Sharp error bounds for the Crank-Nicolson and Saulyev difference scheme in connection with an initial boundary value problem for the inhomogeneous heat equation

For an initial boundary value problem of the inhomogeneous heat equation, the present paper studies the sharpness of error bounds, obtained for approximate solutions via the Crank-Nicolson and Saulyev difference scheme, respectively. Whereas the direct estimates in terms of partial moduli of continuity for partial derivatives of the (exact) solution 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, use is made of some basic properties of the discrete Greenʹs function associated. It may be mentioned that the methods of this paper, though specific, do not rely on any positivity properties of the discrete Greenʹs function, in contrast to our previous investigations which were concerned with boundary value problems for ordinary as well as for elliptic differential equations.