Conjugate function method for numerical conformal mappings (Q455862)

The numerical computation of conformal mappings \(f\) of a domain \(\Omega\subset\mathbb{C}\) into \(\mathbb{C}\) are considered. The domain \(\Omega\) is bounded and there are either one or two simple and non-intersecting boundary curves; i.e., the domain \(\Omega\) is either simply or doubly connected. It is usually convenient to map the domains conformally onto some canonical domains as rectangles \(R_h= \{z\in\mathbb{C}: 0<\text{Re\,}z< 1\), \(0< \text{Im\,}z< h\}\) or annuli \(A_r= \{z\in\mathbb{C}: e^{-r}< |z|< 1\}\).NEWLINENEWLINE While the existence of such conformal mapping is expected due to the Riemann mapping theorem, it is usually not possible to obtain a formula or other representation for the mapping analytically. Several algorithms for numerical computation have been described in the literature. One popular method involves the Schwarz-Cristoffel formula.NEWLINENEWLINE In this paper a new method for constructing numerical conformal mappings is given. This method is based on the harmonic conjugate function and properties of quadrilaterals.