标题： 角奇点的精确根指数收敛速度：闪电加多项式逼近 显示英文标题

标题： Exact root-exponential convergence rates of lightning plus polynomial approximations for corner singularities

摘要： 本文通过有理函数在裂开的单位圆盘上对带有均匀指数聚类极点的角奇异问题进行逼近，建立了严格的关于闪电格式根指数收敛性的分析，这是由 Gopal 和 Trefethen 提出的。出发点是建立裂开单位圆盘上的$z^\alpha$和$z^\alpha\log z$的表示，并发展类似于 Paley-Wiener 定理的结果，由此结合泊松求和公式，在扇形域上原型函数$g(z)z^\alpha$或$g(z)z^\alpha\log z$的逼近中确立了闪电加多项式格式具有精确阶数的根指数收敛性，其中$[0,1]$是一种特殊情形。此外，基于聚类参数的最佳选择，确认了最快的收敛速率。进一步地，基于 Lehman 和 Wasow 对角奇异性的研究以及 Gopal 和 Trefethen 的分解方法，验证了聚类参数的最优选择以及求解拉普拉斯方程时角奇异问题的收敛速率。这种全面的分析为闪电格式和有理逼近提供了坚实的基础。大量的数值实验证明了估计值的最优性和尖锐性。 显示英文摘要

摘要： This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner singularity problems with uniform exponentially clustered poles proposed by Gopal and Trefethen. The start point is to set up the representations of $z^\alpha$ and $z^\alpha\log z$ 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 $g(z)z^\alpha$ or $g(z)z^\alpha\log z$ 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 choice of the clustered parameter and the convergence rate for corner singularity problems in solving Laplace equations are attested based on Lehman and Wasow's study of corner singularities and along with the decomposition of Gopal and Trefethen. The thorough analysis provides a solid foundation for lightning schemes and rational approximation. Ample numerical evidences demonstrate the optimality and sharpness of the estimates.