Stable three-point wavelet bases on general meshes

Summary.

Recently, we introduced a wavelet basis on general, possibly locally refined linear finite element spaces. Each wavelet is a linear combination of three nodal basis functions, independently of the number of space dimensions. In the present paper, we show $H^s$-stability of this basis for a range of $s$, that in any case includes $s=1$, which means that the corresponding additive Schwarz preconditioner is optimal for second order problems. Furthermore, we generalize the construction of the wavelet basis to manifolds. We show that the wavelets have at least one-, and in areas where the manifold is smooth and the mesh is uniform even two vanishing moments. Because of these vanishing moments, apart from preconditioning, the basis can be used for compression purposes: For a class of integral equation problems, the stiffness matrix with respect to the wavelet basis will be close to a sparse one, in the sense that, a priori, it can be compressed to a sparse matrix without the order of convergence being reduced.