In Chap. 10, Sects. 10.1.3– 10.1.4, we’ve already met the important concept of mesh regularity, and took it into account in the refinement step. Here, we continue to discuss it, and introduce a few reliable tests to estimate it. This way, we can make sure that our multilevel refinement is indeed robust: the meshes are not only more and more accurate but also fairly regular.