A fast nonlocal algorithm for solving Neumann-Dirichlet boundary value problems with error control

A method for searching numerical solutions to Neumann-Dirichlet boundary value problems for differential equations of elliptic type is proposed. This method allows reaching a desired accuracy with low consumption of memory and computer time. The method adapts the properties of best polynomial approximations for construction of algorithms without saturation on the basis of nonlocal Chebyshev approximations. A new approach to the approximation of differential operators and to solving the resulting problems of linear algebra is also proposed. Estimates of numerical errors are given. A high convergence rate of the proposed method is substantiated theoretically and is shown numerically in the case of problems with $C^r$-smooth and $C^{\infty}$-smooth solutions. Expressions for arrays approximating the differential operators in problems with various types of boundary conditions are obtained. These expressions allow the reader to quickly implement the method “from scratch”.