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324 7. Discretization of Parabolic Problems

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The collection of conditions (1.32), (1), (2), (3) i), (4) is called (IM).
(IM) implies the inverse monotonicity of A h (Theorem 1.12, (1.39)).
The collection of conditions (IM), (5) is called (CP).
(CP) implies a comparison principle in the sense of Corollary 1.13.
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The collection of conditions (CP),(6) is called (MP) .
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(MP) implies a maximum principle in the form of Theorem 1.10 (1.38).
Alternatively, the collection of conditions (CP) (6) # (see Exercise 1.13) is
called (MP).
(MP) implies a maximum principle in the form of Theorem 1.9 (1.34).
Finally, the collection of conditions (CP), (6), (4) (instead of (4) ), (7) is
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called (SMP).
(SMP) implies a strong maximum principle in the sense of Theorem 1.9.
An L -stability estimate in the sense of Theorem 1.14 is closely related.
∞
This will be taken up in the next section.
In the following we will discuss the above-mentioned properties for the
one-step-theta method, cast into the form (1.31), on the basis of correspond-
ing properties of the underlying elliptic problem and its discretization. It
will turn out that under a reasonable condition (see (7.100)), condition (4) ∗
(and thus (3) ii)) will not be necessary for the elliptic problem. This reflects
the fact that contrary to the elliptic problem, for the parabolic problem also
the case of a pure Neumann boundary condition (where no degrees of free-
dom are given and thus eliminated) is allowed, since the initial condition
acts as a Dirichlet boundary condition.
In assuming that the discretization of the underlying elliptic problem is
of the form (1.31), we return to the notation M = M 1 + M 2 ,where M 2 is
the number of degrees of freedom eliminated, and thus A h ,B h ∈ R M 1 ,M 1 .
We write the discrete problem according to (7.66) as one large system of
equations for the unknown

 1 
u
h
 u 2 
 h 
.  , (7.96)
u h =  .
 . 
u N
h
in which the vector of grid values u ∈ R M 1 are collected to one large
i
h
vector of dimension M 1 := N · M 1. Thus the grid points in Ω × (0,T )are
the points (x j ,t n ),n =1,... ,N, x j ∈ Ω h , e.g., for the finite difference
method. The defining system of equations has the form

C h u h = p , (7.97)
h