On the Sharpness of Error Bounds for the Numerical Solution of Initial Boundary Value Problems by Finite Difference Schemes

Abstract

The present paper studies the sharpness of error bounds obtained for approximate solutions of initial boundary value problems by finite difference schemes. Whereas the direct estimates in terms of partial moduli of continuity for partial derivatives of the (exact) solutions follow by standard methods (stability inequality plus Taylor expansion of the truncation error), the sharpness of these bounds is established by an application of a quantitative extension of the uniform boundedness principle. To verify the relevant resonance condition a general procedure is suggested, in contrast to our previous investigations which were based on rather specific properties of the discrete Green’s functions associated. Exemplarily, details are worked out in connection with Crank–Nicolson, Du Fort-Frankel and Saulyev schemes.