  We show that the celebrated 1956 Lax-Richtmyer linear theorem in Numerical
Analysis - often called the Fundamental Theorem of Numerical Analysis - is in
fact wrong. Here "wrong" does not mean that its statement is false
mathematically, but that it has a limited practical relevance as it
misrepresents what actually goes on in the numerical analysis of partial
differential equations. Namely, the assumptions used in that theorem are
excessive to the extent of being unrealistic from practical point of view. The
two facts which the mentioned theorem gets wrong from practical point of view
are : - the relationship between the convergence and stability of numerical
methods for linear PDEs, - the effect of the propagation of round-off errors in
such numerical methods. The mentioned theorem leads to a result for PDEs which
is unrealistically better than the well known best possible similar result in
the numerical analysis of ODEs. Strangely enough, this fact seems not to be
known well enough in the literature. Once one becomes aware of the above, new
avenues of both practical and theoretical interest can open up in the numerical
analysis of PDEs.
