Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 20

0, all summands appearing in the
sums on the left-hand side are nonnegative. Hence, every summand equals 0. Now
α i =

Since α i < 0 for i = 1 ,...,k and p i − p 0 ≤

0 implies (3.7).
In the following, it is important to note that the discretization can change
the topological structure of .If is connected, it does not follow that h is
connected (with an appropriate definition). The situation shown in Fig. 5 leads to
a system with a reducible matrix. To guarantee that the matrix is irreducible, we
have to use a sufficiently small mesh size.

Fig. 5. Connected domain for which h is not connected

3.4 Definition. h is said to be (discretely) connected provided that between every
pair of points in h , there exists a path of grid lines which remains inside of .
Clearly, using a finite difference method to solve the Poisson equation, we
get a system with an irreducible matrix if and only if h is discretely connected.
We are now in a position to formulate the discrete maximum principle. Note
that the hypotheses for the standard five-point stencil for the Laplace operator are
satisfied.

3.5 Discrete Maximum Principle. Let U be a solution of the linear system which
arises from the discretization of

Lu = f in with f ≤ 0

using a stencil which satisfies the following three conditions at every grid point in
h :
(i) All of the coefficients except for the one at the center are nonpositive.
(ii) The coefficient in one of the directions is negative, say α E < 0 .
(iii) The sum of all of the coefficients is nonnegative.
Then

max
z i ∈ h

U i ≤ max

z j ∈ ∂ h

U j . ( 3 . 8 )

Furthermore, suppose the maximum over all the grid points is attained in the
interior, the coefficients α E ,α S ,α W and α N in all four cardinal directions are
negative, and h is discretely connected. Then U is constant.