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

Abstract: 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.