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Progress In Electromagnetics Research | ISSN: 1070-4698, E-ISSN: 1559-8985 |

Home > Vol. 141 > pp. 671-692
## NEW TRIANGULAR MASS-LUMPED FINITE ELEMENTS OF DEGREE SIX FOR WAVE PROPAGATIONBy W. A. Mulder
Abstract:
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.
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," 4. Mulder, W. A., "A comparison between higher-order finite elements and finite differences for solving the wave equation," 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," 6. Giraldo, F. X. and M. A. Taylor, "A diagonal-mass-matrix triangular-spectral-element method based on cubature points," 7. Lyness, J. N. and D. Jespersen, "Moderate degree symmetric quadrature rules for the triangle," 8. Keast, P., "Cubature formulas for the surface of the sphere," 9. Heo, S. and Y. Xu, "Constructing fully symmetric cubature formulae for the sphere," 10. Keast, P. and J. C. Diaz, "Fully symmetric integration formulas for the surface of the sphere in s dimensions," 11. Keast, P., "Moderate-degree tetrahedral quadrature formulas," 12. Cohen, G., P. Joly, J. E. Roberts, and N. Tordjman, "Higher order triangular finite elements with mass lumping for the wave equation," 13. Mulder, W. A., "Higher-order mass-lumped finite elements for the wave equation," 14. Courant, R., K. Friedrichs, and H. Lewy, "Uber die partiellen Differenzengleichungen der mathematischen Physik," 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," 16. Jund, S. and S. Salmon, "Arbitrary high-order finite element schemes and high-order mass lumping," 17. Lax, P. and B. Wendroff, "Systems of conservation laws," 18. Shubin, G. R. and J. B. Bell, "A modified equation approach to constructing fourth order methods for acoustic wave propagation," 19. Dablain, M. A., "The application of high-order differencing to the scalar wave equation," 20. Chen, J.-B., "Lax-Wendroff and Nystrom methods for seismic modelling," 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," 22. Gilbert, J. C. and P. Joly, "Higher order time stepping for second order hyperbolic problems and optimal CFL conditions," 23. Koornwinder, T., "Two-variable analogues of the classical orthogonal polynomials," 24. Dubiner, M., "Spectral methods on triangles and other domains," 25. Sherwin, S. J. and G. E. Karniadakis, "A new triangular and tetrahedral basis for high-order (hp) finite element methods," 26. Heinrichs, W. and B. I. Loch, "Spectral schemes on triangular elements," 27. Bittencourt, M. L., "Fully tensorial nodal and modal shape functions for triangles and tetrahedra," 28. Xu, Y., "On Gauss-Lobatto integration on the triangle," 29. Taylor, M. A., B. A. Wingate, and R. E. Vincent, "An algorithm for computing Fekete points in the triangle," 30. Bos, L., M. A. Taylor, and B. A. Wingate, "Tensor product Gauss-Lobatto points are Fekete points for the cube," 31. Pasquetti, R. and F. Rapetti, "Spectral element methods on triangles and quadrilaterals: Comparisons and applications," 32. Wingate, B. A. and M. A. Taylor, "Performance of numerically computed quadrature points," 33. Hesthaven, J. S., "From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex," 34. Shen, J., L.-L. Wang, and H. Li, "A triangular spectral element method using fully tensorial rational basis functions," 35. Li, H., J. Sun, and Y. Xu, "Discrete Fourier analysis, cubature and interpolation on a hexagon and a triangle," 36. Riviere, B. and M. F. Wheeler, "Discontinuous finite element methods for acoustic and elastic wave problems," 37. De Basabe, J. D., M. K. Sen, and M. F. Wheeler, "The interior penalty discontinuous Galerkin method for elastic wave propagation: Grid dispersion," 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," 39. Lesage, A. C., R. Aubry, G. Houzeaux, M. Araya Polo, and J. M. Cela, "3D spectral element method combined with H-refinement," |

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