Wavelet bases on the interval with short support and vanishing moments

Abstract

Jia and Zhao have recently proposed a construction of a cubic spline wavelet basis on the interval which satisfies homogeneous Dirichlet boundary conditions of the second order. They used the basis for solving fourth order problems and they showed that Galerkin method with this basis has superb convergence. The stiffness matrices for the biharmonic equation defined on a unit square have very small and uniformly bounded condition numbers. In our contribution, we design wavelet bases with the same scaling functions and different wavelets. We show that our basis has the same quantitative properties as the wavelet basis constructed by Jia and Zhao and additionally the wavelets have vanishing moments. It enables to use this wavelet basis in adaptive wavelet methods and non-adaptive sparse grid methods. Furthermore, we even improve the condition numbers of the stiffness matrices by including lower levels.