Multipole Method for Some Mixed Boundary Value Problems and Its Application to Conformal Mapping

An analytic-numerical multipole method for solving some mixed boundary value problems for the Laplace equation in planar simply connected domains $$g$$ of complex shape with application to conformal mapping of such domains is presented. The method allows one to obtain with high accuracy both the solution and its gradient up to boundary sections near singularities and provides a posterior estimate of the relative error $$\delta $$ in the norm of $$C(\bar {g})$$ . The efficiency of the method was confirmed by examples of numerical implementation of the method for effectiveness mapping of regions with a curvilinear boundary containing reentrant arc corners and narrow slots. In this case, according to the posterior estimate, the error $$\delta $$ was not worse than 10–4 when using only about 100 approximating functions.