 
Summary: Hierarchical Finite Element Bases
for Triangular and Tetrahedral Elements
S. Adjerid
M. Ai a y
J. E. Flaherty z
Abstract
We describe a hierarchical basis for the pversion of the nite element method
in two and three dimensions. The corresponding sti ness matrices are shown to
have good sparsity properties and better conditioning than those generated from
existing hierarchical bases.
1 Introduction
The quality of nite element solutions depends on several factors including the size
and shape of the elements, the approximation properties of the space S of the nite
element solution, and the smoothness of the true solution. From a computational view
point, the choice of a basis is critical to the stability and e ciency of the nite element
procedure. Because of their simplicity, the approximating space S usually consists of
piecewise polynomial functions relative to a partitioning of the problem domain n,
n = 1;2;3, into Nelements j, j = 1;2;::: ;N. Piecewise polynomial bases are usually
constructed by i transforming j to a standard element K Figure 1 by a smooth, one
toone mapping j and ii introducing a basis f^ignp
