On a Hierarchical Three-Point Basis in the Space of Piecewise Linear Functions over Smooth Surfaces

Abstract

In this paper we consider a smooth boundary surface of a three-dimensional domain and the space of piecewise linear functions defined over a uniform triangular grid. We introduce a wavelet basis which is a variant of the well-known three-point hierarchical basis with a simple modification near the boundary points of the global patches of parametrization. Each wavelet is the linear combination of no more than three finite element functions defined over a grid from a hierarchy of triangulations. For the spaces spanned by this basis, the approximation and inverse properties in a certain range of Sobolev spaces are well known. Consequently, the simple basis is a Riesz basis and can be used to precondition operator equations, e.g. boundary element methods. Since the construction of wavelets with and without zero boundary values is part of the setting, the wavelets can also be used for finite element methods.