In lemma 11.1 on P129 you state that L_h^k \omega \leq 0 on the interior of the mesh. Given that is satisfies the approximation of the balck scholes equation is if fair to asume that L_h^k \omega = 0 on the interior of the mesh?

also in Leema 11.2

If first says suppose that max |U_j^n| \leq N for all j and n

and max|f_j^n| leq N for all j and n

firstly is the 2nd term not redundant? Plust by making this assumption are we not effectively assuming stability rather than proving it?

Also comparing equation 3.13 on page 28 with lemma 11.12 the max seems to have been replaced by a + rather than a max function is moving from continuous to discrete time why was this?