Discretization of the Poisson equation using the interpolating scaling function with applications

Dyadic translations of the interpolating scaling function generate a basis that can be used to approximate functions and develop a multiresolution methodology for constructing smooth surfaces or curves. Many wavelet methods for solving par-tial differential equations are also derived from the interpolating scaling function. However, little is done for developing a higher order numerical discretization methodology using the scaling function. In this article, we have employed an iterative interpolation scheme for the construction of scaling functions in a two-dimensional mesh that is a finite collection of rectangles. We have studied the development of a weighted residual collocation method for approximating partial derivatives. We show that the discretization error is controlled by the order of the scaling function. The potential of this novel technique has been verified with some representative examples of the Poisson equation. We have extended the technique for solving nonlinear advection-diffusion equations, and simulated a shear driven flow in a square cavity at CFL = 2.5 (Courant Friedrichs Lewy) and Re = 1 000 (Reynolds number). Agreement with the reference solution at a large CFL = 2.5 explores the potential of this development for advection dominated problems