Fast and accurate poisson solver algorithm in 3D simply and double connected domains with a smooth complex geometry with applications in optics

Abstract

This paper introduces an innovative approach for addressing the Poisson equation in simply and doubly connected 3D domains with irregular surfaces, which has significant implications in various scientific and engineering fields, such as irregular cross-section optical waveguides and electromagnetic wave propagation. The Poisson equation is extensively utilized across disciplines like physics, engineering, and mathematics, and its solution offers insight into diverse physical phenomena. The solution to the Poisson equation is helpful in constructing potentials crucial for the comprehension and design of optical and electromagnetic systems. The application of Radial Basis Functions (RBFs) collocation method with changeable form parameters presents novel opportunities for precise and efficient resolutions of this significant equation. Our methodology is relevant to both simply and doubly connected three-dimensional domains with irregular surfaces, frequently seen in various practical applications, such as complex waveguide geometries. Seven instances are presented for various complex simply and doubly connected 3D domains, illustrating the efficacy of the suggested Poisson solver in generating potentials to improve the precision and efficiency of the method. The proposed method can be considered as a benchmark solver for such type of problems appearing in optics and electromagnetic wave engineering. keyword: Radial Basis Functions, Simply Connected Domains, Double Connected Domains, Variable shape parameter, Three dimensional Laplace equation, Three dimensional Poisson Equation.