An approximate method for determining the harmonic barycentric coordinates for arbitrary polygons

A relation for finding the harmonic barycentric coordinates for an arbitrary polygon is obtained. The solution is approximate analytical. In the proposed statement, the harmonic barycentric coordinates are determined in terms of the logarithmic potential of a double layer by solving the Dirichlet problem by the Fredholm method. The approximate nature of the solution is determined by the expansion of the kernel of the integral Fredholm equation of the second kind for the unknown density of potential on the boundary of the domain in the orthogonal Legendre polynomials and the expansion of Green's function; these expansions are used for the calculation of the potential. An estimate of convergence rate and the error of the solution is obtained. The approximate solutions obtained by the proposed method are compared with the known exact solutions of some benchmark problems.