Results on the Convergence of the Longest-Edge Trisection Method for Tetrahedral Meshes

Abstract

Mesh refinement has been a continued area of research in applied mathematics and engineering applications in last two decades. Especially in 3 Dimensions, it has received much attention for Finite Element Method, where a given volume domain needs to be discretized by using standard elements as tetrahedral or hexahedra prior to be used as an input of the solver module. More concisely, let T(ABCD) be a tetrahedron with vertexes A, B, C and D. The longest-edge trisection of T(ABCD) is as follows: choose the longest edge (say AB) of T(ABCD), let E and F be the points which divide in three AB, then replace T(ABCD) by three tetrahedra T(ACDE), T(CDEF) and T(BCDF). If Longest Edge [3, 4] trisection is iteratively applied to an initial tetrahedron, then it is proved that the diameters of the resulting tetrahedra are between two sharped experimental decreasing functions. The problem of convergence of the triangulations generated by these methods is of importance. We focus on the convergence problem that study how fast the diameters of the resulting tetrahedra tend to zero as repeated Longest Edge trisection is performed. The counterpart problem in 2D have been solved for the Longest Edge Bisection and Trisection, by a series of works by Kearfott, Stynes, Adler, Plaza and Suárez et al. We present empirical evidence of the convergence study in 3D and some other ongoing results will be given for the LE n-section of tetrahedra when n≥4. Our contribution helps to a better understanding of partitioning and refinement methods in 3 Dimensions.