On the deterioration of convergence rate of spectral differentiations for functions with singularities

TL;DR

The paper analyzes the pointwise convergence of Jacobi spectral differentiation for functions with algebraic singularities, revealing that increasing the differentiation order $m$ deteriorates the rate by two orders at endpoints and by one order in smooth regions, with parity-sensitive behavior at interior singularities. It derives sharp asymptotics for Jacobi coefficients of the model singular function, establishes precise remainder bounds $\mathcal{R}_{n}^{m}(x)$, and characterizes the location-dependent convergence exponents $\kappa(x)$. The study extends to Chebyshev interpolation, discusses Lebesgue-lemma sharpness, and identifies superconvergence points, as well as extending the analysis to truncated power functions. The results justify error localization in Jacobi approximations and provide new insights into the convergence behavior of Jacobi spectral methods, with implications for the design and analysis of spectral differentiations in the presence of singularities.

Abstract

Spectral differentiations are basic ingredients of spectral methods. In this work, we analyze the pointwise rate of convergence of spectral differentiations for functions containing singularities and show that the deteriorations of the convergence rate at the endpoints, singularities and other points in the smooth region exhibit different patterns. As the order of differentiation increases by one, we show for functions with an algebraic singularity that the convergence rate of spectral differentiation by Jacobi projection deteriorates two orders at both endpoints and only one order at each point in the smooth region. The situation at the singularity is more complicated and the convergence rate either deteriorates two orders or does not deteriorate, depending on the parity of the order of differentiation, when the singularity locates in the interior of the interval and deteriorates two orders when the singularity locates at the endpoint. Extensions to some related problems, such as the spectral differentiation using Chebyshev interpolation, are also discussed. Our findings justify the error localization property of Jacobi approximation and differentiation and provide some new insight into the convergence behavior of Jacobi spectral methods.

Figures (5)

• Figure 1: Pointwise errors of the zero- and first- and second-order Jacobi differentiations (from bottom to top). Here $n=100$ and the points indicate the errors at the critical points.
• Figure 2: Errors of the first- (left) and second-order (right) Jacobi differentiations as a function of $n$ at $x=1$ ($\Diamond$), $x=-1$ ($\Box$) and $x=-1/4$ ($\circ$), $x=0$ ($\bullet$) and $x=1/4$ (☆). The dashed lines from top to bottom show the rates $O(n^{-s})$ with $s=5/2,7/2,5$ (left) and $s=1/2,3/2,3,4$ (right) .
• Figure 3: Errors of the first- (left) and second-order (right) differentiations of Chebyshev interpolant $p_n(x)$ as a function of $n$ at $x=1/3$ ($\Box$), $x=2/3$ ($\circ$) and $x=1$ ($\bullet$) for $f(x)=|x-1/3|^{3}$. The dashed lines from top to bottom show the rates $O(n^{-s})$ with $s=2,3$ (left) and $s=1,2$ (right) .
• Figure 4: Errors of the first- (left) and second-order (right) differentiations of Chebyshev interpolant $p_n(x)$ as a function of $n$ at $x=-1$ ($\Box$), $x=1/5$ ($\circ$) and $x=1$ ($\bullet$) for the function $f(x)=(1+x)^{5/2}$. The dashed lines from top to bottom show the rates $O(n^{-s})$ with $s=3,5$ (left) and $s=1,3,4$ (right).
• Figure 5: Errors of the zero- (left) and first-order (right) Jacobi spectral differentiations with $n=20$ and $\alpha=\beta=0$ for $f(x)=(1-x)^{5/2}e^{x}$. The points indicates the errors at the superconvergence points $\{x_j^R\}_{j=0}^{n}$ .

Theorems & Definitions (21)

• Remark 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Corollary 2.6
• proof