Comparative analysis of post-processing on spectral collocation methods for non-smooth functions

Abstract

Discretization-based spectral approximation methods provide spectrally accurate reconstruction of an analytic function. The expansion of non-smooth functions is contaminated by high frequency non-diminishing oscillations near discontinuity points, and this behaviour is named as Gibbs phenomenon. This problem can be well resolved by well-chosen post-processing technique, and one possible choice is spectral filtering. In this paper, a comparison scenario of adaptive spectral filtering for resolution of Gibbs phenomenon is presented. Several spectral filter functions are compared using Chebyshev collocation and Legendre collocation spectral methods, in terms of pointwise and $L_2$ normed-wise convergence analysis of computed filtered approximations.

A Appendix

1.1 Weighted space

$L_{w}^{2} [-1,1]$ is a space consisting a class of measurable functions f defined on $[-1,1]\rightarrow R$, and if f satisfies the square-integrable condition [7, 26], such that

$$\begin{aligned} {\displaystyle \int _{-1}^{1} |f(\xi ) |^{2} \, w(\xi ) \, {\text {d}}\xi < +\infty }. \end{aligned}$$

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With weight function defined with respect to certain polynomial space. $L_{w}^{2} [-1,1]$ is Hilbert space endowed the discrete inner product

$$\begin{aligned}{}[f,g]_{L_{w}^{2} [\Omega ],N} = {\displaystyle \sum _{j = 0}^{N} f(\xi _{j}) \, g(\xi _{j}) \, w(\xi _{j})}, \end{aligned}$$

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and induced discrete weighted norm is,

$$\begin{aligned} \Vert g \Vert _{L_{w}^{2} [\Omega ],N}^{2} = [g,g]_{L_{w}^{2} [\Omega ],N}. \end{aligned}$$

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1.2 Maximum norm in weighted $L_{w}^{\infty }$-space

$$\begin{aligned} \Vert f-f_{N} \Vert _{L_{w}^{\infty } } = \underset{\xi \in [-1,1]}{\underbrace{sup}} |f(\xi ) |. \end{aligned}$$

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1.3 Broken Norm

Let $\xi _{1}$ be a discontinuous point and $\xi _{1}^{+}$ and$\xi _{1}^{-}$ are the right and left limit of $\xi$, respectively. Then, the broken norm can be defined as [7, 17, 25],

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