On estimating best approximations of functions defined by differential equations

Finite difference methods are used to estimate the error in approximations to functions defined by differential equations. The problem of minimising the maximum of this error estimate can be solved by linear programming in the linear case, and by a method due to the authors in the nonlinear case. It is shown by examples that this new approach can improve substantially on techniques which minimise the maximum residual in the differential equation, and a convergence result as the mesh spacing tends to zero is given for the linear case.