Estimation of the condition number of a matrix in the direct discretization of the Symm equation on a quasi-uniform grid (Q5932601)

The article is devoted to justifying a direct numerical method for solving an integral equation of the first kind whose kernel has a logarithmic singularity and which is known as the Symm equation. The case of a closed contour is considered. Such equations appear in applications of the method of boundary integral equations to solving boundary value and contact elliptic problems. The author notes that although Symm-type equations are formally equations of the first kind, the problem of solving them is as ill-posed as the problem of numerical differentiation; i.e., the inverse operator decreases the smoothness by one. Therefore, to solve such equations, the direct discretization methods can be applied which do not include the Tikhonov regularization procedure explicitly. The author studies a method that is based on piecewise linear interpolation of the unknown function and determination of the unknown coefficients from the collocation conditions at the same knots. The author exposes a new approach which seems rather universal and rests eventually only on the convexity properties of the kernel of the differential operator. The method fails when considering an arbitrary grid and the result contains some constraints on the value of the quasi-uniformity parameter.