<< . .

. 41
( : 59)



. . >>

2
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)

<< . .

. 41
( : 59)



. . >>