Definition of a P-interpolating space of hierarchical bases of finite elements on the pyramid

Abstract

This article shows in detail how to construct in a simple and ordered way a set of rational functions defined on the pyramid topology. The set of functions is parameterized by an integer p. It is shown that these functions, defined in a hierarchical way, constitute a basis for a complete polynomial interpolation space of degree p on the pyramid domain. In order to help this definition we use a denumerable sequence of orthogonal polynomials defined on an interval of the real line. A priori, any increasing sequence of polynomials in one variable can be used. The bases are constructed to be used in the class C method of finite elements. The rational functions thus defined can be combined to represent any polynomial of degree p. Thus, given an arbitrary number p, one defines a finite element whose geometry is a pyramid that has associated a complete interpolation space of degree p. Moreover, this element is adequate to be used with the p-adaptive technique on heterogeneous meshes of finite elements hierarchical