Page 60 -

1.4. Maximum Principles and Stability 43

Analogously, let ˆ w h ∈ R M 2 be the representation of the function ˆw h de-
∗
fined on ∂Ω . As can be seen from the error representation in Lemma 1.2,
h
− +
statement 4, the difference quotient ∂ ∂ u(x) is exact for polynomials of
second degree. Therefore, we conclude from (1.43) that
ˆ
A h w h = −A h ˆ w h + 1 ≥ 1 ,
which finally implies
a b 1 2 2

|w h | ∞ = w h ∞ ≤ w ∞ = w 1 + w 2 = (a + b ) .
2 2 16
This example motivates the following general procedure to construct w h ∈
R M 1 and a constant C such that (1.41) is fulfilled.
Assume that the boundary value problem under consideration reads in
an abstract form
(Lu)(x) = f(x) for x ∈ Ω ,
(1.44)
(Ru)(x) = g(x) for x ∈ ∂Ω .
Similar to (1.43) we can consider — in case of existence — a solution w
of (1.44) for some f, g, such that f(x) ≥ 1 for all x ∈ Ω,g(x) ≥ 0 for all
x ∈ Ω. If w is bounded on Ω, then
(w h ) i := w(x i ), i =1,... ,M 1 ,

for the (non-Dirichlet) grid points x i , is a candidate for w h . Obviously,
|w h | ∞ ≤ w ∞ .

Correspondingly, we set
( ˆ w h ) i = w(x i ) ≥ 0 , i = M 1 +1,... ,M 2 ,
for the Dirichlet-boundary grid points.
The exact fulfillment of the discrete equations by w h cannot be expected
anymore, but in case of consistency the residual can be made arbitrarily
small for small h. This leads to
Theorem 1.15 Assume that a solution w ∈ C(Ω) of (1.44) exists for data
f ≥ 1 and g ≥ 0. If the discretization of the form (1.31) is consistent with
˜
(1.44) (for these data), and there exists H> 0 so that for some ˜α> 0:
˜
ˆ
−A h ˆ w h + f ≥ ˜α1 for h ≤ H, (1.45)
then for every 0 <α < ˜α there exists H> 0, so that
A h w h ≥ α1 for h ≤ H.

Proof: Set
ˆ
τ h := A h w h + A h ˆ w h − f