Refinement techniques mostly involve the bisection of mesh edges. It is a simple procedure which in practice achieves a smoother grading of the mesh density than is acquired by techniques like the orthogonal projection of a vertex onto the opposite edge of a triangle. The reason can be seen easily in two dimensions. If the angles in a triangular mesh are assumed to be not extremely obtuse, bisection guarantees that the minimum spacing between mesh points does not decrease much more pronounced than by a factor of and no mesh points lie too close together. Obtuse angles might be introduced, but could be eliminated by topological modifications as dicussed later on. Orthogonal projection avoids bad angles, but even under the assumption of an average element quality it cannot be guaranteed that mesh points have a sufficient distance. Too close mesh points induce a high refinement in that region without that the physical problem calls for it. Self-induced refinement may lead to convergence problems of the meshing process if not properly addressed as discussed in Section 6.3.

A new bisection-based refinement method has been introduced by Liu and Joe [92,93]. Interesting research on topics like the longest edge bisection and its ``propagation path'' has been accomplished by Rivara and Hitschfeld [131,132,67]. The order in which several refinement steps are performed has a great effect on the mesh. A lot of research has still to be done in three dimensions.

2000-01-20