Speaker: Dr. Desong Kong (孔德松博士)

Time: 10:00-11:00, 15 May 2025 (Thursday) (Beijing time)

Venue: B415, Lijiao Building

Abstract

Lightning algorithms based on the rational function have been successfully applied to solving the Laplace, Helmholtz and Stokes equations. In this work, we develop an efficient and fast solver for the modified Helmholtz equations in two dimension, specially addressing corner singularities. For modified Helmholtz equations, we begin by representing the solution using a multipole expansion, where the radial coefficients satisfy the modified Bessel's equation. We then employ the modified Bessel function of the second kind, coupled with tapered exponentially clustering poles, to handle the singularities, and the modified Bessel function of the first kind to resolve the smoother components of the solution. As a result, we derive a general solution formula that naturally satisfies the equation, though it may not necessarily satisfy the boundary conditions. Finally, we determine the unknown coefficients by solving a least-squares problem, which is formulated based on boundary sampling. Numerical experiments demonstrate that, for typical problems, the solution can be computed to 6-digit accuracy in less than 1 second on a desktop computer. Several applications on solving partial differential equations in polygonal domains are also considered to illustrate the effectiveness of the proposed algorithm.

About the Speaker

孔德松博士于2023年毕业于中南大学，指导老师是向淑晃教授，攻读博士学位期间受国家留学基金委资助前往新加坡南洋理工大学进行联合培养 ，指导老师是王立联教授。2023年至今在宁波东方理工大学做博士后，指导老师是沈捷教授。孔德松博士的主要研究方向是奇异函数的逼近理论、复杂区域和非标准几何上的高效算法研究，以及发展型方程的时空谱方法研究。