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on G, where a function u : G ’ R is called essentially bounded if the
quantity

sup |u(x)|
u := inf
∞,G
G0 ‚G: |G0 |d =0 x∈G\G0


is ¬nite. For continuous functions, this norm coincides with the usual
maximum norm:

= max |u(x)| , u ∈ C(G) .
u ∞,G
x∈G

For 1 ¤ q ¤ p ¤ ∞, we have Lp (G) ‚ Lq (G), and the embedding is
continuous.
The space Wp (G), l ∈ N, p ∈ [1, ∞], consists of all l-times weakly di¬er-
l

entiable functions from Lp (G) with derivatives in Lp (G). In the special case
p = 2, we also write H l (G) := W2 (G). In analogy to the case of continuous
l

functions, the convention H 0 (G) := L2 (G) is used. The norm in Wp (G) is
l
406 A. Appendices

de¬ned as follows:
1/p

|‚ u| dx p ∈ [1, ∞) ,
± p
u := ,
l,p,G
G
|±|¤l

max |‚ ± u|∞,G .
u :=
l,∞,G
|±|¤l

In H l (G) a scalar product can be de¬ned by

u, v ∈ H l (G) .
‚ ± u‚ ± v dx ,
u, v :=
l,G
G
|±|¤l

· l ∈ N:
The norm induced by this scalar product is denoted by l,G ,


u := u, u .
l,G l,G

For l ∈ N, the symbol | · |l,G stands for the corresponding H l (G)-seminorm:


|u|l,G := |‚ ± u|2 dx .
G
|±|=l


1
The space H0 (G) is de¬ned as the closure (or completion) of C0 (G) in the
norm · 1 of H 1 (G).

Convention: Usually, in the case G = „¦ the speci¬cation of the domain
in the above norms and scalar products is omitted.
In the study of partial di¬erential equations, it is often desirable to speak
of boundary values of functions de¬ned on the domain G. In this respect,
the Lebesgue spaces of functions that are square integrable at the bound-
ary of G are important. To introduce these spaces, some preparations are
necessary.
x
In what follows, a point x ∈ Rd is written in the form x = xd with
x = (x1 , . . . , xd’1 )T ∈ Rd’1 .
A domain G ‚ Rd is said to be located at one side of ‚G if for any x ∈ ‚G
there exist an open neighbourhood Ux ‚ Rd and an orthogonal mapping
Qx in Rd such that the point x is mapped to a point x = (ˆ1 , . . . , xd )T ,
ˆ x ˆ
and so Ux is mapped onto a neighbourhood Ux ‚ R of x, where in the
d
ˆ
ˆ
neighbourhood Ux the following properties hold:
ˆ

(1) The image of Ux © ‚G is the graph of some function Ψx : Yx ‚
Rd’1 ’ R; that is, xd = Ψx (ˆ1 , . . . , xd’1 ) = Ψx (ˆ ) for x ∈ Yx .
ˆ x ˆ x ˆ
(2) The image of Ux © G is “above this graph” (i.e., the points in Ux © G
correspond to xd > 0).
ˆ
A.5. Function Spaces 407

(3) The image of Ux © (Rd \ G) is “below this graph” (i.e., the points in
Ux © (Rd \ G) correspond to xd < 0).
ˆ

A domain G that is located at one side of ‚G is called a C l domain, l ∈ N,
respectively a Lipschitz(ian) domain, if all Ψx are l-times continuously
di¬erentiable, respectively Lipschitz continuous, in Yx .
Bounded Lipschitz domains are also called strongly Lipschitz.
For bounded domains located at one side of ‚G, it is well known (cf.,
e.g. [37]) that from the whole set of neighbourhoods {Ux }x∈‚G there can be
selected a family {Ui }n of ¬nitely many neighbourhoods covering ‚G, i.e.,
i=1
n
n ∈ N and ‚G ‚ i=1 Ui . Furthermore, for any such family there exists a

system of functions {•i }n with the properties •i ∈ C0 (Ui ), •i (x) ∈ [0, 1]
i=1
n
for all x ∈ Ui and i=1 •i (x) = 1 for all x ∈ ‚G. Such a system is called
a partition of unity.
If the domain G is at least Lipschitzian, then Lebesgue™s integral over
the boundary of G is de¬ned by means of those partitions of unity. In cor-
respondence to the de¬nition of a Lipschitz domain, Qi , Ψi , and Yi denote
the orthogonal mapping on Ui , the function describing the corresponding
local boundary, and the preimage of Qi (Ui © ‚G) with respect to Ψi .
A function v : ‚G ’ R is called Lebesgue integrable over ‚G if the
composite functions x ’ v QT Ψ xx ) ˆ belong to L1 (Yi ). The integral
ˆ
i (ˆ
i
is de¬ned as follows:
n
v(s) ds := v(s)•i (s) ds
‚G ‚G
i=1
n
v QT Ψ xx )
ˆ •i QT Ψ xx )
ˆ
:= i i
i (ˆ i (ˆ
Yi
i=1

— |det(‚j Ψi (ˆ )‚k Ψi (ˆ ))d’1 | dˆ .
x x j,k=1 x

A function v : ‚G ’ R belongs to L2 (‚G) i¬ both v and v 2 are Lebesgue
integrable over ‚G.
In the investigation of time-dependent partial di¬erential equations, lin-
ear spaces whose elements are functions of the time variable t ∈ [0, T ],
T > 0, with values in a normed space X are of interest.
A function v : [0, T ] ’ X is called continuous on [0, T ] if for all t ∈ [0, T ]
the convergence v(t + k) ’ v(t) X ’ 0 as k ’ 0 holds.
The space C([0, T ], X) = C 0 ([0, T ], X) consists of all continuous
functions v : [0, T ] ’ X such that

<∞.
sup v(t) X
t∈(0,T )

The space C l ([0, T ], X), l ∈ N, consists of all continuous functions v :
[0, T ] ’ X that have continuous derivatives up to order l on [0, T ] with the
408 A. Appendices

norm
l
v (i) (t)
sup .
X
i=0 t∈(0,T )

The space Lp ((0, T ), X) with 1 ¤ p ¤ ∞ consists of all functions on (0, T )—
„¦ for which
v(t, ·) ∈ X for any t ∈ (0, T ) , F ∈ Lp (0, T ) with F (t) := v(t, ·) .
X

Furthermore,
v := F .
Lp ((0,T ),X) Lp (0,T )
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Index




adjoint, 247 modi¬ed, 237
arti¬cial di¬usion method, 373
adsorption, 12
advancing front method, 179, 180 assembling, 62
element-based, 66, 77
algorithm
node-based, 66
Arnoldi, 235
asymptotically optimal method, 199

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