Patricia Mercedes

doi:10.1016/j.finel.2004.12.010

Cited by 21 publications

(13 citation statements)

References 19 publications

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“…For dispersion analysis, we follow the idea given in [20,19], by setting the source term in (2.7) to zero and neglecting the boundary condition. Also c is assumed to be constant in the dispersion analysis.…”

Section: Numerical Dispersion Of P 1 -Nc Finite Element Solutionsmentioning

confidence: 99%

“…If the three-dimensional DSSY NC element is used, Zyserman and Gauzellino show the following dispersion relation: [19] …”

Section: )mentioning

confidence: 99%

“…Specifically, the following three conforming and nonconforming element methods will be analyzed: (1) the standard Q 1 conforming element (abbreviated as the "Q 1 element"); (2) the DSSY nonconforming element introduced by Douglas et al [5] (abbreviated as the "DSSY NC" element, or the "DSSY" element) which is a modified rotated Q 1 element of Rannacher and Turek [15]; and (3) the P 1 -nonconforming quadrilateral(hexahedron) element [14] (abbreviated as the " P 1 -NC element"). Santos et al [20] and Zyserman et al [19] gave detailed dispersion analyses for solving the Helmholtz equation, and elastic and viscoelastic equations by comparing between the Q 1 conforming and the DSSY NC finite element methods. It is shown in [1,20,19] that the L 2 -error of the DSSY NC element behaves better in reducing numerical dispersion than that of Q 1 element based on the same size of grids.…”

Section: Introductionmentioning

confidence: 99%

“…Santos et al [20] and Zyserman et al [19] gave detailed dispersion analyses for solving the Helmholtz equation, and elastic and viscoelastic equations by comparing between the Q 1 conforming and the DSSY NC finite element methods. It is shown in [1,20,19] that the L 2 -error of the DSSY NC element behaves better in reducing numerical dispersion than that of Q 1 element based on the same size of grids. However, it has been questionable if the DSSY NC element is actually cheaper than the Q 1 conforming element to achieve desired accuracy.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of conforming and nonconforming quadrilateral finite element methods for the Helmholtz equation

Lee¹,

Ha²,

Sheen³

2007

Hokkaido Math. J.

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In this paper we analyze numerical dispersion relation of some conforming and nonconforming quadrilateral finite elements. The elements employed in this analysis are the standard Q 1 conforming finite element, the DSSY nonconforming element [5] and the P 1 -nonconforming quadrilateral finite element [14]. Several aspects of comparative analyses of the above three elements for two or three dimensional problems are shown.

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Section: Numerical Dispersion Of P 1 -Nc Finite Element Solutionsmentioning

confidence: 99%

“…If the three-dimensional DSSY NC element is used, Zyserman and Gauzellino show the following dispersion relation: [19] …”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of conforming and nonconforming quadrilateral finite element methods for the Helmholtz equation

Lee¹,

Ha²,

Sheen³

2007

Hokkaido Math. J.

View full text Add to dashboard Cite

show abstract

“…Many methods have been proposed for studying the dispersive properties of various numerical schemes (e.g., finite difference method (FDM), FEM, and SEM) for many problems, such as wave propagation, linear convection diffusion, and Helmholtz equations. Some methods measure dispersion, such as eigenvalue [13][14][15][16][17][18], wavenumber [12,[19][20][21][22], angular frequency [23][24][25], and error derivation [26][27][28][29] methods. However, such dispersion analysis considers only spatial discretization.…”

Section: Introductionmentioning

confidence: 99%

Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

Geng

Qin

Zhang

et al. 2017

Acoust. Sci. & Tech.

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This paper presents the solution for the continuous space-time spectral element method (CSTSEM) based on Chebyshev polynomials for the acoustic wave equation. Acoustic wave propagation in various dimensions is simulated using quadrilateral, hexahedral, and tesseract elements. The convergence is studied for 1+1-dimensional wave propagation. The extended 2+1-and 3+1-dimensional wave equations are also numerically solved by CSTSEM and the dispersion characteristics are investigated. A fixed-! (! is the angular frequency) method is proposed for computing the dispersion of space-time coupled methods. CSTSEM is verified as a simple, practical isotropic algorithm with low dispersion and high accuracy.

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Dispersion Analysis of Triangle‐Based Finite Element Method for Acoustic and Elastic Wave Simulations

Liu

et al. 2014

Chinese J of Geophysics

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Acoustic and elastic wave equations are discretized by finite element method with triangles and central difference scheme in space and time, respectively. The generalized eigenvalue problems are constructed to analyze the numerical dispersion of the finite element method. Three types of mass finite element method, which are consistent finite element method (CFEM), lumped finite element method (LFEM) and mixed finite element method (MFEM), are implemented to the numerical scheme, and the numerical dispersion is under a comprehensive study. Then, the MFEM is adopted to investigate the behavior of numerical anisotropy in terms of four typical triangle meshes. Comparing the numerical dispersion and anisotropy behavior when MFEM uses different interpolation orders, one can easily find that numerical error decreases as the interpolation order increases. Specifically, the numerical dispersion and anisotropy are negligible when third‐order interpolation is used. Finally, controlling the other impact factors, the dispersive properties of elastic wave propagation in elastic medium with various velocity ratios are analyzed, and the results of numerical dissipation are presented at the end of this paper.

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Dispersion analysis of a nonconforming finite element method for the three-dimensional scalar and elastic wave equations

Cited by 21 publications

References 19 publications

Analysis of conforming and nonconforming quadrilateral finite element methods for the Helmholtz equation

Analysis of conforming and nonconforming quadrilateral finite element methods for the Helmholtz equation

Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

Dispersion Analysis of Triangle‐Based Finite Element Method for Acoustic and Elastic Wave Simulations

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