An approximate solution of first derivatives of the mixed boundary value problem for Laplace’s equation on a rectangle

In a rectangular domain, we discuss about an approximation of the first order derivatives for the solution of the mixed boundary value problem. The boundary values on the sides of the rectangle are supposed to have the second order derivatives satisfying the Hölder condition. Under these conditions for the approximate values of the first derivatives of the solution of mixed boundary problem on a square grid, as the solution of the constructed difference scheme a uniform error estimation of order O(h) (h is the grid size) is obtained. Numerical experiments are illustrated to support the theoretical results.