264 Difference Schemes for Elliptic EquationsProof Let a majorant Y(P) be taken such that
£ Y = I F(P) I, YI,, = 0, Y(P) > 0 for PE w + /
and Y(P) attain its maximum at a node Po E w. As far as Y(Po) =II Y lie
is concerned, the equationD(Po) Y(Po) + B(Po, Q) (Y(Po) - Y(Q)) = I F(Po) I
QEPatt'(Po)yieldsD(Po) Y(Po) < I F(Po) I, Y(P, 0 ) -< I D(Po) F(Po) I < - II D F II e'
thereby completing the proof.Remark Estimate (17) is still valid for the solution of problem (16) pro-
vided that instead of (2) other conditionsIA(P)l#O, IB(P,Q)l#O,
D(P)=IA(P)I- I B(P, Q) I > 0
QEPatt'(P)hold.
Indeed, let I v(P) I > 0 take the maximal value at a node P 0 • Because
of this fact,IA(Po)l · lv(Po)I= I L B(Po,Q)v(Q)+F(Po)I
QEPatt'(Po)< IB(Po,Q)l · lv(Po)l+IF(Po)I,
QEPatt'(Po)whence it follows thatD(Po) I v(Po) I < I F(Po) I, II v lie = I v(Po) I< I~~;:~ I < II ~ lie·
It may happen that D(P) = 0 on a subset /:.; of the grid w and D(P) > 0
on the complement of/:.; tow: /:.; +w* = w. This type of situation is covered
by the following assertion.