Algebraic Analysis of the Hierarchical Basis Preconditioner

Abstract

The use of the hierarchical basis in finite element discretizations of two-dimensional elliptic partial differential equations produces matrices with condition numbers of order $O((\log h^{ - 1} )^2 )$. Standard proofs of this result are functional analytic in style. In this paper, it is shown that for uniform grids the result can be obtained using a purely linear algebraic argument.