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Schwarz Algorithms for Stokes-Stokes Coupling

Alexandros Kyriakis
Boston College UK, Department of Mathematics, Skirbeck Road, PE21 6JF, UK

alex-k@boston.ac.uk

Abstract. In this article, we exhibit the behavior of the Schwarz algorithms for the Steady Stokesequation in the case of two unbounded subdomains at the continuous level. The Schwarz methods havereceived a lot of attention during the last decades with the vast development of parallel computingdevices. Hermann Amandus Schwarz, a German analyst, is considered to be the pioneer of the DomainDecomposition methods. We will closely observe how the overlapping and non overlapping Schwarzmethods work for the steady Stokes problem. This problem has immediate practical application,modeling the flow of an incompressible fluid. For the analysis, we rely on Fourier analysis techniquesand we provide comparison of the exhibited methods.

1. Introduction
Many people have been fascinated by the motion of fluids, and wonder how we are able tosimulate the motion of fluids with such an accuracy. Of course, the answer is simple but at thesame time complicated. Firstly, in order to model various phenomena, we use partial differentialequations( PDEs). PDEs are equations that involve partial derivatives and most of the times weare not able to obtain solutions in closed form. As a result, we use numerical algorithms in orderto obtain the approximate solution of a PDE. This field is called Numerical Analysis of PDEsand it is gaining increasing interest from mathematical and engineering communities worldwide.Especially, the last two decades Domain Decomposition methods [7], [8], [9], [10] are gaining grounddue to the increased use of parallel computing. The pioneer of these methods was the Germananalyst Hermann Schwarz [4], [5], [6] who devised an algorithm to solve the Poisson equation inan irregular domain (union of rectangle and a circle), in order to fix a glitch in Riemann’s mappingtheorem. The algorithm is

∆u
(k)
1 = −f , in Ω1

u
(k)
1 = u

(k−1)
2 , at Γ1

u
(k)
1 = g1, on ∂Ω1 \ Γ1

, then


∆u

(k)
2 = −f , in Ω2

u
(k)
2 = u

(k)
1 , at Γ2

u
(k)
2 = g2, on ∂Ω2 \ Γ2.

(1)
Received: 18 May 2024.
Key words and phrases. Schwarz algorithms; Steady Stokes equation; Convergence analysis.1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 2

Ω1 Ω2Γ1Γ2

Figure 1. Domain decomposition of the global domain into a union of a circle and a rectangle.Having a close look at the above figure we notice the following: Firstly, the Poisson problemis solved in the circle and then in the rectangle, going back and forth, passing the values at theinterfaces Γ1 and Γ2. This iteration process is repeated until the convergence is reached. Theindex (k) denotes the iterations, and f is the source function. This is the so called AlternatingSchwarz algorithm proposed by Schwarz back in 1870. After a significant amount of time, the Fieldsmedalist Pierre Luis Lions [3], [13] proposed a modification in the Alternating Schwarz method (1).After imposing this modification, the algorithm (1) takes the form
∆u

(k)
1 = −f , in Ω1

u
(k)
1 = u

(k−1)
2 , at Γ1

u
(k)
1 = g1, on ∂Ω1 \ Γ1

, and


∆u

(k)
2 = −f , in Ω2

u
(k)
2 = u

(k−1)
1 , at Γ2

u
(k)
2 = g2, on ∂Ω2 \ Γ2.

(2)
In this iterative algorithm, the two local subproblems are solved in parallel passing the Dirichletvalues at the two interfaces. This algorithm (2) is known as the Parallel Schwarz algorithm. Thisiterative scheme provides two great benefits. The first is balancing the computational cost bybreaking the global problem into smaller subproblems. The second benefit is that with the increas-ing amount of computational resources, the Schwarz method (2) is ideal for parallel computations.There has been an avalanche of new research results and there is a great avenue of research onDomain Decomposition methods. In this article, we will observe the behaviour of the Schwarzmethods for the steady Stokes equation, for two unbounded subdomains using Fourier analysistechniques which is a standard approach in the literature [1], [2], [11], [12], [14], [15], [16], [17]. Thesteady Stokes equation is derived from the Navier-Stokes equation, which is a PDE for modelingthe flow of incompressible fluids. It is a generalization of the equations proposed by the Swissmathematician Leonhard Euler in the 18th century. In 1821, Claude-Luis Navier introduced theelement of the viscosity. Later in the mid 19th century, Sir Gabriel Stokes worked extensively onthe equation. The steady Stokes equation in strong form reads

−ν∆~u + Op = ~f in Ω = (−∞,+∞)× (−∞,+∞),

d iv ~u = 0 in Ω,

~u : bounded at ±∞,

p : bounded at ±∞

(3)

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 3where ν > 0 is the kinematic viscosity of the fluid, ~u is the velocity of the fluid and p is thepressure field. The function ~f is called the sink term. The function spaces for the velocity field,pressure field and sink term are (H1(Ω)
)2

, L2(Ω) , (L2(Ω))2 respectively. The space H1(Ω) isclassical Sobolev space, and L2(Ω) is the space of square integrable functions. The (3)2 denotesthe incompressibility condition, with the divergence free velocity field. Furthermore as (3)3, (3)4suggest, the velocity and the pressure field stay bounded at infinity.

2. Parallel Schwarz Method-Dirichlet IC
We decompose the domain Ω = R2 into two subdomains Ω1 = (−∞, H) × (−∞,+∞) and

Ω2 = (0,+∞)× (−∞,+∞). The Parallel Schwarz method in strong form reads

−ν∆−→u1
(k) + Op(k)

1 = ~f in Ω1,

d iv−→u1
(k) = 0 in Ω1,

−→u1
(k) = −→u2

(k−1)at x = H,

−→u1
(k) : bounded at −∞,

p
(k)
1 : bounded at −∞,

, and



−ν∆−→u2
(k) + Op(k)

2 = ~f in Ω2,

d iv−→u2
(k) = 0 in Ω2,

−→u2
(k) = −→u1

(k−1)at x = 0,

−→u2
(k) : bounded at +∞,

p
(k)
2 : bounded at +∞,

(4)

where two initial guesses −→u1
(0), −→u2

(0) are required to start the iterative process.
Theorem 1. The convergence factor of the Parallel Schwarz algorithm using Dirichlet transmission
conditions is given by the formula below

rPSM,D(ξ,H) =
(

1 + 2H2|ξ|2 + 2 |ξ|
√
H2 (1 +H2|ξ|2)

)
e−2|ξ|H (5)

where ξ is the Fourier frequency and H > 0 is the size of the overlap.

Proof. In order to study the convergence behavior of the method, we go back to the local subproblemsin (4) and we consider the homogeneous counterparts taking ~f = ~0. In addition, the velocity fieldsin the two subdomains are −→u1
(k) =

(
u

(k)
1,1 , u

(k)
1,2

) and −→u2
(k) =

(
u

(k)
2,1 , u

(k)
2,2

), where the first indicesdenote the subdomain and the second indices denote the component. Consequently, the ParallelSchwarz method prescribed by (4) can be written in the following form

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 4



∂2u
(k)
1,1

∂x2 +
∂2u

(k)
1,1

∂y2 = 1
ν
∂p

(k)
1
∂x in Ω1,

∂2u
(k)
1,2

∂x2 +
∂2u

(k)
1,2

∂y2 = 1
ν
∂p

(k)
1

∂y in Ω1,

∂u
(k)
1,1

∂x +
∂u

(k)
1,2

∂y = 0 in Ω1,

u
(k)
1,1 = u

(k−1)
2,1 at x = H,

u
(k)
1,2 = u

(k−1)
2,2 at x = H,

u
(k)
1,1 : bounded at −∞,

u
(k)
1,2 : bounded at −∞,

p
(k)
1 : bounded at −∞,

, and



∂2u
(k)
2,1

∂x2 +
∂2u

(k)
2,1

∂y2 = 1
ν
∂p

(k)
2
∂x in Ω2,

∂2u
(k)
2,2

∂x2 +
∂2u

(k)
2,2

∂y2 = 1
ν
∂p

(k)
2

∂y in Ω2,

∂u
(k)
2,1

∂x +
∂u

(k)
2,2

∂y = 0 in Ω2,

u
(k)
2,1 = u

(k−1)
1,1 at x = 0,

u
(k)
2,2 = u

(k−1)
1,2 at x = 0,

u
(k)
2,1 : bounded at +∞,

u
(k)
2,2 : bounded at +∞,

p
(k)
2 : bounded at +∞.

(6)

Going back to (4)1, for ~f = ~0, taking the divergence on both sides for the first subproblem, weobtain
div

(
∆−→u1

(k)
)

=
1

ν
∆p

(k)
1

=

(
∂3u

(k)
1,1

∂x3
+
∂3u

(k)
1,2

∂y∂x2

)
+

(
∂3u

(k)
1,2

∂y3
+
∂3u

(k)
1,1

∂x∂y2

)
= 0 in Ω1

exploiting the equation (4)2 (divergence free velocity in subdomain Ω1). In the same fashion weobtain that div (∆−→u2
(k)
)

= 1
ν ∆p

(k)
2 = 0 in Ω2. As a consequence, we have to solve two Laplaceproblems in each subdomain where the unknown is the pressure field. We deal with ∆p

(k)
1 =

0 in Ω1 and by taking the Fourier transform in the y direction we obtain the homogeneous equation
∂2p̂

(k)
1

∂x2 − |ξ|2p̂(k)
1 = 0. The general solution of this equation is p̂(k)

1 = C(k)
1 e−|ξ|x +D(k)

1 e |ξ|x . By theboundedness assumption of the pressure field in Ω1 as x → −∞, we obtain that p̂(k)
1 = D(k)

1 e |ξ|x .We proceed to solve the equation ∆p
(k)
2 = 0 in Ω2, and the first step is to take the Fourier transformin the y direction. As a result, the equation ∂2p̂

(k)
2

∂x2 − |ξ|2p̂(k)
2 = 0 has a general solution of theform p̂

(k)
2 = C(k)

2 e−|ξ|x + D(k)
2 e |ξ|x .By exploiting the property that the pressure field p̂(k)

2 remainsbounded as x → +∞, we obtain that p̂(k)
2 = C(k)

2 e−|ξ|x . The next move is to go to the two localSchwarz subproblems in (6)1 and to take the Fourier transform in the y direction. This will give
∂2û

(k)
1,1

∂x2
− |ξ|2û(k)

1,1 =
|ξ|D(k)

1 e |ξ|x

ν
, (7)

∂2û
(k)
2,1

∂x2
− |ξ|2û(k)

2,1 =
−|ξ|C(k)

2 e−|ξ|x

ν
. (8)

We solve (7), (8) to obtain the two solutions in closed form
û

(k)
1,1 =

(
B(k)

1 +
x

2ν
D(k)

1

)
e |ξ|x , (9)

û
(k)
2,1 =

(
B(k)

2 +
x

2ν
C(k)

2

)
e−|ξ|x . (10)

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 5We go back to (6)3 and by taking the Fourier trasform in the y direction and exploiting the solutions(9), (10) we obtain
û

(k)
1,2 =

i

ξ

(
|ξ|B(k)

1 +

(
1 + x |ξ|

2ν

)
D(k)

1

)
e |ξ|x , (11)

û
(k)
2,2 =

i

ξ

(
−|ξ|B(k)

2 +

(
1− x |ξ|

2ν

)
C(k)

2

)
e−|ξ|x . (12)

We further proceed, substituting the solutions (9), (10), (11), (12) back to the interface conditions(6)4, (6)5 to obtain the following equations
B(k)

1 +
H

2ν
D(k)

1 =

(
B(k−1)

2 +
H

2ν
C(k−1)

2

)
e−2|ξ|H, (13)

|ξ|B(k)
1 +

(
1 +H|ξ|

2ν

)
D(k)

1 =

(
−|ξ|B(k−1)

2 +

(
1−H|ξ|

2ν

)
C(k−1)

2

)
e−2|ξ|H, (14)

B(k)
2 = B(k−1)

1 , (15)
−|ξ|B(k)

2 +
C(k)

2

2ν
= |ξ|B(k−1)

1 +
D(k−1)

1

2ν
. (16)

We combine the equations (15), (16) to obtain D(k)
1 = C(k+1)

2 − 4|ξ|νB(k+1)
2 . We substitute thecoefficients D(k)

1 back to equation (13) to obtain
B(k+1)

2 (2ν − 4νH|ξ|) +HC(k+1)
2 = B(k−1)

2 2νe−2|ξ|H +HC(k−1)
2 e−2|ξ|H. (17)

In the same spirit, we replace the iteration coefficients D(k)
1 back to (14) to get

B(k+1)
2

(
2ν|ξ|+ 4νH|ξ|2

)
−C(k+1)

2 (1+H|ξ|) = 2ν|ξ|B(k−1)
2 e−2|ξ|H+(H|ξ|−1)C(k−1)

2 e−2|ξ|H. (18)
We take the two equations (17), (18) and write them in matrix form[

(2ν − 4νH|ξ|) H(
2ν|ξ|+ 4νH|ξ|2

)
−(1 +H|ξ|)

][
B(k+1)

2

C(k+1)
2

]
=

[
2νe−2|ξ|H He−2|ξ|H

2ν|ξ|e−2|ξ|H (H|ξ| − 1)e−2|ξ|H

][
B(k−1)

2

C(k−1)
2

]
.

(19)We recast the equation (19) in the form[
B(k+1)

2

C(k+1)
2

]
=

[
(1 + 2H|ξ|)e−2|ξ|H H2|ξ|e−2|ξ|H

ν

8νH|ξ|2e−2|ξ|H (
4H2|ξ|2 − 2H|ξ|+ 1

)
e−2|ξ|H

]
︸ ︷︷ ︸

ΨPSM,D

[
B(k−1)

2

C(k−1)
2

] (20)
where ΨPSM,D is the Schwarz iteration matrix. The Spectrum of ΨPSM,D is σ(ΨPSM,D) =

{λ+, λ−}, where λ+ and λ− are the corresponding eigenvalues given by the formulas
λ+ =

(
1 + 2H2|ξ|2 + 2 |ξ|

√
H2 (1 +H2|ξ|2)

)
e−2|ξ|H ,

λ− =
(

1 + 2H2|ξ|2 − 2 |ξ|
√
H2 (1 +H2|ξ|2)

)
e−2|ξ|H .

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 6Consequently, the convergence factor of the Parallel Schwarz algorithm is
rPSM,D = ρ(ΨPSM,D) = max{|λ+|, |λ−|} =

(
1 + 2H2|ξ|2 + 2 |ξ|

√
H2 (1 +H2|ξ|2)

)
e−2|ξ|H

, where ρ(ΨPSM,D) is the spectral radius of the Schwarz iteration matrix, H is the size of theoverlap between the subdomains, ξ is the Fourier frequency. �

3. Alternating Schwarz Method-Neumann IC
The interface conditions play critical role on the convergence of the Schwarz method. In thissection, we introduce the Alternating Schwarz algorithm employing Neumann interface conditions.We go back to the iterative scheme prescribed by (4) and we modify the transmission conditions in(4)3. As a consequence, the Schwarz method in strong form reads

−ν∆−→u1
(k) + Op(k)

1 = ~f in Ω1,

d iv−→u1
(k) = 0 in Ω1,

νO−→u1
(k)~n − p(k)

1 ~n = νO−→u2
(k−1)~n − p(k−1)

2 ~n at x = H,

−→u1
(k) : bounded at −∞,

p
(k)
1 : bounded at −∞,



−ν∆−→u2
(k) + Op(k)

2 = ~f in Ω2,

d iv−→u2
(k) = 0 in Ω2,

νO−→u2
(k)~n − p(k)

2 ~n = νO−→u1
(k)~n − p(k)

1 ~n at x = 0,

−→u2
(k) : bounded at +∞,

p
(k)
2 : bounded at +∞, (21)where ~n is the outward normal vector. The initial guess νO−→u2

(0)~n − p(0)
2 ~n is required to start theiterative procedure.

Theorem 2. The convergence factor of the Schwarz algorithm using Neumann transmission condi-
tions is given by the formula below

rASM,N(ξ,H) =
∣∣∣(2|ξ|2H2

9
−

2|ξ|H
9

+ 1 +
2
√
|ξ|4H4 + 2|ξ|3H3 + 8 |ξ|2H2

9

)∣∣∣e−2|ξ|H

(22)
where ξ is the Fourier frequency and H > 0 is the size of the overlap.

Proof. As a first step, we go back to the local subproblems in (21) and we consider the homogeneouscounterparts taking ~f = ~0. We recast the method prescribed by (21) in the following form

∂2u
(k)
1,1

∂x2 +
∂2u

(k)
1,1

∂y2 = 1
ν

∂p
(k)
1

∂x in Ω1,

∂2u
(k)
1,2

∂x2 +
∂2u

(k)
1,2

∂y2 = 1
ν

∂p
(k)
1

∂y in Ω1,

∂u
(k)
1,1

∂x +
∂u

(k)
1,2

∂y = 0 in Ω1,

ν ∂
∂x u

(k)
1,1 − p

(k)
1 = ν ∂

∂x u
(k−1)
2,1 − p(k−1)

2 at x = H,

∂
∂x u

(k)
1,2 = ∂

∂x u
(k−1)
2,2 at x = H,

u
(k)
1,1 : bounded at −∞,

u
(k)
1,2 : bounded at −∞,

p
(k)
1 : bounded at −∞,



∂2u
(k)
2,1

∂x2 +
∂2u

(k)
2,1

∂y2 = 1
ν

∂p
(k)
2

∂x in Ω2,

∂2u
(k)
2,2

∂x2 +
∂2u

(k)
2,2

∂y2 = 1
ν

∂p
(k)
2

∂y in Ω2,

∂u
(k)
2,1

∂x +
∂u

(k)
2,2

∂y = 0 in Ω2,

ν ∂
∂x u

(k)
2,1 − p

(k)
2 = ν ∂

∂x u
(k)
1,1 − p

(k)
1 at x = 0,

∂
∂x u

(k)
2,2 = ∂

∂x u
(k)
1,2 at x = 0,

u
(k)
2,1 : bounded at +∞,

u
(k)
2,2 : bounded at +∞,

p
(k)
2 : bounded at +∞.

(23)

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 7We apply the Fourier transform in the y direction to the Schwarz subproblems prescribed by (23).The Fourier transformed velocity components are given by the formulas (9), (10), (11), (12). TheFourier transformed pressure fields are given by the relations: p̂(k)
1 = D(k)

1 e |ξ|x , p̂(k)
2 = C(k)

2 e−|ξ|x .We plug in the Fourier transformed velocities and pressure fields back to the interface conditions(23)4, (23)5 and by doing a little algebra we obtain
2ν|ξ|e |ξ|HB(k)

1 +D(k)
1 (H|ξ| − 1)e |ξ|H = −2ν|ξ|B(k−1)

2 e−|ξ|H − C(k−1)
2 e−|ξ|H(1 +H|ξ|), (24)

2νB(k)
1 |ξ|

2e |ξ|H +D(k)
1 e |ξ|H

(
2|ξ|+H |ξ|2

)
= 2ν|ξ|2B(k−1)

2 e−|ξ|H + C(k−1)
2 e−|ξ|H

(
H|ξ|2 − 2|ξ|

)
,(25)

2ν|ξ|B(k)
2 + C

(k)
2 = −2ν|ξ|B(k)

1 +D(k)
1 , (26)

ν|ξ|2B(k)
2 − |ξ|C(k)

2 = νB(k)
1 |ξ|

2 + |ξ|D(k)
1 . (27)

We multiply (26) by −|ξ| then add (27), and solve with respect to the coefficient B(k)
1 obtaining

B(k)
1 = −

1

3
B(k)

2 −
2

3

1

ν|ξ|C
(k)
2 . (28)

The next step is to obtain a formula for the coefficient D(k)
1 . In order to achieve that, we multiply(26) by |ξ| then add (27) to obtain

D(k)
1 =

4

3
ν|ξ|B(k)

2 −
1

3
C(k)

2 . (29)
We substitute the expressions (28), (29) back to (24) and (25) and this yields

B(k)
2 e |ξ|H

(
6ν|ξ| − 4νH|ξ|2

)
+ C(k)

2 e |ξ|H(3 +H|ξ|) = 6ν|ξ|B(k−1)
2 e−|ξ|H + 3C(k−1)

2 e−|ξ|H(1 +H|ξ|),

B(k)
2 e |ξ|H

(
6ν|ξ|2 + 4νH|ξ|3

)
− C(k)

2 e |ξ|H
(

6|ξ|+H|ξ|2
)

= 6ν|ξ|2B(k−1)
2 e−|ξ|H + 3C(k−1)

2 e−|ξ|H
(
H|ξ|2 − 2|ξ|

)
.

We write the above equations in matrix form and by doing some algebraic manipulations we derivethe stationary iteration[
B(k)

2

C
(k)
2

]
=

−12νH|ξ|3−54ν|ξ|2
54e2|ξ|H |ξ|2ν

H(H|ξ|+4)

9e2|ξ|Hν
8|ξ|2νH
9e2|ξ|H

4|ξ|2H2−2|ξ|H+9

9e2|ξ|H


︸ ︷︷ ︸

ΨASM,N

[
B(k−1)

2

C
(k−1)
2

] (30)
where ΨASM,N is the Schwarz iteration matrix. The spectrum of ΨASM,N is σ(ΨASM,N) = {µ+, µ−},where µ+ and µ− are the eigenvalues of the Schwarz iteration matrix provided by the formulas

µ+ =

(
2|ξ|2H2

9
−

2|ξ|H
9

+ 1 +
2
√
|ξ|4H4 + 2|ξ|3H3 + 8 |ξ|2H2

9

)
e−2|ξ|H,

µ− =

(
2|ξ|2H2

9
−

2|ξ|H
9

+ 1−
2
√
|ξ|4H4 + 2|ξ|3H3 + 8 |ξ|2H2

9

)
e−2|ξ|H.

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 8As a result, the reduction factor of the Schwarz method (Neumann Interface conditions) is given by
rASM,N = ρ(ΨASM,N) = max{|µ+|, |µ−|}

=
∣∣∣(2|ξ|2H2

9
−

2|ξ|H
9

+ 1 +
2
√
|ξ|4H4 + 2|ξ|3H3 + 8 |ξ|2H2

9

)∣∣∣e−2|ξ|H.

�

4. Non-overlapping Optimized Schwarz Algorithm-Robin IC
The domain Ω = R2 is decomposed into two non-overlapping subdomains Ω1 = (−∞, 0) ×

(−∞,+∞) and Ω2 = (0,+∞) × (−∞,+∞).The Optimized Schwarz methods employ mixed in-terface boundary conditions, and more Precisely Robin. In this way, they facilitate both Neumannand Dirichlet conditions and there is a tuning parameter to tune the method accordingly. TheOptimized Schwarz iterative scheme is given in strong form

−ν∆−→u1
(k) + Op(k)

1 = ~f in Ω1,

d iv−→u1
(k) = 0 in Ω1,

νO−→u1
(k)~n − p(k)

1 ~n + γ−→u1
(k) = νO−→u2

(k−1)~n − p(k−1)
2 ~n + γ−→u2

(k−1) at x = 0,

−→u1
(k) : bounded at −∞,

p
(k)
1 : bounded at −∞,

(31)



−ν∆−→u2
(k) + Op(k)

2 = ~f in Ω2,

d iv−→u2
(k) = 0 in Ω2,

νO−→u2
(k)~n − p(k)

2 ~n + γ−→u2
(k) = νO−→u1

(k−1)~n − p(k−1)
1 ~n + γ−→u1

(k−1) at x = 0,

−→u2
(k) : bounded at +∞,

p
(k)
2 : bounded at +∞,

(32)

where γ is the tuning parameter of the method. Two initial guesses are needed for the iterativemethod.
Theorem 3. The contraction factor of the non-overlapping Schwarz algorithm is given by the
mathematical expression

r2
OSM(ξ, ν, γ) =

|3ν2|ξ|2 − 4ν|ξ|γ + γ2|2

|3ν2|ξ|2 + 4ν|ξ|γ + γ2|2 . (33)

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 9

Proof. We recast the local Schwarz subproblems (31), (32) in the form

∂2u
(k)
1,1

∂x2 +
∂2u

(k)
1,1

∂y2 = 1
ν
∂p

(k)
1
∂x in Ω1,

∂2u
(k)
1,2

∂x2 +
∂2u

(k)
1,2

∂y2 = 1
ν
∂p

(k)
1

∂y in Ω1,

∂u
(k)
1,1

∂x +
∂u

(k)
1,2

∂y = 0 in Ω1,

ν ∂
∂x u

(k)
1,1 − p

(k)
1 + γu

(k)
1,1 = ν ∂

∂x u
(k−1)
2,1 − p(k−1)

2 + γu
(k−1)
2,1 at x = 0,

ν ∂
∂x u

(k)
1,2 + γu

(k)
1,2 = ν ∂

∂x u
(k−1)
2,2 + γu

(k−1)
2,2 at x = 0,

u
(k)
1,1 : bounded at −∞,

u
(k)
1,2 : bounded at −∞,

p
(k)
1 : bounded at −∞,

(34)



∂2u
(k)
2,1

∂x2 +
∂2u

(k)
2,1

∂y2 = 1
ν
∂p

(k)
2
∂x in Ω2,

∂2u
(k)
2,2

∂x2 +
∂2u

(k)
2,2

∂y2 = 1
ν
∂p

(k)
2

∂y in Ω2,

∂u
(k)
2,1

∂x +
∂u

(k)
2,2

∂y = 0 in Ω2,

ν ∂
∂x u

(k)
2,1 − p

(k)
2 − γu(k)

2,1 = ν ∂
∂x u

(k−1)
1,1 − p(k−1)

1 − γu(k−1)
1,1 at x = 0,

ν ∂
∂x u

(k)
2,2 − γu

(k)
2,2 = ν ∂

∂x u
(k−1)
1,2 − γu(k−1)

1,2 at x = 0,

u
(k)
2,1 : bounded at +∞,

u
(k)
2,2 : bounded at +∞,

p
(k)
2 : bounded at +∞.

(35)

We employ the Fourier transform for the local Schwarz subproblems (34), (35). The Fourier trans-formed velocity components are given by the mathematical expressions (9), (10), (11), (12). TheFourier transformed pressure fields are given by p̂(k)
1 = D(k)

1 e |ξ|x , p̂(k)
2 = C(k)

2 e−|ξ|x . We substitutethe velocities and pressure fields back to the transmission conditions (34)4, (34)5, (35)4, (35)5, andby doing some algebraic manipulations we obtain
B(k)

1 (2γ + 2ν|ξ|)−D(k)
1 = B(k−1)

2 (2γ − 2ν|ξ| )− C(k−1)
2 , (36)

B(k)
1

(
2ν2|ξ|2 + 2νγ|ξ|

)
+D(k)

1 (2ν|ξ| + γ) = B(k−1)
2

(
2ν2|ξ|2 − 2νγ|ξ|

)
+ C(k−1)

2 (γ − 2ν|ξ|),(37)
B(k)

2 ( 2γ + 2ν|ξ|) + C(k)
2 = B(k−1)

1 ( 2γ − 2ν|ξ|) +D(k−1)
1 , (38)

B(k)
2

(
2νγ|ξ|+ 2ν2|ξ|2

)
− C(k)

2 (2ν|ξ|+ γ) = B(k−1)
1

(
2ν2|ξ|2 − 2νγ|ξ|

)
+D(k−1)

1 (2ν|ξ| − γ).(39)
We pick (38) and we obtain the coefficients

D(k)
1 = B(k+1)

2 (2γ + 2ν|ξ|) + C(k+1)
2 − B(k)

1 ( 2γ − 2ν|ξ|). (40)

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 10We substitiute the coefficients (40) back to (36) and we obtain
4©B(k)

1 = B(k+1)
2 (2γ + 2ν|ξ|) + C(k+1)

2 + B(k−1)
2 (2γ − 2ν|ξ|)− C(k−1)

2 . (41)The equation (40) can take the following form
4©D(k)

1 = B(k+1)
2 (2γ+2ν|ξ|)2+C(k+1)

2 (2γ+2ν|ξ|)−B(k−1)
2 (2γ−2ν|ξ|)2+C(k−1)

2 (2γ−2ν|ξ|) (42)
by employing (41). We take the relation (37), multiply with 4γ, and then plug in (41), (42) to obtain

k1B(k+1)
2 + k2C(k+1)

2 = k3B(k−1)
2 + k4C(k−1)

2 (43)
where k1, k2, k3, k4 are given by the relations below

k1 = 12
(
ν|ξ|+

γ

3

)
(ν|ξ|+ γ)2,

k2 = 6ν2|ξ|2 + 8ν|ξ|γ + 2γ2,

k3 = 12|ξ|3ν3 − 4|ξ|2γν2 − 12|ξ|γ2ν + 4γ3,

k4 = 6ν2|ξ|2 − 8ν|ξ|γ + 2γ2.

In the same fashion, we pick (39), multiply with 4γ, and then exploit the expressions (41), (42) toderive the equation
q1B(k+1)

2 + q2C(k+1)
2 = q3B(k−1)

2 + q4C(k−1)
2 (44)where q1, q2, q3, q4 are provided by the expressions

q1 = 12|ξ|3ν2 + 4|ξ|2γν2 − 12|ξ|γ2ν − 4γ3,

q2 = 6ν2|ξ|2 + 8νγ|ξ|+ 2γ2,

q3 = 12
(
ν|ξ| −

γ

3

)
(ν|ξ| − γ)2,

q4 = 6ν2|ξ|2 − 8νγ|ξ|+ 2γ2.

We take (43), (44) and after some algebraic manipulations we obtain a stationary iteration[
B(k+1)

2

C
(k+1)
2

]
=

[
3ν2|ξ|2−4ν|ξ|γ+γ2

3ν2|ξ|2+4ν|ξ|γ+γ2 0

0 3ν2|ξ|2−4ν|ξ|γ+γ2

3ν2|ξ|2+4ν|ξ|γ+γ2

]
︸ ︷︷ ︸

ΨOSM

[
B(k−1)

2

C
(k−1)
2

]
. (45)

The eigenvalue of the Schwarz iteration matrix ΨOSM of multiplicity two is provided by the formula
µd =

3ν2|ξ|2 − 4ν|ξ|γ + γ2

3ν2|ξ|2 + 4ν|ξ|γ + γ2
.

As a consequence, the contraction factor is
r2
OSM = |µd |2 =

|3ν2|ξ|2 − 4ν|ξ|γ + γ2|2

|3ν2|ξ|2 + 4ν|ξ|γ + γ2|2 .

�

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 115. Non-overlapping Optimized Schwarz Algorithm-Second Order IC
More Sophisticated Schwarz methods arise by the appropriate modification of the interfaceconditions. We can employ the Optimized Schwarz algorithms imposing second order transmissionconditions. More precisely, we go back to the algorithm prescribed by (31), (32), go to the interfaceconditions and instead of γ we use the symbol S , where S = q

(
1 + ξ2

).
Theorem 4. The contraction factor of the non-overlapping Schwarz algorithm (Second order IC) is
given by the mathematical expression

r2
OSM,SOIC(ν, q, ξ) =

|3ν2|ξ|2 − 4ν|ξ|q
(

1 + ξ2
)

+ q2
(

1 + ξ2
)2 |2

|3ν2|ξ|2 + 4ν|ξ|q (1 + ξ2) + q2 (1 + ξ2)2 |2
(46)

where q > 0.

Proof. The calculations follow through in the same spirit as the Optimized Schwarz methods withthe Robin transmission conditions. Instead of γ, the symbol S is used and the convergence factoris obtained naturally. �

Corollary 1. The reduction factor of The Parallel Schwarz method (Dirichlet IC) given by (5)
satisfies the following

rPSM,D(ξ,H) =



1, H = 0

0, |ξ| → +∞

0, H → +∞

< 1, ξ > 0.

(47)

Proof. The result (47)1 occurs by replacing H = 0 back to the formula (5). As a consequence, itmeans that the Schwarz method stagnates without overlap, something which is very usual in theliterature. The (47)2 is obtained by taking the limit of (5) as the Fourier frequency tends to +∞.The (47)3 is coming from the fact that when the overlap is sufficiently large, the convergence factorturns to be zero. The ultimate result (47)4 comes from the fact that for non zero Fourier frequency,the convergence factor is strictly less than 1. �

Corollary 2. The reduction factor of the Alternating Schwarz method (Neumann IC) given by (22),
satisfies the relations

rASM,N(ξ,H) =



1, H = 0

0, |ξ| → +∞

0, H → +∞

< 1, ξ > 0.

(48)

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 12

Proof. The reduction factor is a function that depends on the size of the overlap and the Fourierfrequency. Consequently, for zero overlap, the function becomes one and this leads to stagnation ofthe algorithm ((48)1). When the Fourier number grows large, the function goes to zero as prescribedby (48)2. Moving to (48)3, a big overlap leads to better convergence because the contraction factorrapidly tends to zero. Last but not least, for finite Fourier number, the convergence factor is strictlyless than one ((48)4). �

Corollary 3. The contraction factor of the non-overlapping Optimized Schwarz method (Robin IC)
given by (33) satisfies the properties

r2
OSM(ξ, ν, γ) =



1, γ = 0

1, γ → +∞

1, |ξ| → +∞

< 1, ξ ∈ (0,+∞)

0, γ = γ+ = 3ν|ξ|, γ = γ− = ν|ξ|.

(49)

Proof. The contraction factor depends on the kinematic viscosity, the Fourier frequency and theparameter γ. The first three properties in (49) are straighforward to obtain. Taking the Robinparameter to be zero or tend to infinity gives a stagnant Schwarz algorithm. In addition, when theFourier frequency tends to infinity, the contraction factor becomes 1. For finite Fourier frequency(not growing to infinity) the reduction factor is less than 1. Lastly, the values of the Robin parameterthat make the contraction factor zero are γ+ = 3ν|ξ| and γ− = ν|ξ| and can obtained by solving atrinomial equation appearing in the numerator of the contraction factor. �

Corollary 4. If γ = mν|ξ|, m ∈ Z+ − {1, 3}, then the convergence factor (33) does not depend on
viscosity and Fourier frequency.

Proof. By substitution, we obtain
r2
OSM(ξ, ν, γ) =

|3ν2|ξ|2 − 4ν|ξ|γ + γ2|2

|3ν2|ξ|2 + 4ν|ξ|γ + γ2|2

=
|3ν2|ξ|2 − 4ν|ξ|mν|ξ|+m2|ξ|2ν2|2

|3ν2|ξ|2 + 4ν|ξ|mν|ξ|+m2|ξ|2ν2|2

=
|ν2|ξ|2

(
m2 − 4m + 3

)
|2

|ν2|ξ|2 (m2 + 4m + 3) |2

=
|m2 − 4m + 3|2

|m2 + 4m + 3|2 .

�

Corollary 5. The contraction factor of non-overlapping optimised Schwarz method (Second order
IC) given by (46) satisfies the properties

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Eur. J. Math. Anal. 10.28924/ada/ma.5.6 13

r2
OSM,SOIC(ν, q, ξ) =



1, q = 0

1, q → +∞

1, |ξ| → +∞

< 1, ξ ∈ (0,+∞)

0, q = q+ = 3ν|ξ|
1+ξ2 , q = q− = ν|ξ|

1+ξ2 .

(50)

Proof. The first three relations in (50) can directly be derived by taking the appropriate limits forthe parameter q and the Fourier frequency ξ. For finite Fourier frequency, the reduction factor isless than one. Ultimately, for the indicated parameters q− and q+ the contraction factor becomeszero. �

Corollary 6. If q = mν|ξ|(1 + ξ2)−1, m ∈ Z+ − {1, 3}, then the convergence factor (46) does not
depend on viscosity and Fourier frequency.

Proof. The proof follows by substitution of the q parameter back to (46). The expression obtainedis identical to the one appearing in the corollary 4. �

6. Numerical Evidence-Convergence Curves
In this section, the convergence curves are presented for each one of the Schwarz algorithms. Inthe cases of Oprimised Schwarz methods with Robin and second order transmission conditions, weconsider γ = ν and q = ν. The convergence curves are presented below.

Figure 2. Convergence rate of Schwarz method using Dirichlet IC for varying overlap.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.6 14

Figure 3. Convergence rate of Schwarz method using Neumann IC for varying overlap.

Figure 4. Convergence rate of Schwarz method using Robin IC for varying viscosity.

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 15

Figure 5. Convergence rate of Schwarz method using second order IC for varying viscosity.

Figure 6. Comparison of convergence rates for all Schwarz methods.
Employing the graphs of the convergence rates in FIGURE 2 and FIGURE 3, we can comparethe Schwarz methods using Dirichlet and Neumann Interface conditions. We notice that when

https://doi.org/10.28924/ada/ma.5.6


Eur. J. Math. Anal. 10.28924/ada/ma.5.6 16Neumann transmission conditions are imposed, the convergence rate decays rapidly for increasingFourier modes, whereas using Dirichlet conditions makes the convergence slower. In addition, itis evident that when the overlap between the subdomains is larger then this enhances the overallconvergence which is the expected result when using the Classical Schwarz methods. In figures 4and 5 we have the convergence curves of non-overlapping Optimized Schwarz methods (FIGURE4-Robin IC, FIGURE 5-Second Order IC) for varying values of the viscosity. We notice that thesealgorithms have better convergence for small Fourier frequencies but as the Fourier frequencygrows to infinity, the reduction factor tends to 1. We also notice that when we tune the parametersof these Optimised methods, we can choose values to make the convergence rate equal to zero.Last but not least, we compare all the convergence curves and obtain FIGURE 6, which indicatesthat the Schwarz methods with Neummann and Dirichlet transmission conditions are slower forlow frequencies, and the Optimised methods perform better in this regime. However, the problemsoccur when the frequencies are large which means that the reduction factor tends to 1, which isnot desirable when dealing with Schwarz algorithms.
7. Conclusions

In this work we focused on the convergence analysis of the Schwarz algorithms for Stokes-Stokesconfiguration for varying interface conditions. We carried out the analysis using partial Fouriertransform and we obtained the contraction factors for each one of the methods introduced. Afterconducting the convergence analysis, we notice that the Neumann conditions result in faster decayof reduction factor when ξ grows sufficiently large compared to the Dirichlet IC. The OptimisedSchwarz methods have advantage in the low frequency regime, but as the Fourier number growsthe contraction rate tends to one which is not desirable behavior. The convergence analysis forStokes-Stokes configuration is useful for studying the behavior of Schwarz algorithms and gettinga general insight. So far there is such analysis for Stokes-Darcy coupling [18], as a result thiswork could enrich the existing mathematical literature.
References

[1] M.J. Gander, Optimized Schwarz methods for Helmholtz problems, in: Proceedings of the 13th International Con-ference on Domain Decomposition, CIMNE (2001) 245-252.[2] M.J. Gander, L. Halpern, F. Nataf, Optimized Schwarz methods, in: Proceedings of the 12th International Conferenceon Domain Decomposition, ddm.org (2000) 15-27.[3] P.L. Lions, On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in: T. Chan, R. Glowinski,J. Periaux, O.B. Widlund (Eds.), Third International Symposium on Domain Decomposition Methods for PartialDifferential Equations, SIAM (1990) 202-223.[4] M.J. Gander, G. Wanner, The origins of the alternating Schwarz method, in: Domain Decomposition Methods inScience and Engineering XXI, LNCSE, Springer-Verlag (2014) 487-496.[5] M.J. Gander, Schwarz methods over the course of time, Elec. Trans. Numer. Anal. 31 (2008) 228-255.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.6 17

[6] H.A. Schwarz, Uber einen Grenzubergang durch alternierendes Verfahren, Vierteljahrsschrift der NaturforschendenGesellschaft in Zurich 15 (1870) 272-286.[7] V. Dolean, P. Jolivet, F. Nataf, An introduction to domain decomposition methods: Algorithms, theory, and parallelimplementation, SIAM (2016).[8] G. Ciaramella, M.J. Gander, Iterative methods and preconditioners for systems of linear equations, SIAM (2022).[9] B. Smith, P. Bjorstad, W. Gropp, Domain decomposition: Parallel multilevel methods for elliptic partial differentialequations, Cambridge University Press.[10] A. Quarteroni, A. Valli, Domain decomposition methods for partial differential equations, Oxford Science Publications(1999).[11] O. Ernst, M.J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, in: I. Graham,T. Hou, O. Lakkis, R. Scheichl (Eds.), Numerical Analysis of Multiscale Problems, Springer Verlag (2012) 325-363.[12] M.J. Gander, H. Zhang, Decomposition de domaine et probleme de Helmholtz: Thirty years after and still unique,in: Domain Decomposition Methods in Science and Engineering XXVI, LNCSE, Springer-Verlag (2021).[13] P.-L. Lions, On the Schwarz alternating method. I, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux (Eds.), FirstInternational Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia(1988) 1-42.[14] V. Dolean, M.J. Gander, A. Kyriakis, Optimizing transmission conditions for multiple subdomains in the Magnetotel-luric approximation of Maxwell’s equations, in: Domain Decomposition Methods in Science and Engineering XXVI,LNCSE, Springer-Verlag (2021).[15] A. Kyriakis, Scalable domain decomposition methods for time harmonic wave propagation problems, Ph.D. thesis,University of Strathclyde (2021).[16] V. Dolean, M.J. Gander, A. Kyriakis, Closed form optimized transmission conditions for complex diffusion with manysubdomains, SIAM J. Sci. Comput. 45 (2023) A829-A848.[17] A. Kyriakis, Analysis of Schwarz algorithms for a scalar elliptic problem, Adv. Appl. Math. Sci. 22 (12) (2023)2227-2242.[18] M. Discacciati, G. Giorda, Optimized Schwarz methods for the Stokes-Darcy coupling, IMA J. Numer. Anal. 38 (4)(2018) 1959–1983.

https://doi.org/10.28924/ada/ma.5.6

	1. Introduction
	2. Parallel Schwarz Method-Dirichlet IC
	3. Alternating Schwarz Method-Neumann IC
	4. Non-overlapping Optimized Schwarz Algorithm-Robin IC
	5. Non-overlapping Optimized Schwarz Algorithm-Second Order IC
	6. Numerical Evidence-Convergence Curves
	7. Conclusions
	References

