¯

|cνµ ’ ch | = O(h2 ).

νµ

Because of

νµ νµ

e’» ’ e’»h = e’» 1 ’ e’(»h ’»νµ )t

νµ νµ

t t t

,

and |»νµ ’ »νµ | = O(h2 ) (see (7.61)), also this term is of order O(h2 ) and

h

will be damped exponentially (depending on t and the size of the smallest

eigenvalue »νµ ).

Summarizing, we expect

O(h2 )

to be the dominating error term in the discrete maximum norm · ∞ at the

grid points (cf. de¬nition (1.17)), which will also be damped exponentially

for increasing t. Note that we have given only heuristic arguments and that

the considerations cannot be transferred to the Neumann case, where the

eigenvalue » = 0 appears.

We now turn to the ¬nite element method.

Order of Convergence Estimates for the Finite Element Method

We will investigate the ¬nite element method on a more abstract level as in

the previous subsection, but we will achieve a result (in di¬erent norms) of

similar character. As worked out at the end of Section 7.1, there is a strong

relation between the V -ellipticity of the bilinear form a with the parameter

± and a positive lower bound of the eigenvalues. Here we rely on the results

already achieved in Section 2.3 and Section 3.4 for the stationary case.

For that, we introduce the so-called elliptic projection of the solution u(t)

of (7.32) as a very important tool in the proof.

304 7. Discretization of Parabolic Problems

De¬nition 7.8 For a V -elliptic, continuous bilinear form a : V — V ’ R,

the elliptic, or Ritz, projection Rh : V ’ Vh is de¬ned by

v ’ Rh v ⇐’ a(Rh v ’ v, vh ) = 0 for all vh ∈ Vh .

Theorem 7.9 Under the assumptions of De¬nition 7.8:

(i) Rh : V ’ Vh is linear and continuous.

(ii) Rh yields quasi-optimal approximations; that is,

M

v ’ Rh v ¤ inf v ’ vh ,

V V

± vh ∈Vh

where M and ± are the Lipschitz and ellipticity constants according to

(2.42) and (2.43).

Proof: The linearity of Rh is obvious. The remaining statements

immediately follow from Lemma 2.16 and Theorem 2.17; see Exercise 7.6. 2

Making use of the elliptic projection, we are able to prove the following

result.

Theorem 7.10 Suppose a is a V -elliptic, continuous bilinear form, f ∈

C([0, T ], H), u0 ∈ V, and u0h ∈ Vh . Then if u(t) is su¬ciently smooth,

e’±t + (I ’ Rh )u(t)

uh (t) ’ u(t) ¤ u0h ’ Rh u0

0 0 0

t

e’±(t’s) ds .

(I ’ Rh )u (s)

+ 0

0

Proof: First, the error is decomposed as follows:

uh (t) ’ u(t) = uh (t) ’ Rh u(t) + Rh u(t) ’ u(t) =: θ(t) + (t) .

We take v = vh ∈ Vh in (7.32) and obtain, by the de¬nition of Rh ,

u (t), vh + a(u(t), vh ) = u (t), vh + a(Rh u(t), vh ) = b(t, vh ) .

0 0

Here b(t, ·) is as de¬ned in (7.43).

Subtracting this equation from (7.44), we get

’ u (t), vh

uh (t), vh + a(θ(t), vh ) = 0 ,

0 0

and thus

d

0’ =’

θ (t), vh 0 + a(θ(t), vh ) = u (t), vh Rh u(t), vh (t), vh .

0

dt 0

The application of Lemma 7.4 yields

t

’±t

e’±(t’s) ds .

¤ θ(0)

θ(t) 0e + (s)

0 0

0

Since the elliptic projection is continuous (Theorem 7.9, (i)) and u(t) is

d

su¬ciently smooth, Rh and the time derivative dt commute; that is, (t) =

7.2. Semidiscretization by Vertical Method of Lines 305

(Rh ’ I)u (t). It remains to apply the triangle inequality to get the stated

2

result.

Theorem 7.10 has the following interpretation:

The error norm uh (t) ’ u(t) 0 is estimated by

• the initial error (exponentially decaying in t), which occurs only if

u0h does not coincide with the elliptic projection of u0 ,

• the projection error of the exact solution u(t) measured in the norm

of H,

• the projection error of u (t) measured in the norm of H and integrally

weighted by the factor e’±(t’s) on (0, t).

Remark 7.11 If the bilinear form a de¬nes an elliptic problem such that

for the elliptic projection an error estimate of the type

(I ’ Rh )w ¤ Ch2 w for all w ∈ V © H 2 („¦)

2

0

is valid, if u0h approximates the elliptic projection Rh u0 of the initial value

u0 at least with the same asymptotic quality, and if the solution u of (7.44)

is su¬ciently smooth, then an optimal L2 -error estimate results:

uh (t) ’ u(t) ¤ C(u(t))h2 .

0

We see that in order to obtain semidiscrete error estimates, we need esti-

mates of the projection error measured in the norm of H = L2 („¦). Due to

· 0 ¤ · V , the quasi-optimality of Rh (Theorem 7.9, (ii)) in conjunction

with the corresponding approximation error estimates (Theorem 3.29) al-

ready yield some error estimate. Unfortunately, this result is not optimal.

However, if the adjoint boundary value problem is regular in the sense of

De¬nition 3.36, the duality argument (Theorem 3.37) can be successfully

used to derive an optimal result.

Theorem 7.12 Suppose the bilinear form a is V -elliptic and continuous,

and the solution of the adjoint boundary value problem is regular.

Furthermore, let the space Vh ‚ V be such that for any function w ∈

V © H 2 („¦),

w ’ vh ¤ C h |w|2 ,

inf V

vh ∈Vh

where the constant C > 0 does not depend on h and w. If u0 ∈ V © H 2 („¦),

then for a su¬ciently smooth solution u of (7.44) we have

e’±t

uh (t) ’ u(t) ¤ u0h ’ u0

0 0

t

’±t

e’±(t’s) ds .

2

+ Ch u0 e + u(t) + u (s)

2 2 2

0

306 7. Discretization of Parabolic Problems

Proof: The ¬rst term in the error bound from Theorem 7.10 is estimated

by means of the triangle inequality:

u0h ’ Rh u0 ¤ u0h ’ u0 + (I ’ Rh )u0 0.

0 0

Then the projection error estimate (Theorem 3.37, (1)) yields the given

bounds of the resulting second term as well as of the remaining two terms

2

in the error bound from Theorem 7.10.

Order of Convergence Estimates for the Finite Volume Method

For simplicity we restrict attention to pure homogeneous Dirichlet condi-

tions (“3 = ‚„¦). The idea is similar to the proof given in the ¬nite element

case. However, here we will meet some additional di¬culties, which are

caused by the use of perturbed bilinear and linear forms.

We take v = vh ∈ Vh in (7.32) and subtract the result from (7.48):

’ u (t), vh + ah (uh (t), vh ) ’ a(u(t), vh )

uh (t), vh 0

0,h

’ f (t), vh

= f (t), vh .

0,h 0

In analogy to the ¬nite element method, we introduce the following aux-

iliary problem: Given some v ∈ V, ¬nd an element Rh v ∈ Vh such

that

ah (Rh v, vh ) = a(v, vh ) for all vh ∈ Vh . (7.63)

With this, the above identity can be rewritten as follows:

’ u (t), vh + ah (uh (t) ’ Rh u(t), vh )

uh (t), vh 0

0,h

’ f (t), vh

= f (t), vh .

0,h 0

d

Subtracting from both sides of this relation the term dt Rh u(t), vh 0,h

and assuming that u (t) is a su¬ciently smooth function of x, a slight

rearrangement yields

+ ah (θ(t), vh ) = ’

θ (t), vh (t), vh + u (t), vh (7.64)

0,h 0,h 0

’ u (t), vh ’ f (t), vh

+ f (t), vh ,

0,h 0,h 0

where, as in the ¬nite element case, θ(t) = uh (t) ’ Rh u(t) and (t) =

Rh u(t) ’ u(t). Furthermore, we de¬ne, for v ∈ Vh , b1 (t, v) := u (t), v 0 ’

u (t), v 0,h and b2 (t, v) := f (t), v 0,h ’ f (t), v 0 .

In order to be able to apply the discrete stability estimate (7.50) to this

situation, we need an error estimate for Rh u (t) as in Remark 7.11 and

bounds (consistency error estimates) for |b1 (t, v)|, |b2 (t, v)|.

So we turn to the ¬rst problem. In fact, the estimate is very similar

to the error estimate for the ¬nite volume method given in the proof of

Theorem 6.18.

For an arbitrary function v ∈ V © H 2 („¦) and vh := Rh v ’ Ih (v), we have

by (7.63) that

ah (vh , vh ) = ah (Rh v, vh ) ’ ah (Ih (v), vh ) = a(v, vh ) ’ ah (Ih (v), vh ).

7.2. Semidiscretization by Vertical Method of Lines 307

By partial integration in the ¬rst term of the right-hand side, it follows

that

’ ah (Ih (v), vh ).

ah (vh , vh ) = Lv, vh 0

From [40] an estimate of the right-hand side is known (cf. also (6.22)); thus

1/2

ah (vh , vh ) ¤ Ch v |vh |2 + vh 2

.

2 1 0,h

So Theorem 6.15 yields

1/2

|vh |2 + vh ¤ Ch v

2

2.

1 0,h

By the triangle inequality,

(Rh ’ I)v ¤ Rh v ’ Ih (v) + Ih (v) ’ v 0,h .

0,h 0,h

·

Since the second term vanishes by the de¬nitions of and Ih , we get

0,h

in particular

(Rh ’ I)v ¤ Ch v 2. (7.65)

0,h

Remark 7.13 In contrast to the ¬nite element case (Remark 7.11), this

estimate is not optimal.

To estimate |b1 (t, v)| and |b2 (t, v)|, we prove the following result.

Lemma 7.14 Assume w ∈ C 1 („¦) and v ∈ Vh . Then, if the ¬nite volume

partition of „¦ is a Donald diagram,

| w, v ’ w, v | ¤ Ch|w|1,∞ v 0,h .

0,h 0

Proof: We start with a simple rearrangement of the order of summation:

M

wj vj |„¦j,K | ,

w, v = wj vj mj =

0,h

j=1 K∈Th j:‚K aj

where „¦j,K = „¦j © int K . First, we will consider the inner sum. For any

triangle K ∈ Th with barycentre aS,K , we can write

wj vj |„¦j,K | = [wj ’ w(aS,K )]vj |„¦j,K |

j:‚K aj j:‚K aj

w(aS,K ) vj |„¦j,K | ’

+ v dx

„¦j,K

j:‚K aj

[w(aS,K ) ’ w]v dx +

+ wv dx

„¦j,K „¦j,K

j:‚K aj j:‚K aj

=: I1,K + I2,K + I3,K + wv dx .

K

308 7. Discretization of Parabolic Problems

To estimate I1,K , we apply the Cauchy“Schwarz inequality and get

± 1/2

|I1,K | ¤ |wj ’ w(aS,K )| |„¦j,K |

2

v 0,h,K ,

j:‚K aj

where

± 1/2

vj |„¦j,K |

2

v := .

0,h,K

j:‚K aj

Since |wj ’ w(aS,K )| ¤ hK |w|1,∞ , it follows that

|I1,K | ¤ hK |w|1,∞ |K| v 0,h,K .

Similarly, for I3,K we easily get

|I3,K | = [w(aS,K ) ’ w]v dx

„¦K

¤ w(aS,K ) ’ w ¤ hK |w|1,∞ |K| v

v 0,K .

0,K 0,K

So it remains to consider I2,K . Obviously,

[vj ’ v] dx .

I2,K = w(aS,K )

„¦j,K

j:‚K aj

We will show that if „¦j belongs to a Donald diagram, then the sum van-

ishes. To do so, let us suppose that the triangle under consideration has

the vertices ai , aj , and ak . The set „¦j,K can be decomposed into two sub-

triangles by drawing a straight line between aS,K and aj . We will denote

the interior of these triangles by „¦j,K,i and „¦j,K,k ; i.e.,

„¦j,K,i := int conv{aj , aS,K , aij } , „¦j,K,k := int conv{aj , aS,K , akj } .

On each subtriangle, the integral of v can be calculated exactly by means

of the trapezoidal rule. Since |„¦j,K,i | = |„¦j,K,k | = |K|/6 in the case of the

Donald diagram (cf. also (6.4)), we have

|K| vj + vi vj + vi + vk

v dx = vj + +

18 2 3

„¦j,K,i

|K| 11 5 1

= vj + vi + vk ,

18 6 6 3

|K| 11 5 1

v dx = vj + vk + vi .

18 6 6 3

„¦j,K,k

Consequently,

|K| 11 7 7

v dx = vj + vi + vk ,

18 3 6 6

„¦j,K

7.2. Semidiscretization by Vertical Method of Lines 309

and thus

|K|

v dx = vj .

3

„¦j,K

j:‚K aj j:‚K aj

On the other hand, since 3|„¦j,K | = |K| (cf. (6.4)), we have

|K|

vj dx = vj ,

3

„¦j,K

j:‚K aj j:‚K aj

and so I2,K = 0. In summary, we have obtained the following estimate:

|I1,K + I2,K + I3,K | ¤ hK |w|1,∞ |K| v +v .

0,h,K 0,K

So it follows that

| w, v ’ w, v |¤ |I1,K + I2,K + I3,K |

0,h 0

K∈Th

¤ |K| v

h|w|1,∞ +v .

0,h,K 0,K

K∈Th

By the Cauchy“Schwarz inequality,

1/2 1/2

|K| v ¤ |K| |„¦| v

2

v =

0,h,K 0,h

0,h,K

K∈Th K∈Th K∈Th

and, similarly,

|K| v ¤ |„¦| v 0.

0,K

K∈Th

So we ¬nally arrive at

| w, v ’ w, v | ¤ Ch|w|1,∞ v +v .

0,h 0

0,h 0

· ·

Since the norms and are equivalent on Vh (see Remark 6.16),

0,h 0

we get

| w, v ’ w, v | ¤ Ch|w|1,∞ v 0,h .

0,h 0

2

Now we are prepared to apply the discrete stability estimate (7.50) to

equation (7.64):

’±t

¤

θ(t) θ(0) 0,h e

0,h

t