An exponentially convergent method for the solution of Laplace's equation on polygons

A new approximate method of solving a mixed boundary value problem for Laplace's equation on an arbitrary polygon is presented and substantiated for the case when the right sides in the boundary conditions of the first and second kind on the sides of the polygon are given by algebraic polynomials in the arc length of the boundary of the polygon. By means of this method, an approximate solution of a boundary value problem on a closed polygon can be found with uniform accuracy $\varepsilon>0$ at the expense of $O(|\ln^3\varepsilon|)$ arithmetic operations.
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