Mesh quality improvement and other properties in the four-triangles longest-edge partition

Abstract

The four-triangles longest-edge (4T-LE) partition of a triangle t is obtained by joining the midpoint of the longest edge of t to the opposite vertex and to the midpoints of the two remaining edges. The so-called self-improvement property of the refinement algorithm based on the 4-triangles longest-edge partition is discussed and delimited by studying the number of dissimilar triangles arising from the 4T-LE partition of an initial triangle and its successors. In addition, some geometrical properties such as the number of triangles in each similarity class per mesh level and new bounds on the maximum of the smallest angles and on the second largest angles are deduced.