On the Daubechies-Based Wavelet Differentiation Matrix

The differentiation matrix for a Daubechies-based wavelet basis will be constructed and superconvergence will be proven. That is, it will be proven that under the assumption of periodic boundary conditions that the differentiation matrix is accurate of order 2M, even though the approximation subspace can represent exactly only polynomials up to degree M - 1, where M is the number of vanishing moments of the associated wavelet. It will be illustrated that Daubechies-based wavelet methods are equivalent to finite difference methods with grid refinement in regions of the domain where small- scale structure is present.