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