Justification of the direct projection method for solving an integral equation of the potential theory

Abstract

We justify the direct projection method for solving an integral equation with a logarithmic singularity in the kernel. The equation is treated as a mapping of one Hilbert space into another Hilbert space. The spaces are chosen from conditions ensuring the solution of a broad class of mathematical modeling problems with the use of a simple layer potential. The idea of the projection method is to choose finite-dimensional subspaces into which the exact solution and the right-hand side of the equation are projected. In this case, the problem of finding an approximate solution does not require computing the convolution of kernels. We prove an estimate for the solution error in the norm of the original operator equation.