volume | PIER Journals

Abstract

Dispersion error analysis can help to assess the performance of finite-element discretizations of the wave equation. Although less general than the convergence estimates offered by standard finite-element error analysis, it can provide more detailed insight as well as practical guidelines in terms of the number of elements per wavelength needed for acceptable results. We present eigenvalue and eigenvector error estimates for cubic Hermite elements on an equidistant 1-D mesh and on a regular structured 2-D triangular mesh consisting of squares cut in half. The results show that in 1D, the spectrum consists of 2 modes. If these are unwrapped, the spectrum is effectively doubled. The eigenvalue or dispersion error stays below 7% across the entire spectrum. The error in the corresponding eigenvectors, however, increases rapidly once the number of elements per wavelength decreases to one. In terms of element size, the dispersion error is of order 6 and the eigenvector error of order 4. The latter is consistent with the classic finite-element error estimate. In 2D, we provide eigenvalue and eigenvector errors as a series expansion in the element size and obtain the same orders. 2-D numerical tests in the timeand frequency-domain are included.

1. Mulder, W. A., "A comparison between higher-order finite elements and finite differences for solving the wave equation," Proceedings of the Second ECCOMAS Conference on Numerical Methods in Engineering, 344-350, 1996.

2. Zhebel, Elena, Sara Minisini, Alexey Kononov, and Wim A. Mulder, "A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation," Geophysical Prospecting, Vol. 62, No. 5, 1111-1125, 2014.

3. Geevers, Sjoerd, Wim A. Mulder, and Jaap J. W. van der Vegt, "Dispersion properties of explicit finite element methods for wave propagation modelling on tetrahedral meshes," Journal of Scientific Computing, Vol. 77, No. 1, 372-396, Oct. 2018.

4. Geevers, Sjoerd, Wim A. Mulder, and Jaap J. W. van der Vegt, "New higher-order mass-lumped tetrahedral elements for wave propagation modelling," SIAM Journal on Scientific Computing, Vol. 40, No. 5, A2830-A2857, 2018.

5. Geevers, Sjoerd, Wim A. Mulder, and Jaap J. W. van der Vegt, "Efficient quadrature rules for computing the stiffness matrices of mass-lumped tetrahedral elements for linear wave problems," SIAM Journal on Scientific Computing, Vol. 41, No. 2, A1041-A1065, 2019.

6. Mulder, W. A., "Performance of old and new mass-lumped triangular finite elements for wavefield modelling," Geophysical Prospecting, Vol. 72, No. 3, 885-896, 2024.

7. Mulder, W. A., "Spurious modes in finite-element discretizations of the wave equation may not be all that bad," Applied Numerical Mathematics, Vol. 30, No. 4, 425-445, 1999.

8. Ciarlet, Philippe G. and Pierre-Arnaud Raviart, "General Lagrange and Hermite interpolation in Rn with applications to finite element methods," Archive for Rational Mechanics and Analysis, Vol. 46, No. 3, 177-199, Jan. 1972.

9. Tordjman, N., "Eléments finis d'ordre élevé avec condensation de masse pour l'équation des ondes," Ph.D. dissertation, L’Université Paris IX Dauphine, France, 1995.

10. Cohen, Gary, Patrick Joly, Jean E. Roberts, and Nathalie Tordjman, "Higher order triangular finite elements with mass lumping for the wave equation," SIAM Journal on Numerical Analysis, Vol. 38, No. 6, 2047-2078, 2001.

11. Chin-Joe-Kong, M. J. S., Wim A. Mulder, and M. van Veldhuizen, "Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation," Journal of Engineering Mathematics, Vol. 35, 405-426, 1999.

12. Cohen, Gary, Patrick Joly, Jean E. Roberts, and Nathalie Tordjman, "Higher order triangular finite elements with mass lumping for the wave equation," SIAM Journal on Numerical Analysis, Vol. 38, No. 6, 2047-2078, 2001.

13. Mulder, Wim A., "Higher-order mass-lumped finite elements for the wave equation," Journal of Computational Acoustics, Vol. 9, No. 2, 671-680, 2001.

14. Mulder, William Alexander, "New triangular mass-lumped finite elements of degree six for wave propagation," Progress In Electromagnetics Research, Vol. 141, 671-692, 2013.

15. Cui, Tao, Wei Leng, Deng Lin, Shichao Ma, and Linbo Zhang, "High order mass-lumping finite elements on simplexes," Numerical Mathematics: Theory, Methods and Applications, Vol. 10, No. 2, 331-350, 2017.

16. Liu, Youshan, Jiwen Teng, Tao Xu, and José Badal, "Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling," Journal of Computational Physics, Vol. 336, 458-480, 2017.

17. Mulder, Wim A., "More continuous mass-lumped triangular finite elements," Journal of Scientific Computing, Vol. 92, No. 2, 38, 2022.

18. Felippa, C. A., "Customizing high performance elements by Fourier methods," Trends in Computational Structural Mechanics, 283-296, W. A. Wall, K.-U. Bletzinger, and K. Schweizerhof, Eds., CIMNE, Barcelona, Spain, 2001.

19. Park, K. C. and D. L. Flaggs, "A Fourier analysis of spurious mechanisms and locking in the finite element method," Computer Methods in Applied Mechanics and Engineering, Vol. 46, No. 1, 65-81, 1984.

20. Mullen, Robert and Ted Belytschko, "Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation," International Journal for Numerical Methods in Engineering, Vol. 18, No. 1, 11-29, 1982.

21. Cottrell, J. Austin, Alessandro Reali, Yuri Bazilevs, and Thomas J. R. Hughes, "Isogeometric analysis of structural vibrations," Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 41, 5257-5296, 2006.

22. Boucher, C. R., Zehao Li, C. I. Ahheng, J. D. Albrecht, and L. R. Ram-Mohan, "Hermite finite elements for high accuracy electromagnetic field calculations: A case study of homogeneous and inhomogeneous waveguides," Journal of Applied Physics, Vol. 119, No. 14, 143106, 2016.

23. Shamasundar, Ranjani, "Finite element methods for seismic imaging: Cost reduction through mass matrix preconditioning by defect correction," Delft University of Technology, The Netherlands, 2019.

24. Strang, G. and G. J. Fix, An Analysis of the Finite Element Method, ser. Series in Automatic Computation, XIV, Prentice-Hall, Inc., Englewood Cliffs, NY, 1973.

25. Davis, T. A., Direct Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, 2006.
doi:10.1137/1.9780898718881

26. Wu, Jo-Yu and Robert Lee, "The advantages of triangular and tetrahedral edge elements for electromagnetic modeling with the finite-element method," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 9, 1431-1437, 1997.
doi:10.1109/5.317086

27. Mulder, W. A., E. Zhebel, and S. Minisini, "Time-stepping stability of continuous and discontinuous finite-element methods for 3-D wave propagation," Geophysical Journal International, Vol. 196, No. 2, 1123-1133, 2014.

28. Engwirda, Darren, "Locally optimal Delaunay-refinement and optimisation-based mesh generation," The University of Sydney, Australia, 2014.

Mr. William Alexander Mulder

Dr. Ranjani Shamasundar