Convergence estimates for the wavelet-Galerkin method: superconvergence at the node points

Abstract

In this paper we consider a regular 1-periodic initial value problem and Galerkin approximate solutions in subspaces νt spanned by scaled translates of a basic functionϕ. Our goal is to estimate the error whenϕ is in a class of functions which we name Open image in new window . Herer is a regularity parameter andm is related with a property (the Strang and Fix condition) which determines the best order of accuracy in theL2-norm of approximations from νh. Whenm=r, Open image in new window includes all scaling functions corresponding tor-regular multiresolution analyses ofL2(ℝ). We get the exact node values of the given initial condition as coefficients for the approximate initial data. With this procedure, the coefficients of the resulting Galerkin solution can give a very accurate approximation of the exact solution at the node points, provided thatϕ has many vanishing moments. Since this property is not satisfied in general, we work with another modified basic functionϕ* constructed from the integer translates ofϕ. GlobalL2-estimates are also obtained.