External Barycentric Coordinates for Arbitrary Polygons and an Approximate Method for Calculating Them

In the article, the concept of external barycentric coordinates is introduced to generalize the applicability of the barycentric method for solving external value and initial boundary value problems of mathematical physics. The study is aimed at formulating a simple analytical relation that allows one to calculate barycentric coordinates that are external to a given arbitrary polygonal area with a given accuracy. The corresponding ratio is formed when drawing up an approximate analytical calculation rule, which is based on finding the solution of the external Dirichlet problem for the Laplace equation by the Fredholm method. This solution is based on the decomposition of the kernel of the Fredholm integral equation of the second kind by Legendre polynomials of the first and second orders with use of the Heine formula. As a result, the convergence rate of the external barycentric coordinates obtained by approximate analytical calculation is estimated when establishing exponential convergence in the Hilbert space and polynomial convergence in the space of continuous functions. The algorithmic features of the implementation of an approximate analytical solution with a structured representation of pseudo-codes of algorithms for calculating external barycentric coordinates formed mainly for the MathCad computer algebra system are clarified. The efficiency is demonstrated by specific examples. In conclusion, the author of the article hopes that the detailed results of the algorithmic implementation of calculation of external barycentric coordinates will arouse interest and make the publication material more accessible to a wide range of readers, which would lead to the development of the barycentric method for solving the boundary and initial boundary value problems of mathematical physics.