Multipole Method for Solving the Zaremba Problem in Complex Domains With Application to Construction of Conformal Mapping

This work, continuing the authors’ 2024 article, is devoted to the development of an analytical-numerical multipole method applied to the Zaremba problem, i.e., a mixed boundary value problem with Dirichlet–Neumann boundary conditions for the Laplace equation in planar simply connected domains of complex shape, whose boundary may contain singularities. The method allows obtaining not only the solution but also its derivatives on certain smooth parts of the boundary near singularities. The efficiency of the method was demonstrated by examples of constructing conformal mapping, and in previous works (with other co-authors) – by examples of constructing harmonic mapping of domains with complex curvilinear boundaries.