On an Approach to the Numerical Solution of Dirichlet Problems of Arbitrary Dimensions

A method of the numerical solution of Dirichlet boundary value problems for nonlinear partial differential equations of the elliptic type and of arbitrary dimensions is proposed. It takes little memory and computer time for problems with smooth solutions. The method is based on modified interpolation polynomials with Chebyshev nodes to approximate the sought-for function and on a new approach to constructing and solving the linear algebraic problems corresponding to the initial differential equations. The spectra and condition numbers of the matrices formed by the algorithm are analyzed by using interval methods. Theorems on approximation and stability of the algorithm are proved in the linear case. It is shown that the algorithm provides a considerable decrease in computational costs as compared to the classical collocation and finite difference methods.