Riesz bases of wavelets and applications to numerical solutions of elliptic equations

Abstract

We investigate Riesz bases of wavelets in Sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order. First, we study bi-cubic splines on the unit square with homogeneous boundary conditions. The approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation. Second, we develop a fairly general theory for Riesz bases of Hilbert spaces equipped with induced norms. Under the guidance of the general theory, we are able to construct wavelet bases for Sobolev spaces on the unit square. The condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded. Third, we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient. Finally, we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case.