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44 1. Finite Difference Method for the Poisson Equation

for the consistency error, then
|τ h | ∞ → 0 for h → 0 .
Thus
ˆ
= τ h − A h ˆ w h + f
A h w h
˜
≥ −|τ h | ∞ 1 +˜α1 for h ≤ H
≥ α1 for h ≤ H
and some appropriate H> 0.

Thus a proper choice in (1.41) is
1 1
w h and C := w ∞ . (1.46)
α α
The condition (1.45) is not critical: In case of Dirichlet boundary conditions
ˆ
and (1.32) (5) (for corresponding rows i of A h ) then, due to (f) i ≥ 1, we
can even choose ˜α =1. The discussion of Neumann boundary conditions
following (1.24) shows that the same can be expected.
Theorem 1.15 shows that for a discretization with an inverse monotone
system matrix consistency already implies stability.
To conclude this section let us discuss the various ingredients of (1.32)
or (1.32)* that are sufficient for a range of properties from the inverse
monotonicity up to a strong maximum principle: For the five-point stencil
on a rectangle all the properties are valid for Dirichlet boundary conditions.
If partly Neumann boundary conditions appear, the situation is the same,
but now close and far from the boundary refers to its Dirichlet part. In
the interpretation of the implications one has to take into account that the
heterogeneities of the Neumann boundary condition are now part of the
right-hand side f, as seen, e.g., in (1.26). If mixed boundary conditions are
applied, as
∂ ν u + αu = g on Γ 2 (1.47)
for some Γ 2 ⊂ Γand α = α(x) > 0, then the situation is the same again
if αu is approximated just by evaluation, at the cost that (4)* no longer
holds. The situation is similar if reaction terms appear in the differential
equation (see Exercise 1.10).

Exercises

ˆ
1.11 Give an example of a matrix A h ∈ R M 1 ,M 2 that canbeusedinthe
proof of Theorem 1.12.

1.12 Show that the transposition of an M-matrix is again an M-matrix.