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330 7. Discretization of Parabolic Problems

appropriate for finite element discretizations with mass lumping, see
(7.74)).

7.15 Show the validity of (6) # from Exercise 1.13 for C h if it holds here
for A h and conclude as in Exercise 1.13 a weak maximum principle for the
one-step-theta method.

7.16 Consider the initial-boundary value problem in one space dimension

 u t − εu xx + cu x = f in (0, 1) × (0,T ) ,
u(0,t)= g −(t),u(1,t)= g + (t) , t ∈ (0,T ) ,

u(x, 0) = u 0 (x) , x ∈ (0, 1) ,
where T> 0and ε> 0are constants, and c, f :(0, 1) × (0,T ) → R,
u 0 :(0, 1) → R,and g − ,g + :(0,T ) → R are sufficiently smooth functions
such that the problem has a classical solution.
Define h := 1/m and τ = T/N for some numbers m, N ∈ N. Then the so-
called full-upwind finite difference method for this problem reads as follows:
0
Find a sequence of vectors u ,..., u N by
h h
u n+1 − u n u n+1 − 2u n+1 + u n+1 u n+1 − u n+1 u n+1 − u n+1
i i − ε i+1 i i−1 − c − i+1 i + c + i i−1
τ h 2 h h
= f n+1 , i =1,... ,m − 1,n =0,... ,N − 1,
i
+
0
where c = c − c − with c + =max{c, 0},f n = f(ih, nτ),u = u 0 (ih),
i i
n
u = g − (nτ)and u n = g + (nτ).
0 m
Prove that a weak maximum principle holds for this method.
7.6 Order of Convergence Estimates

Based on stability results already derived, we will investigate the (order
of) convergence properties of the one-step-theta method for different dis-
cretization approaches. Although the results will be comparable, they will
be in different norms, appropriate for the specific discretization method, as
already seen in Chapters 1, 3, and 6.
Order of Convergence Estimates for the Finite Difference Method
From Section 1.4 we know that the investigation of the (order of)
convergence of a finite difference method consists of two ingredients:
• (order of) convergence of the consistency error
• stability estimates.
The last tool has already been provided by Theorem 7.26 and by Theo-
rem 1.14, which together with the considerations of Section 7.5 allow us to
concentrate on the consistency error. Certain smoothness properties will be