Stability of a three-point hierarchical basis

Rob Stevenson

Abstract

Except in one dimension, the standard hierarchical basis of linear finite element spaces is not stable with respect to the H^{1}-norm. The condition number behaves like J^{2} or 2^{J} in 2- or 3D respectively, where J is the number of levels. In our talk we will present a hierarchical basis having a condition number that is bounded uniformly in the number of levels. In contrast to other wavelet bases, this basis can be constructed easily on general, possibly locally refined meshes. Independent of the space dimension, each basis function is a linear combination of only three nodal basis functions on its level, by which the basis-transformation can be performed cheaply. Clearly, due to the stability, an optimal preconditioner for symmetric second order elliptic problems is obtained by taking the inverse of the diagonal of the stiffness matrix corresponding to the hierarchical basis.

Perhaps the most interesting application of the new hierarchical basis can be found in the field of boundary element methods. The basis turned out to be H^{s}- stable for a range of s, yielding optimal preconditioners for several types of integral equations. The fact that the basis functions satisfy so-called moment conditions gives the important advantage that the resulting stiffness matrix is close to a sparse one (matrix compression).