Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
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By W. A. Mulder

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Mass-lumped continuous finite elements allow for explicit time stepping with the second-order wave equation if the resulting integration weights are positive and provide sufficient accuracy. To meet these requirements on triangular and tetrahedral meshes, the construction of higher-degree elements for a given polynomial degree on the edges involves polynomials of higher degrees in the interior. The parameters describing the supporting nodes of the Lagrange interpolating polynomials and the integration weights are the unknowns of a polynomial system of equations, which is linear in the integration weights. To find candidate sets for the nodes, it is usually required that the number of equations equals the number of unknowns, although this may be neither necessary nor sufficient. Here, this condition is relaxed by requiring that the number of equations does not exceed the number of unknowns. This resulted in two new types elements of degree 6 for symmetrically placed nodes. Unfortunately, the first type is not unisolvent. There are many elements of the second type with a large range in their associated time-stepping stability limit. To assess the efficiency of the elements of various degrees, numerical tests on a simple problem with an exact solution were performed. Efficiency was measured by the computational time required to obtain a solution at a given accuracy. For the chosen example, elements of degree 4 with fourth-order time stepping appear to be the most efficient.

W. A. Mulder, "New Triangular Mass-Lumped Finite Elements of Degree Six for Wave Propagation," Progress In Electromagnetics Research, Vol. 141, 671-692, 2013.

1. Fried, I. and D. S. Malkus, "Finite element mass matrix lumping by numerical integration with no convergence rate loss," International Journal of Solids and Structures, Vol. 11, No. 4, 461-466, 1975.

2. Tordjman, N., "Elements finis d'order eleve avec condensation de masse pour l'equation des ondes,", Ph.D. Thesis, L'Universite Paris IX Dauphine, 1995.

3. Cohen, G., P. Joly, and N. Tordjman, "Higher order triangular finite elements with mass lumping for the wave equation," Proceedings of the Third International Conference on Mathematical and Numerical Aspects of Wave Propagation, G. Cohen, E. Becache, P. Joly, and J. E. Roberts, Eds., 270-279, SIAM, Philadelphia, 1995.

4. 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, J.-A. Desideri, E. Onate P. Le Tallec, J. Periaux, E. Stein, (eds.), 344-350, John Wiley & Sons, Chichester, 1996.

5. Chin-Joe-Kong, M. J. S., W. 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, No. 4, 405-426, 1999.

6. Giraldo, F. X. and M. A. Taylor, "A diagonal-mass-matrix triangular-spectral-element method based on cubature points," Journal of Engineering Mathematics, Vol. 56, No. 3, 307-322, 2006.

7. Lyness, J. N. and D. Jespersen, "Moderate degree symmetric quadrature rules for the triangle," Journal of the Institute of Mathematics and Its Applications, Vol. 15, 19-32, 1975.

8. Keast, P., "Cubature formulas for the surface of the sphere," Journal of Computational and Applied Mathematics, Vol. 17, No. 1-2, 151-172, 1987.

9. Heo, S. and Y. Xu, "Constructing fully symmetric cubature formulae for the sphere," Mathematics of Computation, Vol. 70, No. 233, 269-279, 2001.

10. Keast, P. and J. C. Diaz, "Fully symmetric integration formulas for the surface of the sphere in s dimensions," SIAM Journal on Numerical Analysis, Vol. 20, No. 2, 406-419, 1983.

11. Keast, P., "Moderate-degree tetrahedral quadrature formulas," Computer Methods in Applied Mechanics and Engineering, Vol. 55, No. 3, 339-348, 1986.

12. Cohen, G., P. Joly, J. E. Roberts, and N. 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, W. A., "Higher-order mass-lumped finite elements for the wave equation," Journal of Computational Acoustics, Vol. 9, No. 2, 671-680, 2001.

14. Courant, R., K. Friedrichs, and H. Lewy, "Uber die partiellen Differenzengleichungen der mathematischen Physik," Mathematische Annalen, Vol. 100, No. 1, 32-74, 1928.

15. Zhebel, E., S. Minisini, A. Kononov, and W. A. Mulder, "Solving the 3D acoustic wave equation with higher-order mass-lumped tetrahedral finite elements," 73rd EAGE Conference & Exhibition, Extended Abstracts, A010, Vienna, Austria, May 2011.

16. Jund, S. and S. Salmon, "Arbitrary high-order finite element schemes and high-order mass lumping," International Journal of pplied Mathematics and Computer Science, Vol. 17, No. 3, 375-393, 2007.

17. Lax, P. and B. Wendroff, "Systems of conservation laws," Communications on Pure and Applied Mathematics, Vol. 31, No. 2, 217-237, 1960.

18. Shubin, G. R. and J. B. Bell, "A modified equation approach to constructing fourth order methods for acoustic wave propagation," SIAM Journal on Scientific and Statistical Computing, Vol. 8, No. 2, 135-151, 1987.

19. Dablain, M. A., "The application of high-order differencing to the scalar wave equation," Geophysics, Vol. 51, No. 1, 54-66, 1986.

20. Chen, J.-B., "Lax-Wendroff and Nystrom methods for seismic modelling," Geophysical Prospecting, Vol. 57, No. 6, 931-941, 2009.

21. De Basabe, J. D. and M. K. Sen, "Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping," Geophysical Journal International, Vol. 181, No. 1, 577-590, 2010.

22. Gilbert, J. C. and P. Joly, "Higher order time stepping for second order hyperbolic problems and optimal CFL conditions," Computational Methods in Applied Sciences, Vol. 16, 67-93, Springer, Berlin, 2008.

23. Koornwinder, T., "Two-variable analogues of the classical orthogonal polynomials," Theory and Application of Special Functions, R. A. Askey (ed.), 435-495, Academic Press, New York, 1975.

24. Dubiner, M., "Spectral methods on triangles and other domains," Journal of Scientific Computing, Vol. 6, No. 4, 345-390, 1991.

25. Sherwin, S. J. and G. E. Karniadakis, "A new triangular and tetrahedral basis for high-order (hp) finite element methods," International Journal for Numerical Methods in Engineering, Vol. 38, No. 22, 3775-3802, 1995.

26. Heinrichs, W. and B. I. Loch, "Spectral schemes on triangular elements," Journal of Computational Physics, Vol. 173, No. 1, 279-301, 2001.

27. Bittencourt, M. L., "Fully tensorial nodal and modal shape functions for triangles and tetrahedra," International Journal for Numerical Methods in Engineering, Vol. 63, No. 11, 1530-1558, 2005.

28. Xu, Y., "On Gauss-Lobatto integration on the triangle," SIAM Journal on Numerical Analysis, Vol. 49, No. 2, 541-548, 2011.

29. Taylor, M. A., B. A. Wingate, and R. E. Vincent, "An algorithm for computing Fekete points in the triangle," SIAM Journal on Numerical Analysis, Vol. 38, No. 5, 1707-1720, 2000.

30. Bos, L., M. A. Taylor, and B. A. Wingate, "Tensor product Gauss-Lobatto points are Fekete points for the cube," Mathematics of Computation, Vol. 70, No. 236, 1543-1547, 2001.

31. Pasquetti, R. and F. Rapetti, "Spectral element methods on triangles and quadrilaterals: Comparisons and applications," Journal of Computational Physics, Vol. 198, No. 1, 349-362, 2004.

32. Wingate, B. A. and M. A. Taylor, "Performance of numerically computed quadrature points," Applied Numerical Mathematics, Vol. 58, No. 7, 1030-1041, 2008.

33. Hesthaven, J. S., "From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex," SIAM Journal on Numerical Analysis, Vol. 35, No. 2, 655-676, 1998.

34. Shen, J., L.-L. Wang, and H. Li, "A triangular spectral element method using fully tensorial rational basis functions," SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 1619-1650, 2009.

35. Li, H., J. Sun, and Y. Xu, "Discrete Fourier analysis, cubature and interpolation on a hexagon and a triangle," SIAM Journal on Numerical Analysis, Vol. 46, No. 4, 1653-1681, 2008.

36. Riviere, B. and M. F. Wheeler, "Discontinuous finite element methods for acoustic and elastic wave problems," Contemporary Mathematics, Vol. 329, 271-282, 1999.

37. De Basabe, J. D., M. K. Sen, and M. F. Wheeler, "The interior penalty discontinuous Galerkin method for elastic wave propagation: Grid dispersion," Geophysical Journal International, Vol. 175, No. 1, 83-93, 2008.

38. Kaser, M. and M. Dumbser, "An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms," Geophysical Journal International, Vol. 166, No. 2, 855-877, 2006.

39. Lesage, A. C., R. Aubry, G. Houzeaux, M. Araya Polo, and J. M. Cela, "3D spectral element method combined with H-refinement," 72nd EAGE Conference & Exhibition, Extended Abstracts, C047, Barcelona, Spain, Jun. 2010.

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