A nonuniform difference scheme with fourth order of accuracy in a domain with smooth boundary

The paper is devoted to construction and justification of a nonuniform difference scheme with fourth order of accuracy for a two-dimensional boundary-value problem for an elliptic equation of second order in a domain with smooth curvilinear boundary. This scheme is called nonuniform because a stencil of difference operator alternates from node to node. In interior nodes a nine-point and standard five-point stencils are used. The special type of stencil is used near the boundary. The paper contains a description for constructing the scheme and for the proof accuracy. Numerical tests confirm the theoretical conclusions about the fourth order of accuracy for the approximate solution.