On Numerical Solution of a First Kind Integral Equation with a Weak Singularity in the Kernel on a Closed Surface

A direct method (self-regularization method) for the numerical solution of a weakly singular integral equation of the first kind on a closed surface is considered. This equation is an integral formulation of the internal and external three-dimensional Dirichlet problems for the Laplace equation if their solutions are sought in the form of a single-layer potential. It is approximated by a system of linear algebraic equations, which is solved numerically. In this case, a new method of averaging the kernel of the integral operator is used. It preserves the conditional correctness of the discretized problem and significantly increases the rate of convergence of its solution to the exact solution of the integral equation.