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Let t be a triangle in R². We find the Longest Edge (LE) of t, insert n-1 equally-space points in the LE and connect them to the opposite vertex. This yields the generation of n new sub-triangles whose parent is t. Now, continue this process iteratively. Proficient algorithms for mesh refinement using this method are known when n = 2, but less known when n = 3 and completely unknown when n ≤ 4. We prove that the LE n-section of triangles for n ≤ 4 of triangles produces a finite sequence of triangle meshes with guaranteed convergence of diameters. We give upper and lower bounds for the convergence speed in terms of diameter reduction. Then we fill the gap in the analysis of the diameters convergence for general Longest Edge based subdivision. In addition, we give a numerical study for the case of n = 4, the so-called LE quatersection, evidencing its utility in adaptive mesh refinement. [ABSTRACT FROM AUTHOR]