Systematic and generic construction of shape functions for p-adaptive meshes of multidimensional finite elements

Abstract

This paper presents a methodology for generating high-order shape functions for the complete family of finite elements. The geometric entities presented are the point, line, triangle, quadrilateral, tetrahedron, pyramid, prism, and hexahedron. The shape functions constructed are hierarchical and generate continuous approximation spaces. The order of interpolation of the shape function can be determined independently for each edge, face, and volume. The space of interpolation of order p for each element is complete. This definition allows for the construction of hybrid meshes with any combination of these elements, including elements of distinct dimension and/or interpolation order. The reader will observe that our systematic way of constructing shape functions can be extended to elements of higher-dimension.