Overview

P-methods and hp-methods represent solutions to mesh adaptation which provide, in some way, an alternative to h-methods. In brief, a p-method, in contrast to a h-method (Chapter 21), consists of varying the degree p in the approximation while keeping the mesh sizes h unchanged. In this way, the quality (the richness) of the approximation is adapted to the way in which the solution varies.

Well established for some model problems, particularly when the geometry of the domain is relatively simple, p-methods are somewhat delicate to implement when the geometry is really complex. The fact that the size is constant while the degree of the approximation is the sole parameter results in a strong constraint which is not simple to overcome. Therefore, hp-methods have been introduced which combine a p-adaptation with a h-adaptation and thus offer the advantages of both methods while avoiding the unflexibility mentioned above.

This chapter is organized in two distinct non-symmetric parts. The first part concerns the construction of finite elements other than straight (linear) elements ( P ¹ type elements) which are naturally obtained by using automatic mesh generation methods. The second part discusses p and hp-methods, mainly with regard to an adaptive process.

This last aspect will only be dealt with briefly. In contrast, we discuss in greater detail some possible approaches for the construction of finite elements other than P ¹, which are used in p-methods (where