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7.6. Order of Convergence Estimates 331

required for the classical solution u of the initial boundary value problem
(7.1), which in particular makes its evaluation possible at the grid points
x i ∈ Ω h at each instance of time t ∈ [0,T ] and also of various derivatives.
The vector representing the corresponding grid function (for a fixed order-
n
ing of the grid points) will be denoted by U(t), or for short by U := U(t n )
for t = t n . The corresponding grid points depend on the boundary condi-
tion. For a pure Dirichlet problem, the grid points will be from Ω h , but if
Neumann or mixed boundary conditions appear, they are from the enlarged
set
˜
Ω h := Ω h ∩ (Ω ∪ Γ 1 ∪ Γ 2 ) . (7.106)
Then the error at the grid points and each time level is given by
n
n
e := U − u n for n =0,... ,N , (7.107)
h h
n
where u is the solution of the one-step-theta method according to (7.66).
h
The consistency error ˆq h as a grid function on Ω h ×{t 1,...,t N } or corres-
n
pondingly a sequence of vectors ˆ q in R M 1 for n =1,... ,N is then defined
h
by
n+1 1 n+1 n n+1
ˆ q := U − U +ΘA h U
h
τ
n
+ ΘA h U − q ((n +Θ)τ) (7.108)
h
for n =0,... ,N − 1. Then the error grid function obviously satisfies
1 n+1 n n+1 n n+1
e h − e h +ΘA h e h + ΘA h e h = ˆ q h for n =0,... ,N − 1 ,
τ
e 0 h = 0 (7.109)
(or nonvanishing initial data if the initial condition is not evaluated exactly
at the grid points). In the following we estimate the grid function ˆq h in the
discrete maximum norm
:= n
h
ˆq h ∞ max{|(ˆ q ) r || r ∈{1,... ,M 1 } ,n ∈{1,... ,N}}
n
= max{|ˆ q | ∞ | n ∈{1,... ,N}}, (7.110)
h
i.e., pointwise in space and time. An alternative norm would be the discrete
2
L -norm, i.e.,

1/2
1/2
N M 1 N
n 2
n 2
ˆq h 0,h := τ h d |(ˆ q ) r | = τ |ˆ q | , (7.111)
h 0,h
h
n=1 r=1 n=1
2
using the spatial discrete L -norm from (7.59), where the same notation is
employed. If for the sequence of underlying grid points considered there is
a constant C> 0 independent of the discretization parameter h such that
M 1 = M 1 (h) ≤ Ch −d , (7.112)