Convergence estimates for the wavelet-Galerkin method: Superconvergence at the node points
The author studies approximations of a linear partial differential equation in one space variable and in time, the spatial domain being the real line. The data is assumed to be 1-periodic in space, and Galerkin approximations are sought in finite-dimensional subspaces \(V_h\) of \(L^2(0,1)\) which are spanned by the scaled translates of a basic function \(\phi\). That is, Galerkin approximations are of the form \[ u_h(x,t) = \sum_k v_k(t) \phi(Nx - k) \] where \(N\) is a positive integer and \(h = 1/N\) is the mesh size. The mesh is assumed to be regular, and the coefficients are assumed to be 1-periodic with period \(N\). The problem has been previously studied by (*) \textit{V. Thomée} and \textit{B. Wendroff} [SIAM J. Numer. Anal. 11, 1059-1068 (1974; Zbl 0292.65052)], in which \(\phi\) has been taken to be a B-spline. It is possible to obtain a superconvergence result using this approach. The approach of the paper under review is to extend the study in (*) to the larger space \({\mathcal H}^*_{r,m}\), comprising functions which are \(r\)-regular and whose Fourier transforms are non-zero at the origin, and which have zeros of order \(m\) at the points \(2k\pi\), \(k \neq 0\) (this is the so-called Strang-Fix condition). This space includes for \(m = r\), the B-splines as well as \(r\)-regular multiresolution analyses of \(L^2(\mathbb{R})\). Finally, the superposed asterisk denotes the fact that any member \(\phi \in {\mathcal H}^*_{m,r}\) may be modified to obtain a function \(\phi^*\), also in this space, and with the property that its Fourier transform satisfies \(\phi^*(\xi) = 1+O(\xi^{2m+2})\). The first result obtained is an error estimate for smooth 1-periodic functions. Next, it is shown that the following superconvergence result holds: the \(L^2\)-error in the solution to the initial value problem is of order \(\min(2m + 2 - \mu, m + 1)\) where \(r \geq \mu/2\). In the final section of the paper the results are discussed in the context of wavelet analysis. Two examples are given: one concerns splines, and the other scaling functions of Daubechies.
scientific article; zbMATH DE number 826884
scientific article