Element-by-element construction of wavelets satisfying stability and moment conditions

In this paper, we construct a class of locally supported wavelet bases for C-o Lagrange finite element spaces on possibly nonuniform meshes on n-dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H-s for \s\ < 3/2 (\s\ less than or equal to 1 on Lipschitz manifolds), and the wavelets can, in principle, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used, e.g., for constructing an optimal solver of discretized H-s-elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet bases can be implemented efficiently.