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 Lagrange nite element spaces on possibly non uniform meshes on n dimensional domains or manifolds The wavelet bases are stable in the Sobolev spaces Hs for jsj jsj on Lipschitz manifolds and the wavelets can in principal 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 Hs 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 nite 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 re nements of initial elements are uniform We will show that the wavelet bases can be implemented e ciently Introduction Report No Department of Mathematics University of Nijmegen December submitted to SIAM J Numer Anal This paper is concerned with the construction of nite element based wavelet bases with respect to arbitrary initial triangulations This introductory section is devoted to a brief summary of relevant background information which in particular motivates speci c requirements on the wavelet bases concerning stability in Sobolev spaces and moment conditions Motivation and background Let us denote by H s IR or jsj t a scale of Sobolev spaces on an n dimensional domain or su ciently smooth manifold When s the space H is understood to be the dual of H s whose precise structure depends of course also on the boundary conditions incorporated in H Consider the variational problem Given f H r nd u H such that