Superfast Finite Difference Algorithms for Solving Fredholm Equations of the First and Second Kinds

For the Fredholm equations of the first and second kinds, difference schemes with super-power convergence are proposed. They are dramatically more accurate than previously known ones. For the equation of the first kind, a new regularization method is proposed based on adding a stabilizer directly to the matrix of the difference scheme. For the non-self-adjoint problem, the proposed approach reduces the complexity and improves the conditionality of the matrix of the linear system. A new procedure for selecting the regularization parameter is proposed. It is selected so that the systematic error introduced by the stabilizer and the random error due to rounding errors are comparable. A new calculation algorithm with precision control is designed based on grid refining and simultaneously increasing the number precision. The proposed approaches were verified on representative test problems with a known exact solution.