The present article is the last one of a trilogy of papers [see the foregoing entries (Zbl 0723.73074; Zbl 0723.73075)] on the development of an adaptive h-p version of the finite element method.
In this presentation, the authors address the question of how mesh sizes h and spectral orders p can be chosen throughout a finite element mesh. However, it is pointed out that a systematic approach toward generating an optimal distribution of h and p for delivering solutions with a preset value of estimated error is not available. Thus, a simple approximate h-p mesh optimization technique is developed that can be used as an attempt to construct optimal meshes. In restricting the discussion to model classes of one-and two-dimensional elliptic boundary-value problems, a practical and probably efficient approximate scheme is offered leading to a trajectory in space of h-p distributions close to the optimal. Different results of applying the method to several test problems are discussed.