Differential equations with locally perturbed coefficients: High order error estimates for function and derivative

High order error estimates are obtained for both function and derivative when the coefficients and right hand side of a given initial or boundary value problem are replaced by piece-wise polynomial functions. These estimates are given for partition points and also continuously on subintervals. Based on our results for linear case, we propose a new technique to solve nonlinear differential equations. Numerical examples demonstrate the accuracy of our estimates.