The root-exponential convergence of lightning plus polynomial approximation on corner domains

TL;DR

This work provides a rigorous foundation for lightning+polynomial approximations of corner singularities by deriving integral representations for $z^{\alpha}$ and $z^{\alpha}\log z$ on sector domains and constructing LP schemes with tapered exponentially clustered poles. It proves root-exponential convergence with an explicit optimal clustering parameter $\sigma_{opt}=\frac{\sqrt{2(2-\beta)}\pi}{\sqrt{\alpha}}$ and shows uniform error bounds across sector domains, subsequently extending the results to corner and polygonal domains via a Cauchy-type decomposition. The results confirm Conjecture 5.3 in the V-shaped setting and extend the convergence theory to products with analytic functions $g(z)$ using Runge’s theorem, providing explicit rates and practical guidance for parameter choice. Together these findings offer a solid analytic justification for LP schemes as efficient, robust tools for solving PDEs on domains with corners, with direct implications for numerical methods in domains featuring singular corner behavior.

Abstract

This paper builds further rigorous analysis on the root-exponential convergence for lightning schemes approximating corner singularity problems. By utilizing Poisson summation formula, Runge's approximation theorem and Cauchy's integral theorem, the optimal rate is obtained for efficient lightning plus polynomial schemes, newly developed by Herremans, Huybrechs and Trefethen \cite{Herremans2023}, for approximation of $g(z)z^α$ or $g(z)z^α\log z$ in a sector-shaped domain with tapered exponentially clustering poles, where $g(z)$ is analytic on the sector domain. From these results, Conjecture 5.3 in \cite{Herremans2023} on the root-exponential convergence rate is confirmed and the choice of the parameter $σ_{opt}=\frac{\sqrt{2(2-β)}π}{\sqrtα}$ may achieve the fastest convergence rate among all $σ>0$. Furthermore, based on Lehman and Wasow's study of corner singularities \cite{Lehman1954DevelopmentsIT, Wasow}, together with the decomposition of Gopal and Trefethen \cite{Gopal2019}, root-exponential rates for lightning plus polynomial schemes in corner domains $Ω$ are validated, and the best choice of lightning clustering parameter $σ$ for $Ω$ is also obtained explicitly. The thorough analysis provides a solid foundation for lightning schemes.

Figures (11)

• Figure 1: V-shaped domain (left): $V_\beta=\left\{z: \, z=xe^{\pm \frac{\beta\pi}{2}i} \hbox{, with, $x\in[0,1]$}\right\}$ and sector domain (right): $S_{\beta}=\left\{z: \, z=xe^{\pm \frac{\theta\pi}{2}i} \hbox{, with, $x\in [0,1]$ and $\theta\in [0,\beta]$}\right\}$ for fixed $\beta\in [0,2)$. The red points illustrate the distributions of the clustering poles \ref{['eq:tapered2']}.
• Figure 1: The integral contour $\mathfrak{S}$ of \ref{['general integral repres111']}.
• Figure 1: The decay behaviors of $\|I-r_{N_t}\|_{\infty}$ and $\|I_{log}-\widetilde{r}_{N_t}\|_{\infty}$ for $z=xe^{\pm\frac{\theta\pi}{2}}$ with $x\in[0,x^*]$ and $\theta\in[0,\beta]$ with various step length $h_{\ell}=\eta^{-2}_{\ell}h_{opt}$, where $h_{opt}=4\pi^2\alpha$ and we set $\eta^{-2}_{\ell}=0.5\ell,\ \ell=1,2,3$. Additionally, $x^*=e^{\frac{1}{\alpha}(c_0-T)}$, $c_0=\sqrt{M_0h+\frac{1}{4}(2-\beta)^2\alpha^2\pi^2+\delta_0}$ and $M_0$ are defined by \ref{['eq:realM0']} and \ref{['eq:real']}, respectively. We specific $\delta_0=0$ here.
• Figure 1: The decay behaviors of the quadrature errors $\|I-r_{N_t}\|_{\infty}$ for $z^{\alpha}$ (first row) and $\|I_{log}-\widetilde{r}_{N_t}\|_{\infty}$ for $z^{\alpha}\log{z}$ (second row), endowed with $T(\sigma_l)=\frac{\alpha\sigma_l\sqrt{N_t}}{\kappa+1}$, $\lambda(\sigma_l)=\frac{e^{\alpha\sigma_l/2}}{e^{\alpha\sigma_l\eta^2(\sigma_l)/2}-1}$ and $\eta(\sigma_l)=\frac{\sigma_{opt}}{\sigma_l}$ with parameters $\sigma_l,\ l=1,2,3$, which are equivalent to, larger or smaller than the optimal $\sigma_{opt}=\frac{\sqrt{2(2-\beta)}\pi}{\sqrt{\alpha}}$, respectively. The infinite norm $\|\cdot\|_{\infty}$ is evaluated on the sector domain $S_{\beta}$ with $x=1$.
• Figure 1: Gopal2019 A holomorphic function $f(z)$ defined on the corner domain $\Omega$ is decomposed as the sum of $2m$ Cauchy-type integrals: $\sum_{k=1}^mf_k(z)+\sum_{k=1}^mg_k(z)$, 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 ends of those slit contours.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Proof 1
• Theorem 2.2
• Lemma 2.3
• Proof 2
• Lemma 3.1
• Proof 3