Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

Подождите немного. Документ загружается.

348 K. Gürlebeck and W. Sprößig

The operators P

are called general Plemelj projections.

The condition im TL∩ ker Tr ={0} can be seen as a very general formulation

of a maximum principle. In the original papers this condition is called

uniqueness

condition

. We mention her only shortly that there is another approach to this kind of

operator calculus, based on the theory of right-invertible operators (see, e.g., [14, 16]

for the general theory and [20] for many applications).

2.3 Examples of L-Holomorphy

2.3.1 Fractional Calculus

We consider the absolutely continuous function





(t) :=

Γ(α)



(t −τ)

1−α

u(τ ) dτ, (2.2)

which has almost everywhere a derivative in L

[a,1]. Take now for the operator Tr

the restriction of a function to the point t and deﬁne P by

P :=

n−1



k=0

(t −a)

α−k−1

Γ(α−k)

n−k−1

. (2.3)

Furthermore, we introduce

(Lu)(t) :=

Γ(1 −α)



1−α



(t), (T u)(t) :=





(t), (2.4)

and with n =[α]+1 we recognize

(P Tr u)(t) :=

n−1



k=0

(t −a)

α−k−1

Γ(α−k)

n−k−1

n−α

u(t). (2.5)

u)(t) is called Riemann–Liouville fractional integral, and (D

u)(t) is denoted

Riemann–Liouville fractional derivative.

2.4 Quaternionic Analysis

2.4.1 Real and Complex Quaternions

Let H be the algebra of real quaternions and a ∈ H; then a =



k=0

. Further,

let e

=−e

=−1,e

=−e

= e

=−e

= e

=−e

= e

The natural addition (deﬁned as in R

) and the above-deﬁned multiplication turn H

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 349

to a skew-ﬁeld. A quaternionic conjugation, a norm, and a multiplicative inverse are

given by

= e

, e

=−e

(k =1, 2, 3), a =a

−



k=1

−a,

aa = a =|a|

=:|a|

−1

|a|

a, ab =b a.

We denote by H(C) the set of quaternions with complex coefﬁcients, i.e.,

a =



k=0

(α

∈C). (2.6)

For k =0, 1, 2, 3, we have the commutator relation ie

i. Any complex quater-

nion a has the decomposition a = a

+ia

∈ H). Therefore, also the notation

CH is used. We have three possible conjugations:

:=a

−ia

:=a

+ia

:=a

−ia

2.4.2 D

-Holomorphy

Let a :=ia

with a

∈R. Then



k=1

∂

+a (2.7)

is called

Dirac operator, where e

(k =1, 2, 3) are the quaternionic units. Functions

u ∈C

(G) are called D

-holomorphic iff D

u =0.

2.4.3 Spaces of H (HC)-Valued Functions

We will say that u ∈X iff u

∈X and u =



i=0

.LetG ⊂R

be a bounded do-

main with sufﬁciently smooth boundary. C

k,λ

(G) denotes the space of all k-times

continuously differentiable functions such that the kth derivatives are λ-Hölder-

continuous. A norm is given by

u

k,λ

= sup

|s|≤k,x∈G



u(x)



+ sup

|s|=k,x∈G,x=y

u(x) −D

u(y)|

|x −y|

. (2.8)

350 K. Gürlebeck and W. Sprößig

Let k =0, 1,... and p ≥1; then the quaternionic (Clifford algebra-valued) Sobolev

space is deﬁned by

(G) =



u ∈L

(G) :D

u ∈L

(G);|s|≤k



. (2.9)

A norm is given by u

p,k



|s|≤k

D

u

)

.Let

(u) =



G×G

|u(x) −u(y)|

|x −y|

k+λp

dx dy (2.10)

with k =[k]+λ;λ ∈(0, 1). Then we can deﬁne

(G) =



u ∈L

(G) :D

u ∈L

;|s|≤k;I





< ∞



. (2.11)

A norm is given by

u

p,k



u

p,[k]



|s|=[k]







. (2.12)

Such spaces are called to be of quaternionic (Clifford algebra-valued)

Sobolev–

Slobodetzkij

type.

2.5 Bergman–Hodge Decomposition

Consider the real Clifford algebra C

0,n

generated by e

,...,e

with basis

,...,e

n−1

;...;e

...e

The operator D =



i=1

∂

is called Dirac operator with zero mass. Functions of

the class

ker D(G) ∩C

m,λ

(G) (2.13)

are called D

-holomorphic or simply holomorphic.

Theorem 2.5 [7] The sets ker D(G) ∩C

m,λ

(G) and ker D(G) ∩W

(G) are closed

subsets in C

m,λ

(G), W

(G), respectively, and are called Bergman spaces.

Proof Two facts are important: We need a mean value formula and Weierstrass’

theorem for sequences of holomorphic functions. 

A very important tool is a suitable decomposition of the quaternionic Hilbert

space. This can be obtained in the following way. We introduce in L

(G) the inner

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 351

product

(u, v)



u(x)v(x)dx ∈H, (2.14)

which fulﬁls the relations

λ(u, v)

=(λu, v)

,(u,v)

λ =(u, vλ)

(λ ∈H). (2.15)

We then ﬁnd the orthogonal decomposition

(G) =L

(G) ∩ker D ⊕D

(G) (Bergman–Hodge decomposition).

(2.16)

The orthoprojection

P :L

(G)

onto

→ ker D ∩L

(G) (2.17)

is called

Bergman-type projection relative to the inner product (u, v)

. We denote by

Q :=I −P :L

(G)

onto

→ D

(G). (2.18)

It can be proved that

P :=F

(trTF

)

−1

Remark For sufﬁciently smooth H-holomorphic functions, the operator trTF

co-

incides with the single-layer potential V

u :=

4π



|y−x|

dΓ .

Theorem 2.6 (Trace theorem) We have the following isomorphism:

trV

k−

(Γ ) ∩im P

→W

k+1−

(Γ ) ∩im Q

. (2.19)

This result can be found in [8]. Analogous results can be proved for the general-

ized Cauchy–Riemann operator ∂ =∂

+D.

2.6 Quaternionic Operator Calculus

Let X = W

(G), Y = W

k+1

(G), Z = W

k+1−

(Γ ), k = 0, 1, 2,...;1 <p<∞.

Further, let

L := D +iα

(Dirac operator with mass), (2.20)

(T u)(x) := −



iα

(x −y)u(y)dy (Teodorescu type transform), (2.21)

352 K. Gürlebeck and W. Sprößig

(P u)(x) :=



iα

(x −y)n(y)u(y)dΓ

(CF-type operator), (2.22)

(Tr u)(ξ) := n.-t.- lim

z→ξ∈Γ,z∈G

u(z). (2.23)

n.-t.-lim stands for the nontangential limit, and the abbreviation “CF” means

Cauchy–Fueter. The disturbed Cauchy–Fueter kernel is given by

iα(x)

:=−



iα

2π



(3/2)



|x|

−1/2

3/2



α|x|



ω −K

1/2



α|x|



, (2.24)

where ω ∈S

, and K

(t) denotes Macdonald’s function.

2.7 Discrete Quaternionic Analysis

For an introduction of a discrete calculus, we have to deﬁne the following lattice

sets:



(ih, j h, kh) :i, j, k integer,h>0



:=G ∩R



x ∈G

:dist(x, co G

) ≤

√



Let V

i,h

x be the shift of x by ±h in the x

-direction. Then

h,(r)



x ∈Γ

:∃i :V

i,h

x/∈G



left (right) side planes



h,(r);i



x ∈Γ

i,h

x/∈G



h,(r);i,j

:= Γ

h,(r);i

∩Γ

h,(r);j



left (right) edges



h,(r);i,j,k

:= Γ

h,(r);i,j

∩Γ

h,(r);k



left (right) corners



Let X =W

2,h

), Y =L

2,h

), Z =W

2,h

). Then

(Lu)(x) :=





(x) =



i=1





i,h



−u(x)



, (2.25)

(T u)(x) :=





(x)





int G

∪Γ

h,(r)



left (right) corners

−



left (right) edges



×e

(x −y)u(y)h

, (2.26)

where e

are the discrete fundamental solutions of D

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 353

We have

(P Tr u)(x) :=





(x) =



i=1



−



−



ij k





x −V

∓

i,h



n(y)u(y)h



i=1



y∈Γ

h,(r);m,j,k

m=j=k

(x −y)e

u(y)h

where s

=Γ

h,;i

∪Γ

h,r;i

:=Γ

h,;j

−V

i,h

h,

, s

ij k

:=Γ

h,;j,k

−V

i,h

h,;i,k

In [7] the following mean value theorem is proved:

Theorem 2.7

u(x) =





(x) +T

u(x). (2.27)

These formulae are also called (generalized or discrete)

Borel–Pompeiu formulae.

To be constructive we need a representation for the fundamental solutions of

the discrete Dirac operators. Following [5] and [8], E

(x) is the solution of the

difference equation

−Δ

(x) =−



i=1

−

i,h

(x) =δ

(x) =



−3

,x=0,

0,x∈R

\{0}.

We get, with the Fourier transform F ,

(x) =

√

2π





. (2.28)

The function d is deﬁned by



sin

hξ

+sin

hξ

+sin

hξ



, (2.29)

and R

u is the restriction of the continuous function u onto the lattice R

. Due to

the factorization D

j,h

∓

j,h

=Δ

, we get

(x) := D

∓

j,h

(x). (2.30)

3 Fluid Flow Problems

3.1 A Brief History of Fluid Dynamics

In 1730 Daniel Bernoulli made the observation that the pressure p of a ﬂuid de-

creases if its speed increases. Nowadays, this is called

Bernoulli’s principle.Eleven

354 K. Gürlebeck and W. Sprößig

years later, Leonard Euler, who was invited by Frederick the Great to Potsdam, was

charged by him with the construction of a water fountain. In 1755, after thorough

studies of the motion of the ﬂuid, he formulated Newton’s law for the rate of change

of the momentum of a ﬂuid element. This is a set of equations that exactly represents

the ﬂow of a ﬂuid as long as one can suppose that the ﬂuid is inviscid:

∂

u +(u ·∇)u =−

∇p

(Euler’s equations). (3.1)

So he derived the equations of an

ideal ﬂuid (no viscous effects included). In 1822

Claude-Louis Navier derived an equation which also considers the inner friction of

a ﬂowing ﬂuid without understanding the character of a viscous ﬂuid. His deriva-

tion was based on a molecular theory of attraction and repulsion between molecules.

J.D. Anderson wrote in his

A History of Aerodynamics: The irony is that although

Navier had no conception of shear stress and did not set out to obtain equations that

would describe motion involving friction, he nevertheless arrived at the proper form

for such equations. Navier’s equations were several times rediscovered (Cauchy

1828, Denise Poisson 1828, and Barré de Saint-Venant 1843). Saint-Venant’s model

even includes the turbulence case. George Stokes published in 1845 a strong math-

ematical derivation and explained the equations in the sense that is currently un-

derstood. Therefore these equations are called nowadays

Navier–Stokes equations

(NSE)

given by

∂

u −Δu +

(u ·∇)u +

∇p =f in G (3.2)

for some bounded domain in R

3.2 Stationary Linear Stokes Problem

Let G ⊂R

be a bounded domain with sufﬁciently smooth boundary Γ . The linear

Stokes system reads as follows:

−Δu +

∇p =

f in G, (3.3)

div u = f

in G, (3.4)

u = g on Γ. (3.5)

Here η is the

viscosity, and ρ the density of the ﬂuid. We have to look for the velocity

u and the hydrostatical pressure p. Between f

and g, we have to fulﬁl the relation



dx =



ngdΓ. (3.6)

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 355

For g =0, the measure of the compressibility f

satisﬁes the identity



dx =0. (3.7)

For all such real functions f

, the unique solution (p is unique up to a real constant)

can be represented as follows:

Theorem 3.1 [7] Let f :=f

+f ∈W

(G,H)(k≥0, 1 <p<∞). Then we have

u =

Vec T

f −

Vec F

(tr

Vec F

)

−1

Vec T

f −T

(3.8)

p = ρ ScT

f −ρ Sc F

(tr

Vec F

)

−1

Vec T

f +ηf

. (3.9)

In that way we strongly separate velocity and pressure!

3.3 Nonlinear Stokes Problem

If the compressibility depends on the velocity and the nonlinear outer forces, then

the equations read as follows:

−Δu +

∇p =Λf ( u) in G, (3.10)

div



−1



= 0inG, (3.11)

u =0onΓ. (3.12)

The viscosity η(η>0) depends on the position. Suppose that f ∈L

(G,H), p ∈

(G), η ∈C

∞

(G).

Theorem 3.2 (Representation formula [7]) Every solution of the nonlinear Stokes

problem permits the representation

u = ΛRBf −RBDp,

0 =ScΛQT

Bf −Sc QT

BDp.

Here the operator B stands for the multiplication by η

−1

, and R :=T

Theorem 3.3 (Iteration procedure [7]) Suppose that f(u), B, and Λ satisfy the

following estimates:

(i) f(u)−f(v)

≤Lu −v

2,1

for u

2,1

, v

2,1

≤1,

(ii) B

≤K for positive constants K,L,

(iii) Λ<{T 

imQ∩L

T 

KL}

−1

356 K. Gürlebeck and W. Sprößig

Then the iteration procedure

= ΛRBf (u

n−1

) −RBDp

Λ Sc DBRf (u

n−1

) = Sc DBRDp

with u



2,1

≤ 1 (u

∈

(G,H)), converges to the unique solution {u, p}∈

(G,H) ∩ ker(divB) × L

(G) of (3.10)–(3.12), where p is unique up to a real

constant.

3.4 Stationary Navier–Stokes Problem

As we have already seen, the stationary NSE are given by

−Δu +

(u ·∇)u +

∇p =f in G, (3.13)

div u = 0inG, (3.14)

u =0onΓ. (3.15)

In dependence on the special application, terms can be dropped out:

• νΔu ≈0 ... supersonic motion around an airfoil,

• ∂

u ≈0... geostrophic ﬂow (steady-state case),

• (u ·∇)u ≈0... creeping ﬂow in ground.

Theorem 3.4 [8] The main result is now the following:

1. Let f ∈L

(G,H) and p ∈W

(G). Every solution permits the operator integral

representation

u =−

RM(u) −

Qp, (3.16)

0 =

ScQT

M(u) −

ScQp. (3.17)

2. The above system has a unique solution {u, p}∈

(G,H) ∩ ker(divB) ×

(G), where p is unique up to a real constant, if

(i) f 

≤(18K

)

−1

with K :=

T



∩imQ,W

]

T 

]

(ii) u

∈

(G,H) ∩ker(divB) with u



2,1

≤min(V ,

4KC

+W) hold.

Corollary 3.5 Setting V := (2KC

)

−1

, W := [(4KC

)

−2

−

ρf 

ηC

]

, and C

C, where C is the embedding constant from W

in L

, the iteration procedure

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 357

(starting with u

)

RM(u

n−1

) −

RDp

ScQT

M(u

n−1

) =−

ScQp

∈

(G,H) ∩ker div

converges in W

(G,H) ×L

(G).

3.5 Navier–Stokes Equations with Heat Conduction

We will now consider the ﬂow of a viscous ﬂuid under the inﬂuence of temperature.

The corresponding equations read as follows:

−Δu +

(u ·∇)u +

∇

gw = f in G, (3.18)

−∇w +

(u ·∇)w =

h in G, (3.19)

div u = 0inG, (3.20)

u,w = 0onΓ. (3.21)

Here, γ is the Grashof number, κ the temperature conductivity, and m stands for

the Prandtl number. Applying the quaternionic operator calculus, we can rewrite the

previous system equivalently:

u =−R



M(u) −



−

Qp, (3.22)

0 =ScDR



M(u) −



−

Qp, (3.23)

w =−

R Sc(uD)w +Rg, (3.24)

where M(u) :=

(u ·grad)u +f(u)−F .

Remark The Grashof number approximates the ratio of the buoyancy forces to the

viscous forces in a ﬂuid,

γ =Gr =

gαΔwL

buoyancy forces

viscous forces

, (3.25)

where g is the

gravitational acceleration constant, α is the (thermal) volume expan-

sion coefﬁcient

of the ﬂuid, Δw is the temperature difference between the ﬂuid and

the wall, L is the

characteristic length, and μ is the kinematic viscosity of the ﬂuid.