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 equati on, the present paper studies the sharpness of error bounds, obtained for approximate solutions via the Crank-Nicolson and Saulyev differenc e scheme, respectively. Whereas the direct estimates in terms of parti al moduli of continuity for partial derivatives of the (exact) solutio n follow by standard methods (stability inequality plus Taylor expansi on of the truncation error), the sharpness of these bounds is establis hed by an application of a quantitative extension of the uniform bound edness principle. To verify the relevant resonance condition, use is m ade of some basic properties of the discrete Green's function associat ed. It may be mentioned that the methods of this paper, though specifi c, do not rely on any positivity properties of the discrete Green's fu nction, in contrast to our previous investigations which were concerne d with boundary value problems for ordinary as well as for elliptic di fferential equations.