The higher-order finite-element scheme with mass lumping for triangles and tetrahedra is an efficient method for solving the wave equation. A number of lower-order elements have already been found. Here the search for elements of higher order is continued. Elements are constructed in a systematic manner. The nodes are chosen in a symmetric way. Integration rules must be exact up to a certain degree to maintain an overall accuracy that is the same as without mass lumping. First, for given integration degrees, consistent rule structures are derived for which integration formulas are likely to exist. Then, as each rule structure corresponds to a potential element of certain order, the position of element nodes and the integration weights can be found by solving the related system of nonlinear equations. With this systematic approach, a number of new sixth-order triangular elements and a new fourth-order tetrahedral element have been found.
Chin-Joe-Kong, M. J. S., Mulder, W. A., & van Veldhuizen, M. (1999). Higher-order triangular and tetrahedal finite elements with mass lumping for solving the wave equation. Journal of Engineering Mathematics, 35, 405-426. https://doi.org/10.1023/A:1004420829610