On using the Lagrange coefficients for a posteriori error estimation

A posteriori error estimation of the goal functional is considered using a differential presentation of a finite difference scheme and adjoint equations. The local approximation error is presented as a Tailor series remainder in the Lagrange form. The field of the Lagrange coefficients is determined by a high accuracy finite difference stencil affecting results of computation. The feasibility of using the Lagrange coefficients for the refining solution and estimation of its uncertainty are considered.