Commun. Comput. Phys. doi: 10.4208/cicp.OA-2020-0052 Vol. 30, No. 1, pp. 210-226 July 2021 A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm Jiuyang Liang 1 , Pei Liu 2 and Zhenli Xu 1, ∗ 1 School of Mathematical Sciences, Institute of Natural Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China. 2 School of Mathematics, University of Minnesota, Twin Cities, Minneapolis, MN 55455, USA. Received 21 March 2020; Accepted (in revised version) 20 November 2020 Abstract. Poisson’s equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations, molecular dynamics simulations and computational astrophysics. In this paper, a fast and highly accurate algorithm is presented for the solution of the Poisson’s equation in a cuboidal domain with boundary conditions of mixed type. This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-space Green’s function within a sphere containing the cuboid, and another surface integration over the spherical sur- face. Numerical quadratures are introduced to approximate the integrals, resulting in the solution represented by a summation of point sources in free space, which can be accelerated by means of the fast multipole algorithm. The complexity of the algo- rithm is linear to the number of quadrature points, and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function. AMS subject classifications: 35J08, 35Q70, 33F05, 78M16 Key words: Fast algorithm, Poisson’s equation, boundary integral method, image charge, mixed boundary condition, fast multipole method. 1 Introduction The solution of Poisson’s equation plays an essential role in scientific computing as well as many physical and engineering applications such as molecular simulations, electric structure calculations, computational astrophysics, and fluid dynamics for both particle simulations [1–4] and continuum-theory calculations [5–8]. The development of efficient ∗ Corresponding author. Email addresses: liangjiuyang@sjtu.edu.cn (J. Liang), liu01304@umn.edu (P. Liu), xuzl@sjtu.edu.cn (Z. Xu) http://www.global-sci.com/cicp 210 c 2021 Global-Science Press