Postprocessing Galerkin method using quadratic spline wavelets and its efficiency

Abstract

The wavelet-Galerkin method is a useful tool for solving differential equations mainly because the condition number of the stiffness matrix is independent of the matrix size and thus the number of iterations for solving the discrete problem by the conjugate gradient method is small. We have recently proposed a quadratic spline wavelet basis that has a small condition number and a short support. In this paper we use this basis in the Galerkin method for solving the second-order elliptic problems with Dirichlet boundary conditions in one and two dimensions and by an appropriate post-processing we achieve the L2-error of order Oh4 and the H1-error of order Oh3, where h is the step size. The rate of convergence is the same as the rate of convergence for the Galerkin method with cubic spline wavelets. We show theoretically as well as numerically that the presented method outperforms the Galerkin method with other quadratic or cubic spline wavelets. Furthermore, we present local post-processing for example of the equation with Dirac measure on the right-hand side.