Abstract: The cubic Hermite collocation method is used in discretization of an elliptic problem with a discontinuous solution.The discontinuous points of the solution are taken as grid points,the left and right limits of the solution at the discontinuous points as unknowns.The jumpings of the solution at the discontinuous points are combined to dicrete the original problem.The test indicates that the method has the convergence of order O(h4).

Key words: elliptic problem, Hermite interpolation, collocation method