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

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 $g(z)z^\alpha$ or $g(z)z^\alpha\log z$ 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 $\sigma_{opt}=\frac{\sqrt{2(2-\beta)}\pi}{\sqrt{\alpha}}$ may achieve the fastest convergence rate among all $\sigma>0$. 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 $\Omega$ are validated, and the best choice of lightning clustering parameter $\sigma$ for $\Omega$ is also obtained explicitly. The thorough analysis provides a solid foundation for lightning schemes.