30 Years of Newest Vertex Bisection

One aspect of adaptive mesh refinement in the finite element method for solving partial differential equations is the method by which elements are refined. In the early 1980’s the dominant method for refining triangles was the red-green algorithm of Bank and Sherman. The red refinements are the desired refinements, but will result in an incompatible mesh when used alone. The green refinements are used to recover compatibility for stability of the finite element discretization, and are removed before the next adaptive step. Prof. Bob Skeel raised the question as to whether it is possible to perform adaptive refinement of triangles without this complicated patching/unpatching process. As a result, a new triangle refinement method, called newest vertex bisection, was devised as an alternative to red-green refinement in the mid 1980’s. The new approach is simpler and maintains compatibility of the mesh at all times, avoiding the patching/unpatching of the green refinement. In this historical paper we review the development of the newest vertex bisection method for adaptive refinement, and subsequent extensions of the method. c © xxxx European Society of Computational Methods in Sciences and Engineering