Fast Computing of Conformal Mapping and Its Inverse of Bounded Multiply Connected Regions onto Second, Third and Fourth Categories of Koebe’s Canonical Slit Regions

Abstract

This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded multiply connected region onto a disk with spiral slits region $$\varOmega _1$$Ω1. This extends the methods that have recently been given for mappings onto annulus with spiral slits region $$\varOmega _2$$Ω2, spiral slits region $$\varOmega _3$$Ω3, and straight slits region $$\varOmega _4$$Ω4 but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto all four regions $$\varOmega _1$$Ω1, $$\varOmega _2$$Ω2, $$\varOmega _3$$Ω3, and $$\varOmega _4$$Ω4 as well as their inverses. The integral equations are solved numerically using combination of Nystrom method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is $$O((m + 1)n)$$O((m+1)n), where $$m+1$$m+1 is the multiplicity of the multiply connected region and n is the number of nodes on each boundary component. Previous algorithms require $$O((m+1)^3 n^3)$$O((m+1)3n3) operations. The algorithm is tested on several test regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method.