On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions

The finite element method usually requires regular or strongly regular families of partitions in order to get guaranteed a priori or a posteriori error estimates. In this paper we examine the recently invented longest-edge bisection algorithm that always produces only face-to-face simplicial partitions. First, we prove that the regularity of the family of partitions generated by this algorithm is equivalent to its strong regularity in any dimension. Second, we present a number of 3d numerical tests, which demonstrate that the technique seems to produce regular (and therefore strongly regular) families of tetrahedral partitions. However, a mathematical proof of this statement is still an open problem.