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Within the framework of the collocation boundary element method, a semi-analytical approximation of a double layer heat potential near the boundary of a two-dimensional domain is obtained, which provides uniform quadratic convergence of approximate solutions of heat conduction problems in the domain. The proposed approximation is based on the use of analytical integration over some component of the distance from the interior point of the domain to the boundary point of integration. Such analytical integration is carried out in cases when the noted distance does not exceed approximately one third of the radius of the Lyapunov circle. The results of the numerical solution of a heat conduction problem near the boundary of a circular domain are presented, demonstrating the high accuracy of the proposed approximation in comparison with standard quadrature formulas. [ABSTRACT FROM AUTHOR]