The consistency error qh as a grid function on „¦h — {t1 , . . . , tN } or corres-

ˆ

pondingly a sequence of vectors q n in RM1 for n = 1, . . . , N is then de¬ned

ˆh

by

1

U n+1 ’ U n + ˜Ah U n+1

q n+1

ˆh :=

„

+ ˜Ah U n ’ q h ((n + ˜)„ ) (7.108)

for n = 0, . . . , N ’ 1. Then the error grid function obviously satis¬es

1 n+1

eh ’ en + ˜Ah en+1 + ˜Ah en for n = 0, . . . , N ’ 1 ,

q n+1

ˆh

=

h h

h

„

e0 = 0 (7.109)

h

(or nonvanishing initial data if the initial condition is not evaluated exactly

at the grid points). In the following we estimate the grid function qh in the

ˆ

discrete maximum norm

max{|(ˆ n )r | | r ∈ {1, . . . , M1 } , n ∈ {1, . . . , N }}

qh

ˆ := qh

∞

max{|ˆn |∞ | n ∈ {1, . . . , N }},

= qh (7.110)

i.e., pointwise in space and time. An alternative norm would be the discrete

L2 -norm, i.e.,

1/2 1/2

M1

N N

2

|(ˆ n )r | |ˆ n |2

hd

qh

ˆ := „ qh = „ q h 0,h , (7.111)

0,h

n=1 r=1 n=1

using the spatial discrete L2 -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

M1 = M1 (h) ¤ Ch’d , (7.112)

332 7. Discretization of Parabolic Problems

then obviously,

¤ (CT )1/2 qh

qh

ˆ ˆ ,

∞

0,h

so that the L2 -norm is weaker than the maximum norm. Condition (7.112)

is satis¬ed for such uniform grids, as considered in Section 1.2. A norm in

between is de¬ned by

:= max {|ˆ n |0,h | n = 1, . . . , N } ,

qh

ˆ qh (7.113)

∞,0,h

which is stronger than (7.111) and in the case of (7.112) weaker than the

maximum norm.

Analogously to Section 1.4, we denote U n amended by the eliminated

ˆn ˜n

boundary values U h ∈ RM2 by the vector U ∈ RM .

For simplicity we restrict attention, at the beginning, to the case of pure

Dirichlet data. Taking into account (7.98) and assuming that f ((n’1+˜)„ )

is derived from the continuous right-hand side by evaluation at the grid

points, we get

1 n+1 d

’ U n) ’

q n+1

ˆh = (U U (tn + ˜„ )

„ dt

˜ ˜ n+1 + ˜Ah U n ’ (LU )(tn + ˜„ )

˜˜

+ ˜Ah U

=: S 1 + S 2 , (7.114)

so that S 1 , consisting of the ¬rst two terms, is the consistency error for the

time discretization.

d

Here dt U and LU are the vectors representing the grid functions corre-

d

sponding to dt u and Lu, which requires the continuity of these functions

as in the notion of a classical solution. We make the following assumption:

The spatial discretization has the order of consistency ± measured in

· ∞ (according to (1.17)) if the solution of the stationary problem (7.6)

is in C p („¦) for some ± > 0 and p ∈ N.

For example, for the Dirichlet problem of the Poisson equation and the

¬ve-point stencil discretization on a rectangle, we have seen in Chapter 1

that ± = 2 is valid for p = 4. If we assume for u(·, t), u being the solution

of (7.1), that

the spatial derivatives up to order p exist continuously

and are bounded uniformly in t ∈ [0, T ] , (7.115)

then there exists a constant C > 0 such that

|(Ah U (t))i ’ (Lu(·, t))(xi )| ¤ Ch±

˜˜ (7.116)

for every grid point xi ∈ „¦h and t ∈ [0, T ].

In the case of Neumann or mixed boundary conditions, then some of the

equations will correspond to discretizations of these boundary conditions.

This discretization may be directly a discretization of (7.3) or (7.4) (typ-

ically, if one-sided di¬erence quotients are used) or a linear combination

7.6. Order of Convergence Estimates 333

of the discretizations of the di¬erential operator at xi ∈ „¦h and of the

˜

boundary di¬erential operator of (7.3) or (7.4) (to eliminate “arti¬cial”

grid points) (see Section 1.3).

Thus we have to take xi ∈ „¦h and interpret Lu in (7.116) as this modi¬ed

˜

di¬erential operator for xi ∈ “1 ∪ “2 just described to extend all the above

reasoning to the general case.

The estimation of the contribution S 2 on the basis of (7.116) is directly

possible for ˜ = 0 or ˜ = 1, but requires further smoothness for ˜ ∈ (0, 1).

We have

S2 = S3 + S4 ,

where

˜ ˜ n+1 ’ (LU )(tn+1 )) + ˜(Ah U n ’ (LU )(tn )) ,

˜˜

S3 := ˜(Ah U

:= ˜(LU )(tn+1 ) + ˜(LU )(tn ) ’ (LU )(tn + ˜„ ) .

S4

By Taylor expansion we conclude for a function v ∈ C 2 [0, T ] that

2

˜2 ˜

2

v (t1 ) + ˜ v (t2 )

˜v(tn+1 ) + ˜v(tn ) = v(tn + ˜„ ) + „ ˜ n n

2 2

for some t1 ∈ (tn , tn + ˜„ ), t2 ∈ (tn + ˜„, tn+1 ), so that

n n

|S 4 |∞ ¤ C„ 2 (7.117)

for some constant C > 0 independent of „ and h if for ˜ ∈ (0, 1) the

solution u of (7.1) satis¬es

‚2

‚

Lu ∈ C(QT ) .

Lu , (7.118)

‚t2

‚t

This is a quite severe regularity assumption, which often does not hold.

For S 3 we conclude directly from (7.116) that

|S 3 |∞ ¤ Ch± . (7.119)

1

and ˜ = 1 : If

To estimate S 1 we have to distinguish between ˜ = 2 2

‚2 ‚3

‚ 1

u ∈ C(QT ) and for ˜ = u ∈ C(QT ),

u, also (7.120)

‚t2 ‚t3

‚t 2

then Lemma 1.2 implies (for ˜ = 0, 1, 1 , for ˜ ∈ (0, 1) again with a Taylor

2

expansion)

|S 1 |∞ ¤ C„ β (7.121)

for some constant C, independent of „ and h, with β = 1 for ˜ = 1 and

2

β = 2 for ˜ = 1 .

2

Thus, under the additional regularity assumptions (7.115), (7.118),

(7.120), and if the spatial discretization has order of consistency ± in

334 7. Discretization of Parabolic Problems

the maximum norm, i.e., (7.116), then the one-step-theta method has the

following order of consistency:

¤ C(h± + „ β )

qh

ˆ (7.122)

∞

for some constant C, independent of „ and h, with β as in (7.121).

By using a weaker norm one might hope to achieve a higher order of

convergence. If this is, for example, the case for the spatial discretization,

e.g., by considering the discrete L2 -norm · 0,h instead of · ∞ , then

instead of (7.116) we have

˜˜

Ah U (t) ’ Lu(·, t) ¤ Ch± , (7.123)

0,h

where the terms in the norm denote the corresponding grid functions.

Then again under (weaker forms of) the additional regularity assump-

tions (7.115), (7.118), (7.120) and assuming (7.112), we have

¤ C(h± + „ β ) .

qh

ˆ (7.124)

0,h

By means of Theorem 7.26 we can conclude the ¬rst order of convergence

result:

Theorem 7.29 Consider the one-step-theta method and assume that the

spatial discretization matrix Ah has a basis of eigenvectors wi with eigen-

values »i ≥ 0, i = 1, . . . , M1 , orthogonal with respect to the scalar product

·, · h , de¬ned in (7.58). The spatial discretization has order of consistency

± in · 0,h for solutions in C p („¦). If „ is such that the method is sta-

ble according to (7.95), then for a su¬ciently smooth solution u of (7.1)

(e.g., (7.115), (7.118), (7.120)), and for a sequence of grid points satisfying

(7.112), the method converges in the norm · ∞,0,h with the order

O(h± + „ β ) ,

1

where β = 2 for ˜ = and β = 1 otherwise.

2

Proof: Due to Theorem 7.26 and (7.109) we have to estimate the

N

consistency error in a norm de¬ned by „ n=1 |ˆ n |0,h (i.e., a discrete L1 -

qh

2

L -norm), which is weaker than qh 0,h , in which the estimate has been

ˆ

2

veri¬ed in (7.124).

Again we see here a smoothing e¬ect in time: The consistency error has

to be controlled only in a discrete L1 -sense to gain a convergence result in

a discrete L∞ -sense.

If a consistency estimate is provided in · ∞ as in (7.122), a convergence

estimate still needs the corresponding stability. Instead of constructing a

vector as in Theorem 1.14 for the formulation (7.97), we will argue directly

with the help of the comparison principle (Theorem 7.28, 1)), which would

have been possible also in Section 1.4 (see Exercise 1.14).

7.6. Order of Convergence Estimates 335

Theorem 7.30 Consider the one-step-theta method and assume that the

spatial discretization matrix Ah satis¬es (1.32) (1), (2), (3) (i) and assume

its L∞ -stability by the existence of vectors wh ∈ RM1 and a constant C > 0

independent of h such that

Ah w h ≥ 1 |wh |∞ ¤ C .

and (7.125)

The spatial discretization has order of consistency ± in · ∞ for solutions

in C p („¦). If (7.100) is satis¬ed, then for a su¬ciently smooth solution u

of (7.1) (e.g., (7.115), (7.118), (7.120)) the method converges in the norm

· ∞ with the order

O(h± + „ β ) ,

1

where β = 2 for ˜ = and β = 1 otherwise.

2

Proof: From (7.122) we conclude that

’C(h± + „ β )1 ¤ q n ¤ C(h± + „ β )1

ˆ ˆ

ˆh for n = 1, . . . , N

ˆ

for some constant C independent of h and „.

Thus (7.109) implies

1 n+1

eh ’ en + ˜Ah en+1 + ˜Ah en ¤ ˆ

C(h± + „ β )1 ,

h h

h

„

e0 = 0.

h

ˆ

Setting wn := C(h± + „ β )wh with wh from (7.125), this constant sequence

h

of vectors satis¬es

1

wn+1 ’ w n + ˜Ah wn+1 + ˜Ah wn ≥ C(h± + „ β )1 .

ˆ

h h

h h

„

Therefore, the comparison principle (Theorem 7.28, (1)) implies

en ¤ w n = C(h± + „ β )w h

ˆ

h h

for n = 0, . . . , N, and analogously, we see that

’C(h± + „ β )wh ¤ en ,

ˆ h

so that

(en )j ¤ C(h± + „ β )(w h )j

ˆ (7.126)

h

for all n = 0, . . . , N and j = 1, . . . , M1 , and ¬nally,

|en |∞ ¤ C(h± + „ β )|w h |∞ ¤ C(h± + „ β )C

ˆ ˆ

h

2

with the constant C from (7.125).

Note that the pointwise estimate (7.126) is more precise, since it also

takes into account the shape of w h . In the example of the ¬ve-point stencil

with Dirichlet conditions on the rectangle (see the discussion around (1.43))

336 7. Discretization of Parabolic Problems

the error bound is smaller in the vicinity of the boundary (which is to be

expected due to the exactly ful¬lled boundary conditions).

Order of Convergence Estimates for the Finite Element Method

We now return to the one-step-theta method for the ¬nite element method

as introduced in (7.72). In particular, instead of considering grid functions

as for the ¬nite di¬erence method, the ¬nite element method allows us to

consider directly a function U n from the ¬nite-dimensional approximation

space Vh and thus from the underlying function space V .

In the following, an error analysis for the case ˜ ∈ [ 1 , 1] under the as-

2

sumption u ∈ C 2 ([0, T ], V ) will be given. In analogy with the decomposition

of the error in the semidiscrete situation, we write

u(tn ) ’ U n = u(tn ) ’ Rh u(tn ) + Rh u(tn ) ’ U n =: (tn ) + θn .

The ¬rst term of the right-hand side is the error of the elliptic projection at

the time tn , and for this term an estimate is already known. The following

identity is used to estimate the second member of the right-hand side, which

immediately results from the de¬nition of the elliptic projection:

1 n+1

’ θn ), vh + a(˜θn+1 + ˜θn , vh )

(θ

„ 0

1

((Rh u(tn+1 ) ’ Rh u(tn )), vh + a(˜Rh u(tn+1 ) + ˜Rh u(tn ), vh )

=

„ 0

1 n+1

’ ’ U n ), vh ’ a(˜U n+1 + ˜U n , vh )

(U

„ 0

1

(Rh u(tn+1 ) ’ Rh u(tn )), vh + a(˜u(tn+1 ) + ˜u(tn ), vh )

=

„ 0

’ bn+˜ (vh )

1

(Rh u(tn+1 ) ’ Rh u(tn )), vh ’ ˜u (tn+1 ) + ˜u (tn ), vh 0

=

„ 0

= wn , vh 0 ,

where

1

(Rh u(tn+1 ) ’ Rh u(tn )) ’ ˜u (tn+1 ) ’ ˜u (tn ) .

wn :=

„

Taking into consideration the inequality a(vh , vh ) ≥ 0 , the particular choice

of the test function as vh = ˜θn+1 + ˜θn yields

+ (1 ’ 2˜) θn , θn+1 ’ ˜ θn ¤ „ wn , ˜θn+1 + ˜θn 0 .

˜ θn+1 2 2

0

0 0

For ˜ ∈ [ 1 , 1] we have (1 ’ 2˜) ¤ 0, and hence

2

’ θn

θn+1 ˜ θn+1 + ˜ θn

0 0 0 0

+ (1 ’ 2˜) θn 0 θn+1 0 ’ ˜ θn

˜ θn+1 2 2

= 0 0

¤ + (1 ’ 2˜) θn , θn+1 0 ’ ˜ θn 2

˜ θn+1 2

0 0

¤ „ wn ˜ θn+1 + ˜ θn .

0 0 0

7.6. Order of Convergence Estimates 337

Dividing each side by the expression in the square brackets, we get

¤ θn

θn+1 + „ wn 0.

0 0

The recursive application of this inequality leads to

n

¤θ

n+1 0

wj

θ +„ . (7.127)

0 0 0

j=0

That is, it remains to estimate the terms wj 0 . A simple algebraic

manipulation yields

1 1

((Rh ’ I)u(tn+1 ) ’ (Rh ’ I)u(tn )) + (u(tn+1 ) ’ u(tn ))

wn :=

„ „

’ ˜u (tn+1 ) ’ ˜u (tn ) . (7.128)

Taylor expansion with integral remainder implies

tn+1

(tn+1 ’ s)u (s) ds

u(tn+1 ) = u(tn ) + u (tn )„ +

tn

and

tn

u(tn ) = u(tn+1 ) ’ u (tn+1 )„ + (tn ’ s)u (s) ds .

tn+1

Using the above relations we get the following useful representations of the

di¬erence quotient of u in tn :

tn+1

1 1

(u(tn+1 ) ’ u(tn )) (tn+1 ’ s)u (s) ds ,

= u (tn ) +