Galerkin Method with Trigonometric Basis on Stable Numerical Differentiation

Abstract

This paper considers the $ p $ ($ p=1,2,3 $) order numerical differentiation on function $ y $ in $ (0,2\pi) $. They are transformed into corresponding Fredholm integral equation of the first kind. Computational schemes with analytic solution formulas are designed using Galerkin method on trigonometric basis. Convergence and divergence are all analysed in Corollaries 5.1, 5.2, and a-priori error estimate is uniformly obtained in Theorem 6.1, 7.1, 7.2. Therefore, the algorithm achieves the optimal convergence rate $ O( \delta^{\frac{2\mu}{2\mu+1}} ) \ (\mu = \frac{1}{2} \ \textrm{or} \ 1)$ with periodic Sobolev source condition of order $ 2\mu p $. Besides, we indicate a noise-independent a-priori parameter choice when the function $ y $ possesses the form of \begin{equation*} \sum^{p-1}_{k=0} a_k t^k + \sum^{N_1}_{k=1} b_k \cos k t + \sum^{N_2}_{k=1} c_k \sin k t, \ b_{N_1}, c_{N_2} \neq 0, \end{equation*} In particular, in numerical differentiations for functions above, good filtering effect (error approaches 0) is displayed with corresponding parameter choice. In addition, several numerical examples are given to show that even derivatives with discontinuity can be recovered well.