Hierarchical Bases and the Finite Element Method

The choice of basis functions for a nite element space has important consequences in the practical implementation of the nite element method. A traditional choice is the nodal or Lagrange basis. Many of the computational advantages of this basis derive from the property of compact support enjoyed by the basis functions. Here we study a second choice, the hierarchical basis, and examine its application to some specialized computations in nite element analysis. In particular, we examine the computation of a posteriori error estimates using hierarchical basis functions, and multilevel iterative methods for solving large sparse linear systems of nite element equations.