ńņš. 16 |

simple example shows:

01

For A = , we have that A = 0 but (A) = 0.

00

However, for any matrix norm Ā· the following relation is valid:

(A) ā¤ A . (A3.8)

Very often, matrices and vectors simultaneously appear as a product

Ax. In order to be able to handle such situations, there should be a certain

correlation between matrix and vector norms.

A matrix norm Ā· is called mutually consistent or compatible with the

vector norm | Ā· | if the inequality

|Ax| ā¤ A |x| (A3.9)

is valid for all x ā Rn and all A ā Rn,n .

A.3. Linear Algebra 397

Examples of mutually consistent norms are

with |x|ā ,

A or A ā

G

with |x|1 ,

A or A

G 1

with |x|2 .

A or A

G F

In many cases, the bound for |Ax| given by (A3.9) is not sharp enough;

i.e., for x = 0 we just have that

|Ax| < A |x| .

Therefore, the question arises of how to ļ¬nd, for a given vector norm, a

compatible matrix norm such that in (A3.9) the equality holds for at least

one element x = 0.

Given a vector norm |x|, the number

|Ax|

|Ax|

A := sup = sup

xāRn \{0} |x| xāRn : |x|=1

is called the induced or subordinate matrix norm.

The induced norm is a compatible matrix norm with the given vector

norm. It is the smallest norm among all matrix norms that are compatible

with the given vector norm |x|.

To illustrate the deļ¬nition of the induced matrix norm, the matrix norm

induced by the Euclidean vector norm is derived:

:= max |Ax|2 = max xT (AT A)x = Ī»max (AT A) = (AT A) .

A 2

|x|2 =1 |x|2 =1

(A3.10)

The matrix norm A 2 induced by the Euclidean vector norm is also called

the spectral norm. This term becomes understandable in the special case of

a symmetric matrix A. If Ī»1 , . . . , Ī»n denote the real eigenvalues of A, then

the matrix AT A = A2 has the eigenvalues Ī»2 satisfying

i

= |Ī»max (A)| .

A 2

For symmetric matrices, the spectral norm coincides with the spectral ra-

dius. Because of (A3.8), it is the smallest possible matrix norm in that

case.

As a further example, the maximum row sum A ā is the matrix norm

induced by the maximum norm |x|ā .

The number

Aā’1

Īŗ(A) := A

is called the condition number of the matrix A with respect to the matrix

norm under consideration.

398 A. Appendices

The following relation holds:

1 ā¤ I = AAā’1 ā¤ A Aā’1 .

For | Ā· | = | Ā· |p , the condition number is also denoted by Īŗp (A). If all

eigenvalues of A are real, the number

Īŗ(A) := Ī»max (A)/Ī»min (A)

is called the spectral condition number. Hence, for a symmetric matrix A

the equality Īŗ(A) = Īŗ2 (A) is valid.

Occasionally, it is necessary to estimate small perturbations of nonsin-

gular matrices. For this purpose, the following result is useful (perturbation

lemma or Neumannā™s lemma). Let A ā Rn,n satisfy A < 1 with respect

to an arbitrary, but ļ¬xed, matrix norm. Then the inverse of I ā’ A exists

and can be represented as a convergent power series of the form

ā

ā’1

(I ā’ A) Aj ,

=

j=0

with

1

(I ā’ A)ā’1 ā¤ . (A3.11)

1ā’ A

Special Matrices

The matrix A ā Rn,n is called an upper, respectively lower, triangular

matrix if its entries satisfy aij = 0 for i > j, respectively aij = 0 for i < j.

A matrix H ā Rn,n is called an (upper) Hessenberg matrix if it has the

following structure:

ļ£« ļ£¶

h11

ā—

ļ£¬ ļ£·

ļ£¬ h21 . . . ļ£·

ļ£¬ ļ£·

ļ£¬ ļ£·

.. ..

H := ļ£¬ ļ£·

. .

ļ£¬ ļ£·

ļ£¬ ļ£·

.. ..

ļ£¬ ļ£·

. .

ļ£ ļ£ø

0 hnnā’1 hnn

(that is, hij = 0 for i > j + 1).

The matrix A ā Rn,n satisļ¬es the strict row sum criterion (or is strictly

row diagonally dominant) if it satisļ¬es

n

|aij | < |aii | for all i = 1, . . . , n .

j=1

j=i

It satisļ¬es the strict column sum criterion if the following relation holds:

n

|aij | < |ajj | for all j = 1, . . . , n .

i=1

i=j

A.4. Some Deļ¬nitions and Arguments of Linear Functional Analysis 399

The matrix A ā Rn,n satisļ¬es the weak row sum criterion (or is weakly row

diagonally dominant) if

n

|aij | ā¤ |aii | holds for all i = 1, . . . , n

j=1

j=i

and the strict inequality ā<ā is valid for at least one number

i ā {1, . . . , n} .

The weak column sum criterion is deļ¬ned similarly.

The matrix A ā Rn,n is called reducible if there exist subsets N1 , N2 ā‚

{1, . . . , n} with N1 ā© N2 = ā…, N1 = ā… = N2 , and N1 āŖ N2 = {1, . . . , n} such

that the following property is satisļ¬ed:

For all i ā N1 , j ā N2 : aij = 0 .

A matrix that is not reducible is called irreducible.

A matrix A ā Rn,n is called an L0 -matrix if for i, j ā {1, . . . , n} the

inequalities

aii ā„ 0 and aij ā¤ 0 (i = j)

are valid. An L0 -matrix is called an L-matrix if all diagonal entries are

positive.

A matrix A ā Rn,n is called monotone (or of monotone type) if the

relation Ax ā¤ Ay for two (otherwise arbitrary) elements x, y ā Rn implies

x ā¤ y. Here the relation sign is to be understood componentwise.

A matrix of monotone type is invertible.

A matrix A ā Rn,n is a matrix of monotone type if it is invertible and

all entries of the inverse are nonnegative.

An important subclass of matrices of monotone type is formed by the

so-called M-matrices.

A monotone matrix A with aij ā¤ 0 for i = j is called an M-matrix.

Let A ā Rn,n be a matrix with aij ā¤ 0 for i = j and aii ā„ 0 (i, j ā

{1, . . . , n}). In addition, let A satisfy one of the following conditions:

(i) A satisļ¬es the strict row sum criterion.

(ii) A satisļ¬es the weak row sum criterion and is irreducible.

Then A is an M-matrix.

A.4 Some Deļ¬nitions and Arguments of Linear

Functional Analysis

Working with vector spaces whose elements are (classical or generalized)

functions, it is desirable to have a measure for the ālengthā or āmagnitudeā

of a function, and, as a consequence, for the distance of two functions.

400 A. Appendices

Let V be a real vector space (in short, an R vector space) and let Ā·

be a real-valued mapping Ā· : V ā’ R.

The pair (V, Ā· ) is called a normed space (āV is endowed with the norm

Ā· ā) if the following properties hold:

u ā„ 0 for all u ā V , u = 0 ā” u = 0, (A4.1)

Ī±u = |Ī±| u for all Ī± ā R , u ā V , (A4.2)

u+v ā¤ u + v for all u, v ā V . (A4.3)

The property (A4.1) is called deļ¬niteness; (A4.3) is called the triangle

inequality. If a mapping Ā· : V ā’ R satisļ¬es only (A4.2) and (A4.3), it

is called a seminorm. Due to (A4.2), we still have 0 = 0, but there may

exist elements u = 0 with u = 0.

A particularly interesting example of a norm can be obtained if the space

V is equipped with a so-called scalar product. This is a mapping Ā·, Ā· :

V Ć— V ā’ R with the following properties:

(1) Ā·, Ā· is a bilinear form, that is,

for all u, v1 , v2 ā V ,

u, v1 + v2 = u, v1 + u, v2

(A4.4)

for all u, v ā V, Ī± ā R ,

u, Ī±v = Ī± u, v

and an analogous relation is valid for the ļ¬rst argument.

(2) Ā·, Ā· is symmetric, that is,

for all u, v ā V .

u, v = v, u (A4.5)

(3) Ā·, Ā· is positive, that is,

u, u ā„ 0 for all u ā V . (A4.6)

(4) Ā·, Ā· is deļ¬nite, that is,

u, u = 0 ā” u = 0 . (A4.7)

A positive and deļ¬nite bilinear form is called positive deļ¬nite.

A scalar product Ā·, Ā· deļ¬nes a norm on V in a natural way if we set

1/2

v := v, v . (A4.8)

In absence of the deļ¬niteness (A4.7), only a seminorm is induced.

A norm (or a seminorm) induced by a scalar product (respectively by a

symmetric and positive bilinear form) has some interesting properties. For

example, it satisļ¬es the Cauchyā“Schwarz inequality, that is,

| u, v | ā¤ u for all u, v ā V ,

v (A4.9)

and the parallelogram identity

+ uā’v + v 2 ) for all u, v ā V .

2 2 2

u+v = 2( u (A4.10)

A.4. Linear Functional Analysis 401

Typical examples of normed spaces are the spaces Rn equipped with one

of the p -norms (for some ļ¬xed p ā [1, ā]). In particular, the Euclidean

norm (A3.3) is induced by the Euclidean scalar product

(x, y) ā’ x Ā· y for all x, y ā Rn . (A4.11)

On the other hand, inļ¬nite-dimensional function spaces play an important

role (see Appendix A.5).

If a vector space V is equipped with a scalar product Ā·, Ā· , then, in

analogy to Rn , an element u ā V is said to be orthogonal to v ā V if

u, v = 0 . (A4.12)

Given a normed space (V, Ā· ), it is easy to deļ¬ne the concept of convergence

of a sequence (ui )i in V to u ā V :

ui ā’ u for i ā’ ā āā’ ui ā’ u ā’ 0 for i ā’ ā . (A4.13)

Often, it is necessary to consider function spaces endowed with diļ¬erent

norms. In such situations, diļ¬erent kinds of convergence may occur. How-

ever, if the corresponding norms are equivalent, then there is no change

in the type of convergence. Two norms Ā· 1 and Ā· 2 in V are called

equivalent if there exist constants C1 , C2 > 0 such that

ā¤u ā¤ C2 u for all u ā V .

C1 u (A4.14)

1 2 1

If there is only a one-sided inequality of the form

ā¤C u for all u ā V

u (A4.15)

2 1

with a constant C > 0, then the norm Ā· 1 is called stronger than the

norm Ā· 2 .

In a ļ¬nite-dimensional vector space, all norms are equivalent. Examples

can be found in Appendix A.3. In particular, it is important to observe that

the constants may depend on the dimension n of the ļ¬nite-dimensional

vector space. This observation also indicates that in the case of inļ¬nite-

dimensional vector spaces, the equivalence of two diļ¬erent norms cannot

be expected, in general.

As a consequence of (A4.14), two equivalent norms Ā· 1 , Ā· 2 in V yield

the same type of convergence:

ui ā’ u w.r.t. Ā· ā” ui ā’ u ā’0

1 1

ā” ui ā’ u ā’0 ā” ui ā’ u w.r.t.

Ā· 2.

2

(A4.16)

In this book, the ļ¬nite-dimensional vector space R is used in two as-

n

pects: For n = d, it is the basic space of independent variables, and for

n = M or n = m it represents the ļ¬nite-dimensional trial space. In the

ļ¬rst case, the equivalence of all norms can be used in all estimates without

any side eļ¬ects, whereas in the second case the aim is to obtain uniform

402 A. Appendices

estimates with respect to all M and m, and so the dependence of the

equivalence constants on M and m has to be followed thoroughly.

Now we consider two normed spaces (V, Ā· V ) and (W, Ā· W ). A mapping

f : V ā’ W is called continuous in v ā V if for all sequences (vi )i in V with

vi ā’ v for i ā’ ā we get

f (vi ) ā’ f (v) for i ā’ ā.

Note that the ļ¬rst convergence is measured in Ā· V and the second

one in Ā· W . Hence a change of the norm may have an inļ¬‚uence on the

continuity. As in classical analysis, we can say that

f is continuous in all v ā V āā’

(A4.17)

f ā’1 [G] is closed for each closed G ā‚ W .

Here, a subset G ā‚ W of a normed space W is called closed if for any

sequence (ui )i from G such that ui ā’ u for i ā’ ā the inclusion u ā

G follows. Because of (A4.17), the closedness of a set can be veriļ¬ed by

showing that it is a continuous preimage of a closed set.

The concept of continuity is a qualitative relation between the preimage

and the image. A quantitative relation is given by the stronger notion of

Lipschitz continuity:

A mapping f : V ā’ W is called Lipschitz continuous if there exists a

constant L > 0, the Lipschitz constant, such that

f (u) ā’ f (v) ā¤ L uā’v for all u, v ā V . (A4.18)

W V

slope: L

admissible region for f(y)

f

slope: -L

x

Figure A.1. Lipschitz continuity (for V = W = R).

A Lipschitz continuous mapping with L < 1 is called contractive or a

contraction; cf. Figure A.1.

Most of the mappings used are linear; that is, they satisfy

f (u + v) = f (u) + f (v) ,

for all u, v ā V and Ī» ā R . (A4.19)

f (Ī»u) = Ī»f (u) ,

For a linear mapping, the Lipschitz continuity is equivalent to the

boundedness; that is, there exists a constant C > 0 such that

ā¤C u for all u ā V .

f (u) (A4.20)

W V

A.4. Linear Functional Analysis 403

In fact, for a linear mapping f, the continuity at one point is equivalent

to (A4.20). Linear, continuous mappings acting from V to W are also

called (linear, continuous) operators and are denoted by capital letters,

for example S, T, . . . .

In the case V = W = Rn , the linear, continuous operators in Rn are

the mappings x ā’ Ax deļ¬ned by matrices A ā Rn,n . Their boundedness,

for example with respect to Ā· V = Ā· W = Ā· ā , is an immediate

consequence of the compatibility property of the Ā· ā -norm. Moreover,

since all norms in Rn are equivalent, these mappings are bounded with

respect to any norms in Rn .

Similarly to (A4.20), a bilinear form f : V Ć— V ā’ R is continuous if it is

bounded, that is, if there exists a constant C > 0 such that

|f (u, v)| ā¤ C u for all u, v ā V .

v (A4.21)

V V

In particular, due to (A4.9) any scalar product is continuous with respect

to the induced norm of V ; that is,

ui ā’ u , vi ā’ v ā’ ui , vi ā’ u, v . (A4.22)

Now let (V, Ā· V ) be a normed space and W a subspace that is (addi-

tionally to Ā· V ) endowed with the norm Ā· W . The embedding from

(W, Ā· W ) to (V, Ā· V ) , i.e., the linear mapping that assigns any element

of W to itself but considered as an element of V, is continuous iļ¬ the norm

Ā· W is stronger than the norm Ā· V (cf. (A4.15)).

The collection of linear, continuous operators from (V, Ā· V ) to (W, Ā· W )

forms an R vector space with the following (argumentwise) operations:

for all u ā V ,

(T + S)(u) := T (u) + S(u)

for all u ā V ,

(Ī»T )(u) := Ī»T (u)

for all operators T, S and Ī» ā R. This space is denoted by

L[V, W ] . (A4.23)

In the special case W = R, the corresponding operators are called linear,

continuous functionals, and the notation

V := L[V, R] (A4.24)

is used. The R vector space L[V, W ] can be equipped with a norm, the

so-called operator norm, by

uāV , u ā¤1 for T ā L[V, W ] . (A4.25)

T := sup T (u) W V

Here T is the smallest constant such that (A4.20) holds. Speciļ¬cally, for

a functional f ā V , we have that

f = sup |f (u)| ā¤1 .

u V

404 A. Appendices

For example, in the case V = W = Rn and u V = u W , the norm of a

linear, bounded operator that is represented by a matrix A ā Rn,n coincides

with the corresponding induced matrix norm (cf. Appendix A.3).

Let (V, Ā· V ) be a normed space. A sequence (ui )i in V is called a Cauchy

sequence if for any Īµ > 0 there exists a number n0 ā N such that

ui ā’ uj ā¤Īµ for all i, j ā N with i, j ā„ n0 .

V

The space V is called complete or a Banach space if for any Cauchy sequence

(ui )i in V there exists an element u ā V such that ui ā’ u for i ā’ ā. If

the norm Ā· V of a Banach space V is induced by a scalar product, then

V is called a Hilbert space.

A subspace W of a Banach space is complete iļ¬ it is closed. A basic

problem in the variational treatment of boundary value problems consists

in the fact that the space of continuous functions (cf. the preliminary deļ¬-

nition (2.7)), which is required to be taken as a basis, is not complete with

respect to the norm ( Ā· l , l = 0 or l = 1). However, if in addition to the

normed space (W, Ā· ), a larger space V is given that is complete with

respect to the norm Ā· , then that space or the closure

W := W (A4.26)

(as the smallest Banach space containing W ) can be used. Such a com-

pletion can be introduced for any normed space in an abstract way. The

problem is that the ānatureā of the limiting elements remains vague.

If the relation (A4.26) is valid for some normed space W, then W is

called dense in W . In fact, given W, all āessentialā elements of W are

already captured. For example, if T is a linear, continuous operator T from

(W , Ā· ) to another normed space, then the identity

T (u) = 0 for all u ā W (A4.27)

is suļ¬cient for

for all u ā W .

T (u) = 0 (A4.28)

The space of linear, bounded operators is complete if the image space is

complete. In particular, the space V of linear, bounded functionals on the

normed space V is always complete.

A.5 Function Spaces

In this section G ā‚ Rd denotes a bounded domain.

The function space C(G) contains all (real-valued) functions deļ¬ned on

G that are continuous in G. By C l (G), l ā N, the set of l-times continuously

diļ¬erentiable functions on G is denoted. Usually, for the sake of consistency,

ā

the conventions C 0 (G) := C(G) and C ā (G) := l=0 C l (G) are used.

A.5. Function Spaces 405

Functions from C l (G), l ā N0 , and C ā (G) need not be bounded, as for

d = 1 the example f (x) := xā’1 , x ā (0, 1) shows.

To overcome this diļ¬culty, further spaces of continuous functions are

introduced. The space C(G) contains all bounded and uniformly contin-

uous functions on G, whereas C l (G), l ā N, consists of functions with

bounded and uniformly continuous derivatives up to order l on G. Here the

ā

conventions C 0 (G) := C(G) and C ā (G) := l=0 C l (G) are used, too.

The space C0 (G), respectively C0 (G), l ā N, denotes the set of all those

l

continuous, respectively l-times continuously diļ¬erentiable, functions, the

supports of which are contained in G. Often this set is called the set of

functions with compact support in G. Since G is bounded, this means that

0

the supports do not intersect boundary points of G. We also set C0 (G) :=

C0 (G) and C0 (G) := C0 (G) ā© C ā (G).

ā

The linear space Lp (G), p ā [1, ā), contains all Lebesgue measurable

functions deļ¬ned on G whose pth power of their absolute value is Lebesgue

integrable on G. The norm in Lp (G) is deļ¬ned as follows:

1/p

|u| dx p ā [1, ā) .

p

u := ,

0,p,G

G

In the case p = 2, the speciļ¬cation of p is frequently omitted; that is,

u 0,G = u 0,2,G. The L2 (G)-scalar product

u, v ā L2 (G) ,

u, v := uv dx ,

0,G

G

induces the L2 (G)-norm by setting u := u, u 0,G .

0,G

The space Lā (G) contains all measurable, essentially bounded functions

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

quantity

sup |u(x)|

u := inf

ā,G

G0 ā‚G: |G0 |d =0 xāG\G0

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

maximum norm:

= max |u(x)| , u ā C(G) .

u ā,G

xāG

For 1 ā¤ q ā¤ p ā¤ ā, we have Lp (G) ā‚ Lq (G), and the embedding is

continuous.

The space Wp (G), l ā N, p ā [1, ā], consists of all l-times weakly diļ¬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

deļ¬ned as follows:

1/p

|ā‚ u| dx p ā [1, ā) ,

Ī± p

u := ,

l,p,G

G

|Ī±|ā¤l

max |ā‚ Ī± u|ā,G .

u :=

l,ā,G

|Ī±|ā¤l

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

u, v ā H l (G) .

ā‚ Ī± uā‚ Ī± v dx ,

u, v :=

l,G

G

|Ī±|ā¤l

Ā· l ā N:

The norm induced by this scalar product is denoted by l,G ,

u := u, u .

l,G l,G

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

|u|l,G := |ā‚ Ī± u|2 dx .

G

|Ī±|=l

ā

1

The space H0 (G) is deļ¬ned as the closure (or completion) of C0 (G) in the

norm Ā· 1 of H 1 (G).

Convention: Usually, in the case G = ā„¦ the speciļ¬cation of the domain

in the above norms and scalar products is omitted.

In the study of partial diļ¬erential equations, it is often desirable to speak

of boundary values of functions deļ¬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 ā Rd is written in the form x = xd with

x = (x1 , . . . , xdā’1 )T ā Rdā’1 .

A domain G ā‚ Rd is said to be located at one side of ā‚G if for any x ā ā‚G

there exist an open neighbourhood Ux ā‚ Rd and an orthogonal mapping

Qx in Rd such that the point x is mapped to a point x = (Ė1 , . . . , xd )T ,

Ė x Ė

and so Ux is mapped onto a neighbourhood Ux ā‚ R of x, where in the

d

Ė

Ė

neighbourhood Ux the following properties hold:

Ė

(1) The image of Ux ā© ā‚G is the graph of some function ĪØx : Yx ā‚

Rdā’1 ā’ R; that is, xd = ĪØx (Ė1 , . . . , xdā’1 ) = ĪØx (Ė ) for x ā Yx .

Ė x Ė x Ė

(2) The image of Ux ā© G is āabove this graphā (i.e., the points in Ux ā© G

correspond to xd > 0).

Ė

A.5. Function Spaces 407

(3) The image of Ux ā© (Rd \ G) is ābelow this graphā (i.e., the points in

Ux ā© (Rd \ G) correspond to xd < 0).

Ė

A domain G that is located at one side of ā‚G is called a C l domain, l ā N,

respectively a Lipschitz(ian) domain, if all ĪØx are l-times continuously

diļ¬erentiable, respectively Lipschitz continuous, in Yx .

Bounded Lipschitz domains are also called strongly Lipschitz.

For bounded domains located at one side of ā‚G, it is well known (cf.,

e.g. [37]) that from the whole set of neighbourhoods {Ux }xāā‚G there can be

selected a family {Ui }n of ļ¬nitely many neighbourhoods covering ā‚G, i.e.,

i=1

n

n ā N and ā‚G ā‚ i=1 Ui . Furthermore, for any such family there exists a

ā

system of functions {Ļ•i }n with the properties Ļ•i ā C0 (Ui ), Ļ•i (x) ā [0, 1]

i=1

n

for all x ā Ui and i=1 Ļ•i (x) = 1 for all x ā ā‚G. Such a system is called

a partition of unity.

If the domain G is at least Lipschitzian, then Lebesgueā™s integral over

the boundary of G is deļ¬ned by means of those partitions of unity. In cor-

respondence to the deļ¬nition of a Lipschitz domain, Qi , ĪØi , and Yi denote

the orthogonal mapping on Ui , the function describing the corresponding

local boundary, and the preimage of Qi (Ui ā© ā‚G) with respect to ĪØi .

A function v : ā‚G ā’ R is called Lebesgue integrable over ā‚G if the

composite functions x ā’ v QT ĪØ xx ) Ė belong to L1 (Yi ). The integral

Ė

i (Ė

i

is deļ¬ned as follows:

n

v(s) ds := v(s)Ļ•i (s) ds

ā‚G ā‚G

i=1

n

v QT ĪØ xx )

Ė Ļ•i QT ĪØ xx )

Ė

:= i i

i (Ė i (Ė

Yi

i=1

Ć— |det(ā‚j ĪØi (Ė )ā‚k ĪØi (Ė ))dā’1 | dĖ .

x x j,k=1 x

A function v : ā‚G ā’ R belongs to L2 (ā‚G) iļ¬ both v and v 2 are Lebesgue

integrable over ā‚G.

In the investigation of time-dependent partial diļ¬erential equations, lin-

ear spaces whose elements are functions of the time variable t ā [0, T ],

T > 0, with values in a normed space X are of interest.

A function v : [0, T ] ā’ X is called continuous on [0, T ] if for all t ā [0, T ]

the convergence v(t + k) ā’ v(t) X ā’ 0 as k ā’ 0 holds.

The space C([0, T ], X) = C 0 ([0, T ], X) consists of all continuous

functions v : [0, T ] ā’ X such that

<ā.

sup v(t) X

tā(0,T )

The space C l ([0, T ], X), l ā N, consists of all continuous functions v :

[0, T ] ā’ 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 tā(0,T )

The space Lp ((0, T ), X) with 1 ā¤ p ā¤ ā consists of all functions on (0, T )Ć—

ā„¦ for which

v(t, Ā·) ā X for any t ā (0, T ) , F ā Lp (0, T ) with F (t) := v(t, Ā·) .

X

Furthermore,

v := F .

Lp ((0,T ),X) Lp (0,T )

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Index

adjoint, 247 modiļ¬ed, 237

artiļ¬cial diļ¬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

CG, 223

multigrid iteration, 243

nested iteration, 253 Banach space, 404

Newtonā™s method, 357 Banachā™s ļ¬xed-point theorem, 345

algorithmic error, 200 barycentric coordinates, 117

angle condition, 173 basis of eigenvalues

angle criterion, 184 orthogonal, 300

anisotropic, 8, 139 best approximation error, 70

ansatz space, 56, 67 BICGSTAB method, 238

nested, 240 bifurcation, 363

properties, 67 biharmonic equation, 111

approximation bilinear form, 400

superconvergent, 193 bounded, 403

approximation error estimate, 139, continuous, 93

144 deļ¬nite, 400

for quadrature rules, 160 positive, 400

one-dimensional, 137 positive deļ¬nite, 400

approximation property, 250 symmetric, 400

aquifer, 7 V -elliptic, 93

Armijoā™s rule, 357 Vh -elliptic, 156

Arnoldiā™s method, 235 block-Gaussā“Seidel method, 211

algorithm, 235 block-Jacobi method, 211

416 Index

Bochner integral, 289 conjugate gradient, see CG

boundary, 393 connectivity condition, 173

boundary condition, 15 conormal, 16

Dirichlet, 15 conormal derivative, 98

ļ¬‚ux, 15 conservative form, 14

homogeneous, 15 conservativity

inhomogeneous, 15 discrete global, 278

mixed, 15 consistency, 28

Neumann, 16 consistency error, 28, 156

boundary point, 393 constitutive relationship, 7

boundary value problem, 15 continuation method, 357, 363

adjoint, 145 continuity, 402

regular, 145 continuous problem, 21

weak solution, 107 approximation, 21

Brambleā“Hilbert lemma, 135 contraction, 402

bulk density, 12 contraction number, 199

control domain, 257

Cantorā™s function, 53 control volume, 257

capillary pressure, 10 convection

Cauchy sequence, 404 forced, 5, 12

Cauchyā“Schwarz inequality, 400 natural, 5

CG method, 221 convection-diļ¬usion equation, 12

algorithm, 223 convection-dominated, 268

error reduction, 224 convective part, 12

with preconditioning, 228 convergence, 27

CGNE method, 235 global, 343

CGNR method, 234 linear, 343

characteristics, 388 local, 343

Chebyshev polynomial, 225 quadratic, 343

Cholesky decomposition, 84 superlinear, 343

incomplete, 231 with order of convergence p, 343

modiļ¬ed with respect to a norm, 401

incomplete, 232 correction, 201

chord method, 354 Crank-Nicolson method, 313

circle criterion, 184 cut-oļ¬ strategy, 187

closure, 393 Cuthillā“McKee method, 89

coarse grid correction, 242, 243

coeļ¬cient, 16 Darcy velocity, 7

collocation method, 68 Darcyā™s law, 8

collocation point, 68 decomposition

column sum criterion regular, 232

strict, 398 deļ¬niteness, 400

comparison principle, 40, 328 degree of freedom, 62, 115, 120

completion, 404 Delaunay triangulation, 178, 263

complexity, 88 dense, 96, 288, 404

component, 5 density, 7

condition number, 209, 397 derivative

spectral, 398 generalized, 53

conjugate, 219 material, 388

Index 417

weak, 53, 289 duality argument, 145

diagonal ļ¬eld, 362

diagonal scaling, 230 edge swap, 181

diagonal swap, 181 eigenfunction, 285

diļ¬erence quotient, 23 eigenvalue, 285, 291, 394

backward, 23 eigenvector, 291, 394

forward, 23 element, 57

symmetric, 23 isoparametric, 122, 169

diļ¬erential equation element stiļ¬ness matrix, 78

convection-dominated, 12, 368 element-node table, 74

degenerate, 9 ellipticity

elliptic, 17 uniform, 100

homogeneous, 16 embedding, 403

H k (ā„¦) in C(ā„¦), 99

ĀÆ

hyperbolic, 17

inhomogeneous, 16 empty sphere criterion, 178

linear, 16 energy norm, 218

nonlinear, 16 energy norm estimates, 132

order, 16 energy scalar product, 217

parabolic, 17 equidistribution strategy, 187

quasilinear, 16 error, 201

semilinear, 16, 360 error equation, 68, 242

type of, 17 error estimate

diļ¬erential equation model a priori, 131, 185

instationary, 8 anisotropic, 144

linear, 8 error estimator

stationary, 8 a posteriori, 186

diļ¬usion, 5 asymptotically exact, 187

diļ¬usive mass ļ¬‚ux, 11 eļ¬cient, 186

diļ¬usive part, 12 reliable, 186

Dirichlet domain, 262 residual, 188

Dirichlet problem dual-weighted, 194

solvability, 104 robust, 187

discrete problem, 21 error level

discretization, 21 relative, 199

ļ¬ve-point stencil, 24 Euler method

upwind, 372 explicit, 313

discretization approach, 55 implicit, 313

discretization parameter, 21 extensive quantity, 7

divergence, 20 extrapolation factor, 215

divergence form, 14 extrapolation method, 215

domain, 19, 394

C l , 407 face, 123

C k -, 96 family of triangulations

C ā -, 96 quasi-uniform, 165

Lipschitz, 96, 407 regular, 138

strongly, 407 Fickā™s law, 11

domain of (absolute) stability, 317 ļ¬ll-in, 85

Donald diagram, 265 ļ¬nite diļ¬erence method, 17, 24

dual problem, 194 ļ¬nite element, 115, 116

418 Index

C 1 -, 115, 127 Lebesgue integrable, 407

aļ¬ne equivalent, 122 measurable, 393

Bognerā“Foxā“Schmit rectangle, 127 piecewise continuous, 48

C 0 -, 115 support, 394

cubic ansatz on simplex, 121 functional, 403

cubic Hermite ansatz on simplex, functional matrix, 348

126 functions

d-polynomial ansatz on cuboid, 123 equal almost everywhere, 393

equivalent, 122

Hermite, 126 Galerkin method, 56

Lagrange, 115, 126 stability, 69

linear, 57 unique solvability, 63

linear ansatz on simplex, 119 Galerkin product, 248

quadratic ansatz on simplex, 120 Galerkin/least squaresā“FEM, 377

simplicial, 117 Gaussā™s divergence theorem, 14, 47,

ļ¬nite element code 266

assembling, 176 Gaussā“Seidel method, 204

kernel, 176 convergence, 204, 205

post-processor, 176 symmetric, 211

ļ¬nite element discretization Gaussian elimination, 82

conforming, 114 generating function, 316

condition, 115 GMRES method, 235

nonconforming, 114 truncated, 238

ļ¬nite element method, 18 with restart, 238

characterization, 67 gradient, 20

convergence rate, 131 gradient method, 218

maximum principle, 175 error reduction, 219

mortar, 180 gradient recovery, 192

ļ¬nite volume method, 18 graph

cell-centred, 258 dual, 263

cell-vertex, 258 grid

node-centred, 258 chimera, 180

semidiscrete, 297 combined, 180

ļ¬ve-point stencil, 24 hierarchically structured, 180

ļ¬xed point, 342 logically structured, 177

ļ¬xed-point iteration, 200, 344 overset, 180

consistent, 200 structured, 176

convergence theorem, 201 in the strict sense, 176

ļ¬‚uid, 5 in the wider sense, 177

Fourier coeļ¬cient, 287 unstructured, 177

Fourier expansion, 287 grid adaptation, 187

Friedrichsā“Keller triangulation, 64 grid coarsening, 183

frontal method, 87 grid function, 24

full discretization, 293 grid point, 21, 22

full upwind method, 373 close to the boundary, 24, 327

function far from the boundary, 24, 327

almost everywhere vanishing, 393 neighbour, 23

continuous, 407

essentially bounded, 405 harmonic, 31

Index 419

heat equation, 9 L0 -matrix, 399

Hermite element, 126 L-matrix, 399

Hessenberg matrix, 398 Lagrange element, 115, 126

Hilbert space, 404 Lagrangeā“Galerkin method, 387

homogenization, 6 Lagrangian coordinate, 387

hydraulic conductivity, 8 Lanczos biorthogonalization, 238

Langmuir model, 12

IC factorization, 231 Laplace equation, 9

ill-posedness, 16 Laplace operator, 20

ILU factorization, 231 lemma

existence, 232 Brambleā“Hilbert, 135

ILU iteration, 231 CĀ“aā™s, 70

e

inequality ļ¬rst of Strang, 155

of Kantorovich, 218 lexicographic, 25

Friedrichsā™, 105 linear convergence, 199

inverse, 376 Lipschitz constant, 402

of PoincarĀ“, 71

e Lipschitz continuity, 402

inļ¬‚ow boundary, 108 load vector, 62

inhomogeneity, 15 LU factorization, 82

initial condition, 15 incomplete, 231

initial-boundary value problem, 15

inner product M-matrix, 41, 399

on H 1 (ā„¦), 54 macroscale, 6

integral form, 14 mapping

integration by parts, 97 bounded, 402

interior, 394 continuous, 402

interpolation contractive, 402

local, 58 linear, 402

interpolation error estimate, 138, 144 Lipschitz continuous, 402

one-dimensional , 136 mass action law, 11

interpolation operator, 132 mass average mixture velocity, 7

interpolation problem mass lumping, 314, 365

local, 120 mass matrix, 163, 296, 298

isotropic, 8 mass source density, 7

iteration matrix

inner, 355 band, 84

outer, 355 bandwidth, 84

iteration matrix, 200 consistently ordered, 213

iterative method, 342 Hessenberg, 398

hull, 84

Jacobi matrix, 348 inverse monotone, 41

Jacobiā™s method, 203 irreducible, 399

convergence, 204, 205 L0 -, 399

jump, 189 L-, 399

jump condition, 14 LU factorizable, 82

M-, 399

Krylov (sub)space, 222 monotone, 399

Krylov subspace of monotone type, 399

method, 223, 233 pattern, 231

420 Index

positive deļ¬nite, 394 ļ¬nite diļ¬erence, 24

proļ¬le, 84 full upwind, 373

reducible, 399 Galerkin, 56

row bandwidth, 84 Gaussā“Seidel, 204

row diagonally dominant GMRES, 235

strictly, 398 iterative, 342

weakly, 399 Jacobiā™s, 203

sparse, 25, 82, 198 Krylov subspace, 223, 233

symmetric, 394 Lagrangeā“Galerkin, 387

triangular linear stationary, 200

lower, 398 mehrstellen, 30

upper, 398 moving front, 179

matrix norm multiblock, 180

compatible, 396 multigrid, 243

induced, 397 Newtonā™s, 349

mutually consistent, 396 of bisection, 182

submultiplicative, 396 stage number of, 182

subordinate, 397 one-step, 316

matrix polynomial, 394 one-step-theta, 312

matrix-dependent, 248 overlay, 177

max-min-angle property, 179 PCG, 228, 229

maximum angle condition, 144 r-, 181

maximum column sum, 396 relaxation, 207

maximum principle Richardson, 206

strong, 36, 39, 329 Ritz, 56

weak, 36, 39, 329 Rotheā™s, 294

maximum row sum, 396 semi-iterative, 215

mechanical dispersion, 11 SOR, 210

mesh width, 21 SSOR, 211

method streamline upwind Petrovā“

advancing front, 179, 180 Galerkin, 375

algebraic multigrid, 240 streamline-diļ¬usion, 377

Arnoldiā™s , 235 method of conjugate directions, 219

artiļ¬cial diļ¬usion, 373 method of lines

asymptotically optimal, 199 horizontal, 294

BICGSTAB, 238 vertical, 293

block-Gaussā“Seidel, 211 method of simultaneous

block-Jacobi, 211 displacements, 203

CG, 221 method of successive displacements,

classical Ritzā“Galerkin, 67 204

collocation, 68 MIC decomposition, 232

consistent, 28 micro scale, 5

convergence, 27 minimum angle condition, 141

Crank-Nicolson, 313 minimum principle, 36

Cuthillā“McKee, 89 mobility, 10

reverse, 90 molecular diļ¬usivity, 11

Euler explicit, 313 monotonicity

Euler implicit, 313 inverse, 41, 280

extrapolation, 215 monotonicity test, 357

Index 421

moving front method, 179 equivalent, 395

multi-index, 53, 394 numbering

length, 53, 394 columnwise, 25

order, 53, 394 rowwise, 25

multiblock method, 180

multigrid iteration, 243 octree technique, 177

algorithm, 243 one-step method, 316

multigrid method, 243 A-stable, 317

algebraic, 240 strongly, 319

L-stable, 319

neighbour, 38 nonexpansive, 316

nested iteration, 200, 252 stable, 320

algorithm, 253 one-step-theta method, 312

Neumannā™s lemma, 398 operator, 403

Newtonā™s method, 349 operator norm, 403

algorithm, 357 order of consistency, 28

damped, 357 order of convergence, 27

inexact, 355 orthogonal, 401

simpliļ¬ed, 353 orthogonality of the error, 68

nodal basis, 61, 125 outer unit normal, 14, 97

nodal value, 58 outļ¬‚ow boundary, 108

node, 57, 115 overlay method, 177

adjacent, 127 overrelaxation, 209

degree, 89 overshooting, 371

neighbour, 63, 89, 211

norm, 400 parabolic boundary, 325

discrete L2 -, 27 parallelogram identity, 400

equivalence of, 401 Parsevalā™s identity, 292

Euclidean, 395 particle velocity, 7

Frobenius, 396 partition, 256

induced by a scalar product, 400 partition of unity, 407

p -, 395 PCG

matrix, 395 method, 228, 229

maximum, 395 PĀ“clet number

e

maximum , 27 global, 12, 368

maximum column sum, 396 grid, 372

maximum row sum, 396 local, 269

of an operator, 403 permeability, 8

spectral, 397 perturbation lemma, 398

streamline-diļ¬usion, 378 phase, 5

stronger, 401 immiscible, 7

total, 396 phase average

vector, 395 extrinsic, 6

Īµ-weighted, 374 intrinsic, 6

normal derivative, 98 k-phase ļ¬‚ow, 5

normal equations, 234 (k + 1)-phase system, 5

normed space piezometric head, 8

complete, 404 point

norms boundary, 40

422 Index

close to the boundary, 40 relative permeability, 9

far from the boundary, 40 relaxation method, 207

Poisson equation, 8 relaxation parameter, 207

Dirichlet problem, 19 representative elementary volume, 6

polynomial residual, 188, 189, 201, 244

characteristic, 395 inner, 355

matrix, 394 restriction, 248

pore scale, 5 canonical, 247

pore space, 5 Richards equation, 10

porosity, 6 Richardson method, 206

porous medium, 5 optimal relaxation parameter, 208

porous medium equation, 9 Ritz method, 56

preconditioner, 227 Ritz projection, 304

preconditioning, 207, 227 Ritzā“Galerkin method

from the left, 227 classical, 67

from the right, 227 root of equation, 342

preprocessor, 176 Rotheā™s method, 294

pressure row sum criterion

global, 10 strict, 204, 398

principle of virtual work, 49 weak, 205, 399

projection 2:1-rule, 181

elliptic, 303, 304

prolongation, 246, 247 saturated, 10

canonical, 246 saturated-unsaturated ļ¬‚ow, 10

pyramidal function, 62 saturation, 7

saturation concentration, 12

quadrature points, 80 scalar product, 400

quadrature rule, 80, 151 energy, 217

accuracy, 152 Euclidean, 401

Gaussā“(Legendre), 153 semi-iterative method, 215

integration points, 151 semidiscrete problem, 295

nodal, 152 semidiscretization, 293

trapezoidal rule, 66, 80, 153 seminorm, 400, 406

weights, 151 separation of variables, 285

quadtree technique, 177 set

closed, 393, 402

range, 343 connected, 394

reaction convex, 394

homogeneous, 13 open, 394

inhomogeneous, 11 set of measure zero, 393

surface, 11 shape function, 59

recovery operator, 193 cubic ansatz on simplex, 121

red mblack ordering, 212 d-polynomial ansatz on cube, 123

reduction strategy, 187 linear ansatz on simplex, 120

reference element, 58 quadratic ansatz on simplex, 121

standard simplicial, 117 simplex

reļ¬nement barycentre, 119

iterative, 231 degenerate, 117

red/green, 181 face, 117

Index 423

regular d-, 117 superposition principle, 16

sliver element, 179 surface coordinate, 119

smoothing system of equations

barycentric, 181 positive real, 233

Laplacian, 181

weighted barycentric, 181 test function, 47

smoothing property, 239, 250 theorem

smoothing step, 178, 242 of Aubin and Nitsche, 145

a posteriori, 243 of Kahan, 212

a priori, 243 of Laxā“Milgram, 93

smoothness requirements, 20 of Ostrowski and Reich, 212

Sobolev space, 54, 94 of PoincarĀ“, 71

e

solid matrix, 5 Trace, 96

solute concentration, 11 Thiessen polygon, 262

solution three-term recursion, 234

classical, 21 time level, 312

of an (initial-) boundary value time step, 312

problem, 17 tortuosity factor, 11

variational, 49 trace, 97

weak, 49, 290 transformation

uniqueness, 51 compatible, 134

solvent, 5 isoparametric, 168

SOR method, 210, 213 transformation formula, 137

convergence, 212 transmission condition, 34

optimal relaxation parameter, 213 triangle inequality, 400

sorbed concentration, 12 triangulation, 56, 114

source term, 14 anisotropic, 140

space conforming, 56, 125

normed, 400 element, 114

space-time cylinder, 15 properties, 114

bottom, 15 reļ¬nement, 76

lateral surface, 15 truncation error, 28

spectral norm, 397 two-grid iteration, 242

spectral radius, 395 algorithm, 242

spectrum, 395

split preconditioning, 228 underrelaxation, 209

SSOR method, 211 unsaturated, 10

stability function, 316 upscaling, 6

stability properties, 36 upwind discretization, 372

stable, 28 upwinding

static condensation, 128 exponential, 269

stationary point, 217 full, 269

step size, 21

stiļ¬ness matrix, 62, 296, 298 V-cycle, 244

element entries, 76 V -elliptic, 69

streamline upwind Petrovā“Galerkin variation of constants, 286

method, 375 variational equation, 49

streamline-diļ¬usion method, 377 equivalence to minimization

streamline-diļ¬usion norm, 378 problem, 50

424 Index

solvability, 93 regular, 262

viscosity, 8

volume averaging, 6 W-cycle, 244

volumetric ļ¬‚uid velocity, 7 water pressure, 8

volumetric water content, 11 weight, 30, 80

Voronoi diagram, 262 well-posedness, 16

Voronoi polygon, 262 Wignerā“Seitz cell, 262

Voronoi tesselation, 178

Z 2 ā“estimate, 192

Voronoi vertex, 262

degenerate, 262 zero of function f , 342

ńņš. 16 |