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Exact convergence rates of lightning plus polynomial approximation for branch singularities with uniform exponentially clustered poles

Shuhuang Xiang, Yanghao Wu, Shunfeng Yang

TL;DR

The paper provides a rigorous framework for exact root-exponential convergence of lightning-plus-polynomial (LP) rational approximations for branch-type singularities in sector domains with uniformly exponentially clustered poles. By deriving integral representations for $z^{\alpha}$ and $z^{\alpha}\log z$, and constructing LPs with $N_2=O(\sqrt{N_1})$ using exponential clustering, it proves sharp convergence rates via Paley–Wiener-type results and Poisson summation. It removes previous restrictions on the sector parameter and extends the theory to multipliers $g(z)$ multiplying the singular terms, yielding explicit error bounds and optimal parameter choices $\sigma_{opt}$. The methods are then applied to conformal mappings and Laplace problems on corner domains, demonstrating practical impact for solving PDEs with corner singularities.

Abstract

This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner (branch) singularity problems with uniform exponentially clustered poles proposed by Gopal and Trefethen. The start point is to set up the integral representations of $z^α$ and $z^α\log z$ in the slit disk and develop results akin to Paley-Wiener theorem, from which, together with the Poisson summation formula, the root-exponential convergence of the lightning plus polynomial scheme with an exact order for each clustered parameter is established in approximation of prototype functions $z^α$ or $z^α\log z$ on a sector-shaped domain, which includes $[0,1]$ as a special case. In addition, the fastest convergence rate is confirmed based upon the best choice of the clustered parameter. Furthermore, the optimal selection of the clustered parameter is employed in conformal mappings through solving Laplace problems on corner domains, building upon Lehman and Wasow's analysis of corner singularities and incorporating the domain decomposition method proposed by Gopal and Trefethen.

Exact convergence rates of lightning plus polynomial approximation for branch singularities with uniform exponentially clustered poles

TL;DR

The paper provides a rigorous framework for exact root-exponential convergence of lightning-plus-polynomial (LP) rational approximations for branch-type singularities in sector domains with uniformly exponentially clustered poles. By deriving integral representations for and , and constructing LPs with using exponential clustering, it proves sharp convergence rates via Paley–Wiener-type results and Poisson summation. It removes previous restrictions on the sector parameter and extends the theory to multipliers multiplying the singular terms, yielding explicit error bounds and optimal parameter choices . The methods are then applied to conformal mappings and Laplace problems on corner domains, demonstrating practical impact for solving PDEs with corner singularities.

Abstract

This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner (branch) singularity problems with uniform exponentially clustered poles proposed by Gopal and Trefethen. The start point is to set up the integral representations of and in the slit disk and develop results akin to Paley-Wiener theorem, from which, together with the Poisson summation formula, the root-exponential convergence of the lightning plus polynomial scheme with an exact order for each clustered parameter is established in approximation of prototype functions or on a sector-shaped domain, which includes as a special case. In addition, the fastest convergence rate is confirmed based upon the best choice of the clustered parameter. Furthermore, the optimal selection of the clustered parameter is employed in conformal mappings through solving Laplace problems on corner domains, building upon Lehman and Wasow's analysis of corner singularities and incorporating the domain decomposition method proposed by Gopal and Trefethen.

Paper Structure

This paper contains 13 sections, 11 theorems, 176 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Integral representations of $z^\alpha$ and $z^\alpha \log z$
  3. Principles of LPs \ref{['eq:rat']} for $z^{\alpha}$ and $z^{\alpha}\log z$
  4. Exponential transformation
  5. Truncation errors
  6. Construction of the rational functions for $z^\alpha$ and $z^\alpha\log z$
  7. Runge's approximation theorem
  8. LPs with $N_2=\mathcal{O}(\sqrt{N_1})$ for $z^{\alpha}$ and $z^{\alpha}\log z$
  9. Extension of LPs to $g(z)z^{\alpha}$ and $g(z)z^{\alpha}\log{z}$
  10. Paley-Wiener type theorems and best convergence rate for trapezoidal rule
  11. Quadrature errors for the rational approximations
  12. Proof of Theorem \ref{['mainthm']}
  13. Applications to conformal mappings on planar corner domains

Key Result

Theorem 1.1

Gopal2019 Let $f$ be a bounded analytic function in the slit disk $S_\beta$ that satisfies $f(z) =\mathcal{O}(|z|^\delta)$ as $z\rightarrow 0$ for some $\delta>0$ and let $\beta\in (0,1)$ be fixed. Then for some $\hat{\eta}\in (0, 1)$ depending on $\beta$ but not $f$, there exist type $(N-1, N)$ rat as $n\rightarrow \infty$ for some $C_0>0$, where $\Omega=\hat{\eta}S_\beta$.

Figures (17)

  • Figure 1: Curvy domains with an interior angle $\varphi_k\pi$, determined by the tangent rays extending from the common vertex. Additionally, all these domains can be covered by a sufficiently large sector domain centered at the vertex, with a radius angle $\beta_k\pi$ coinciding with or larger than the interior angle $\varphi_k\pi$. The red points illustrate the distribution of the clustering poles around vertex $w_k$.
  • Figure 2: This figure is cited from Gopal2019: A holomorphic function $f(z)$ defined in the corner domain $\Omega$ is decomposed as the sum of $2m$ Cauchy-type integrals: $\sum_{k=1}^m f_k(z)+\sum_{k=1}^m g_k(z)$, with $f_k(z)=\frac{1}{2\pi i}\int_{\Lambda_k}\frac{f(\zeta)}{\zeta-z}d\zeta$ along the two sides of an exterior bisector slit to each corner, and $g_k(z)=\frac{1}{2\pi i}\int_{\Gamma_k}\frac{f(\zeta)}{\zeta-z}d\zeta$ along each line segment connecting the beginnings and ends of those slit contours.
  • Figure 3: V-shaped domain (left): $V_\beta=\{z: \, z=xe^{\pm \frac{\beta\pi}{2}i}$ with $x\in[0,1]\}$ and sector domain (right): $S_{\beta}=\{z: \, z=xe^{\pm \frac{\theta\pi}{2}i}$ with $x\in [0,1]$ and $\theta\in [0,\beta]\}$ for fixed $\beta\in [0,2)$. The red points illustrate the distributions of the clustering poles \ref{['eq:uniform0']}.
  • Figure 4: Various corner domains: pentagon (first), curvy pentagon (second) and quincunx-shaped (third) domains. The red points illustrate the distributions of the clustering poles.
  • Figure 5: Decay rates of approximation errors $\|z^{\alpha}-\check{r}_N(z)\|_{C(S_{\beta})}$ of Gopal and Trefethen's interpolation (GTs) in Gopal2019 are compared with the LP \ref{['eq:rat']} with $N_2={\rm ceil}(1.3\sqrt{N_1})$ on $S_{\beta}$ with various values of $\sigma$ as well as $\alpha$, $\beta$, where $N$ is the degree of rational approximation. The lightning parameter $\sigma_2=\sigma_{\mathrm{opt}}\left(=\frac{\pi\sqrt{2-\beta}}{\sqrt{\alpha}}\right)$ is the optimal choice among all of $\sigma>0$ to get the corresponding fastest convergence rate.
  • ...and 12 more figures

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • ...and 13 more