ABSTRACT

Chapter 2

Hierarchic master elements

of arbitrary order

The rst step in the technology of hierarchic higher-order nite element meth-

ods is to design suitable master elements of arbitrary polynomial order. We

will consider the most commonly used reference domains, equip them with

appropriate scalar and vector-valued polynomial spaces and dene hierarchic

higher-order shape functions. Some of the constructions are actually quite ex-

citing, particularly in vector-valued spaces in higher spatial dimensions, but

in each case this chapter is intended merely as a database of formulae rather

than information for systematic study. The reader may nd it interesting to

read about the De Rham diagram (Section 2.1) which relates the spaces H

,

H(curl), H(div) and L

by means of dierential operators, since nite ele-

ments in these spaces have to respect the diagram as well. Then she/he may

visit a paragraph that discusses a particular nite element of interest.