V

Lemma 7.4 Let a be a V -elliptic, continuous bilinear form, u0 ∈ H, and

f ∈ C ([0, T ], H), and suppose the considered boundary conditions are ho-

mogeneous. Then, for the solution u(t) of (7.32) the following estimate

holds:

t

’±t

e’±(t’s) ds

¤ u0 for all t ∈ (0, T ) .

u(t) e + f (s)

0 0 0

0

Proof: The following equations are valid almost everywhere in (0, T ).

Setting v = u(t), (7.32) reads as

u (t), u(t) + a(u(t), u(t)) = f (t), u(t) .

0 0

Using the relation

1d 1d d

2

u (t), u(t) = u(t), u(t) = u(t) = u(t) u(t)

0 0

0

0 0

2 dt 2 dt dt

and the V -ellipticity, it follows that

d

¤ f (t), u(t)

2

u(t) u(t) + ± u(t) .

0 0 V 0

dt

Now the simple inequality

¤ u(t)

u(t) 0 V

and the Cauchy“Schwarz inequality

¤ f (t)

f (t), u(t) u(t)

0 0

0

7.1. Problem Setting and Solution Concept 291

yield, after division by u(t) 0 , the estimate

d

¤ f (t)

u(t) + ± u(t) 0.

0 0

dt

Multiplying this relation by e±t , the relation

d ±t d

(e u(t) 0 ) = e±t + ±e±t u(t)

u(t) 0 0

dt dt

leads to

d ±t

(e u(t) 0 ) ¤ e±t f (t) .

0

dt

The integration over (0, t) results in

t

’ u(0) ¤

±t

e±s f (s)

e u(t) ds

0 0 0

0

for all t ∈ (0, T ). Multiplying this by e’±t and taking into consideration

the initial condition, we get the asserted relation

t

’±t ’±(t’s)

¤ u0

u(t) 0e + f (s) 0e ds .

0

0

2

As a consequence of this lemma, the uniqueness of the solution of (7.32)

is obtained.

Corollary 7.5 Let a be a V -elliptic, continuous bilinear form. Then there

exists at most one solution of (7.32).

Proof: Suppose there are two di¬erent solutions u1 (t), u2 (t) ∈ V. Then

the di¬erence v(t) := u1 (t) ’ u2 (t) solves a homogeneous problem of the

type (7.32) (i.e., with f = 0, u0 = 0). Lemma 7.4 immediately implies

v(t) 0 = 0 in [0, T ); that is, u1 (t) = u2 (t) for all t ∈ [0, T ). 2

There is a close relation between Lemma 7.4 and solution representations

such as (7.18) (with the sum being in¬nite). The eigenvalue problem (7.12)

is de¬ned as follows in its variational form (see also the end of Section 2.2):

De¬nition 7.6 A number » ∈ R is called an eigenvalue for the eigenvector

w ∈ V, w = 0, if

for all v ∈ V .

a(w, v) = » w, v 0

Assume that additionally to our assumptions the bilinear form is symmetric

and the embedding of V into H is compact (see [26]), which is the case here.

Then there are enough eigenvectors in the sense that a sequence (wi , »i ),

292 7. Discretization of Parabolic Problems

0 < »1 ¤ »2 ¤ . . . , exists such that the wi are orthonormal with respect

to ·, · 0 and every v ∈ V has a unique representation (in H) as

∞

v= ci wi . (7.33)

i=1

As in (7.25) the Fourier coe¬cients ci are given by

ci = v, wi . (7.34)

0

In fact, (7.33) gives a rigorous framework to the speci¬c considerations

in (7.16) and subsequent formulas. From (7.33) and (7.34) we conclude

Parseval™s identity

∞

| v, wi 0 |2 .

2

v = (7.35)

0

i=1

’1/2

Furthermore, the sequence vi := »i wi is orthogonal with respect to

a(·, ·), and a representation corresponding to (7.33), (7.34) holds such that

∞ ∞ ∞

»’1 |a(v, wi )|2

|a(v, vi )| = »i | v, wi 0 |2 .

2

a(v, v) = = (7.36)

i

i=1 i=1 i=1

From (7.35) and (7.36) we see that the ellipticity constant can be inter-

preted as the smallest eigenvalue ». In fact, the solution representation

(7.18) (with the sum being in¬nite in H) also holds true under the as-

sumptions mentioned and also leads to the estimate of Lemma 7.4. But

note that the proof there does not require symmetry of the bilinear form.

Exercises

7.1 Consider the initial-boundary value problem

ut ’ uxx = in (0, ∞) — (0, ∞) ,

0

h(t) , t ∈ (0, ∞) ,

u(0, t) =

x ∈ (0, ∞) ,

u(x, 0) = 0,

where h : (0, ∞) ’ R is a di¬erentiable function, the derivative of which

has at most exponential growth.

(a) Show that the function

∞

x2

2 ’s2 /2

h t’ 2

u(x, t) = e ds

√

π 2s

x/ 2t

is a solution.

(b) Is ut bounded in the domain of de¬nition? If not, give conditions on

h that guarantee the boundedness of ut .

7.2. Semidiscretization by the Vertical Method of Lines 293

7.2 Consider the initial-boundary value problem in one space dimension

ut ’ uxx = in (0, π) — (0, ∞) ,

0

t ∈ (0, ∞) ,

u(0, t) = u(π, t) = 0,

x ∈ (0, π) .

u(x, 0) = u0 (x) ,

(a) Solve it by means of the method of separation.

(b) Give a representation for ut (t) 0 .

(c) Consider the particular initial condition u0 (x) = π’x and investigate,

using the result from subproblem (b), the asymptotic behaviour of

ut (t) 0 near t = 0.

7.3 Let the domain „¦ ‚ Rd be bounded by a su¬ciently smooth boundary

and set V := H0 („¦), H := L2 („¦). Furthermore, a : V — V ’ R is a

1

continuous, V -elliptic, symmetric bilinear form and u0 ∈ H. Prove by using

the so-called energy method (cf. the proof of Lemma 7.4) the following a

priori estimate for the solution u of the initial boundary value problem

+ a(u(t), v) = 0 for all v ∈ V, t ∈ (0, T ) ,

ut (t), v 0

u(0) = u0 .

t t

ds ¤ M

2 2 2

(a) ±t u(t) +2 s ut (s) u(s) 1 ds .

1 0

0 0

M1

¤

(b) ut (t) u0 0 .

0

2± t

Here M and ± denote the corresponding constants in the continuity and

ellipticity conditions, respectively.

7.2 Semidiscretization

by the Vertical Method of Lines

For solving parabolic equations numerically, a wide variety of methods

exists. The most important classes of these methods are the following:

• Full discretizations:

“ Application of ¬nite di¬erence methods to the classical initial

boundary value problem (as of the form (7.1)).

“ Application of so-called space-time ¬nite element methods to a

variational formulation that includes the time variable, too.

• Semidiscretizations:

“ The vertical method of lines: Here the discretization starts with

respect to the spatial variable(s) (e.g., by means of the ¬nite dif-

294 7. Discretization of Parabolic Problems

ference method, the ¬nite element method, or the ¬nite volume

method).

“ The horizontal method of lines (Rothe™s method): Here the

discretization starts with respect to the time variable.

As the name indicates, a semidiscretization has to be followed by a further

discretization step to obtain a full discretization, which may be one of

the above-mentioned or not. The idea behind semidiscretization methods

is to have intermediate problems that are of a well-known structure. In

the case of the vertical method of lines, a system of ordinary di¬erential

equations arises for the solution of which appropriate solvers are often

available. Rothe™s method generates a sequence of elliptic boundary value

problems for which e¬cient solution methods are known, too.

The attributes “vertical” and “horizontal” of the semidiscretizations are

motivated by the graphical representation of the domain of de¬nition of the

unknown function u = u(x, t) in one space dimension (i.e., d = 1), namely,

assigning the abscissa (horizontal axis) of the coordinate system to the

variable x and the ordinate (vertical axis) to the variable t, so that the

spatial discretization yields problems that are setted along vertical lines.

In what follows, the vertical method of lines will be considered in more

detail.

In the following, and similarly in the following sections, we will de-

velop the analogous (semi)discretization approaches for the ¬nite di¬erence

method, the ¬nite element method, and the ¬nite volume method. This

will allow us to analyze these methods in a uniform way, as far as only the

emerging (matrix) structure of the discrete problems will play a role. On

the other hand, di¬erent techniques of analysis as in Chapters 1, 3 and 6

will further elucidate advantages and disadvantages of the methods. Read-

ers who are interested only in a speci¬c approach may skip some of the

following subsections.

The Vertical Method of Lines for the Finite Di¬erence Method

As a ¬rst example we start with the heat equation (7.8) with Dirichlet

boundary conditions on a rectangle „¦ = (0, a) — (0, b). As in Sec-

tion 1.2 we apply the ¬ve-point stencil discretizations at the grid points

x ∈ „¦h (according to (1.5)) for every ¬xed t ∈ [0, T ]. This leads to the

approximation

1

’ ui,j’1 (t) ’ ui’1,j (t) + 4uij (t) ’ ui+1,j (t) ’ ui,j+1 (t)

‚t uij (t) +

h2

i = 1, . . . , l ’ 1, j = 1, . . . , m ’ 1, t ∈ (0, T ) ,

= fij (t) ,

(7.37)

i ∈ {0, l}, j = 0, . . . , m ,

uij (t) = gij (t),

(7.38)

j ∈ {0, m}, i = 0, . . . , l .

7.2. Semidiscretization by Vertical Method of Lines 295

Here we use

fij (t) := f (ih, jh, t) ,

(7.39)

gij (t) := g(ih, jh, t) ,

and the index 3 in the boundary condition is omitted. Additionally, the

initial condition (at the grid points) will be prescribed, that is,

uij (0) = u0 (ih, jh), (ih, jh) ∈ „¦h . (7.40)

The system (7.37), (7.38), (7.40) is a system of (linear) ordinary di¬erential

equations (in the “index” (i, j)). If, as in Section 1.2, we ¬x an ordering of

the grid points, the system takes the form

d

t ∈ (0, T ) ,

uh (t) + Ah uh (t) = q h (t) ,

(7.41)

dt

uh (0) = u0 ,

with Ah , q h as in (1.10), (1.11) (but now q h = q h (t) because of the t-

dependence of f and g).

The unknown is the function

uh : [0, T ] ’ RM1 , (7.42)

which means that the Dirichlet boundary conditions are eliminated as in

Section 1.2.

For a simpli¬cation of the notation we use in the following M instead

of M1 , which also includes the eliminated degrees of freedom. Only in

Sections 7.5 and 7.6 will we return to the original notation.

More generally, if we consider a ¬nite di¬erence approximation, which

applied to the stationary problem (7.6) will lead to the system of equations

Ah uh = q h ,

with uh ∈ RM , then the same method applied to (7.1) for every ¬xed

t ∈ (0, T ) leads to (7.41). In particular, the system (7.41) has a unique

solution due to the theorem of Picard“Lindel¨f (cf. [26]).

o

The Vertical Method of Lines for the Finite Element Method

We proceed as for the ¬nite di¬erence method by now applying the ¬nite

element method to (7.1) in its weak formulation (7.32) for every ¬xed t ∈

(0, T ), using the abbrevation

b(t, v) := f (t), v + g1 (·, t)v dσ + g2 (·, t)v dσ . (7.43)

0

“1 “2

So let Vh ‚ V denote a ¬nite-dimensional subspace with dim Vh = M =

M (h) and let u0h ∈ Vh be some approximation to u0 . Then the semidiscrete

problem reads as follows:

296 7. Discretization of Parabolic Problems

Find uh ∈ L2 ((0, T ), Vh ) with uh ∈ L2 ((0, T ), H) , uh (0) = u0h and

d

+a(uh (t), vh ) = b(t, vh ) for all vh ∈ Vh , t ∈ (0, T ) . (7.44)

uh (t), vh

dt 0

To gain a more speci¬c form of (7.44), again we represent the unknown

uh (t) by its degrees of freedom:

M

Let {•i }M be a basis of Vh , uh (t) = i=1 ξi (t) •i and u0h =

i=1

M

i=1 ξ0i •i . Then for any t ∈ (0, T ), the discrete variational equality (7.44)

is equivalent to

M M

dξj (t)

a(•j , •i ) ξj (t) = b(t, •i ) for all i ∈ {1, . . . , M } .

•j , •i +

0 dt

j=1 j=1

ˆ

Denoting by Ah := (a(•j , •i ))ij the sti¬ness matrix , by Bh := •j , •i 0 ij

the mass matrix, and by

β h (t) := (b(t, •i ))i ,

respectively ξ 0h := (ξ0i )i , the vectors of the right-hand side and of the

initial value, we obtain for ξ h (t) := (ξi (t))i the following system of linear

ordinary di¬erential equations with constant coe¬cients:

d

t ∈ (0, T ) ,

ˆ

Bh ξ (t) + Ah ξ h (t) = β h (t) ,

dt h (7.45)

ξ h (0) = ξ 0h .

Since the matrix Bh is symmetric and positive de¬nite, it can be factored

T

(e.g., by means of Cholesky™s decomposition) as Bh = Eh Eh . Introducing

the new variable uh := Eh ξ h (to maintain the possible de¬niteness of Ah ),

the above system (7.45) can be written as follows:

d

t ∈ (0, T ) ,

uh (t) + Ah uh (t) = q h (t) ,

(7.46)

dt

uh (0) = uh0 ,

’T ˆ ’1 ’T

where Ah := Eh Ah Eh is an RM -elliptic matrix and q h := Eh β h ,

uh0 := Eh ξ 0h .

Thus again the discretization leads us to a system (7.41).

Remark 7.7 By means of the same arguments as in the proof of

Lemma 7.4, an estimate of uh (t) 0 can be derived.

The Vertical Method of Lines for the Finite Volume Method

Based on the ¬nite volume methods introduced in Chapter 6, in this sub-

section a ¬nite volume semidiscretization is given for the problem (7.1)

in its weak formulation (7.32) for every ¬xed t ∈ (0, T ) in the special

case “3 = ‚„¦ and of homogeneous Dirichlet boundary conditions. As in

Chapter 6, the only essential di¬erence to problem (7.1) is that here the

7.2. Semidiscretization by Vertical Method of Lines 297

di¬erential expression L is in divergence form, i.e.,

Lu := ’∇ · (K ∇u ’ c u) + r u = f ,

where the data K, c, r, and f are as in (7.2).

Correspondingly, the bilinear form a in the weak formulation (7.32) is to

be replaced by

[(K ∇u ’ c u) · ∇v + ruv] dx .

a(u, v) = (7.47)

„¦

In order to obtain a ¬nite volume semidiscretization of the problem (7.1)

in divergence form, and of (7.32) with the modi¬cation (7.47), we recall

the way that it was done in the elliptic situation. Namely, comparing the

weak formulation of the elliptic problem (see De¬nition 2.2) with the ¬nite

volume method in the discrete variational formulation (6.11), we see that

the bilinear form a and the linear form b(·) := f, · 0 have been replaced by

certain discrete forms ah and f, · 0,h , respectively. This formal procedure

can be applied to the weak formulation (7.32) of the parabolic problem,

too.

So let Vh ‚ V denote a ¬nite-dimensional subspace as introduced in Sec-

tion 6.2 with dim Vh = M = M (h) and let u0h ∈ Vh be some approximation

to u0 . Then, the semidiscrete ¬nite volume method reads as follows:

Find uh ∈ L2 ((0, T ), Vh ) with uh ∈ L2 ((0, T ), H) , uh (0) = u0h and

d

for all vh ∈ Vh , t ∈ (0, T ) ,

uh (t), vh +ah (uh (t), vh ) = f (t), vh 0,h

dt 0,h

(7.48)