Scalability analysis for two dimensional chains of fixed size subdo-

In this subsection we study the scalability properties of OSMs for a two dimensional chain of fixed size subdomains. For the one dimensional chain analysis, we refer the interested reader to [28, Section 4.1]. Let us considerL>0 andδ, 0<δ<^L₂, and define the grid points a_j for j=1, . . . ,N+1 andb_j for j=0, . . . ,N as shown in Figure 1.3. The j-th subdomain of the chain is a rectangle of dimensionΩj :=(aj,bj)×(0,bL), andΩ:= ∪^N_j=1Ωj. We are interested in the solution to

−∆u=f inΩ, u=g on∂Ω. (1.4.1)

We consider directly the error equation and we define the errorseⁿ_j :=u−uⁿ_j, whereuⁿ_j are the iterates of the OSM. In the error form, the overlapping OSM with Robin transmission conditions with parameterpis given by

−∆eⁿ_j =0 inΩj, eⁿ_j(·, 0)=0,eⁿ_j(·,L)b =0,

∂xeⁿ_j(aj,·)−peⁿ_j(aj,·)=∂xeⁿ_j−1⁻¹(aj,·)−peⁿ_j−1⁻¹(aj,·),

∂xeⁿ_j(b_j,·)+peⁿ_j(b_j,·)=∂xeⁿ_j+⁻₁¹(b_j,·)+peⁿ_j+⁻₁¹(b_j,·),

(1.4.2)

forj=2, . . . ,N−1, and

−∆ 1 = Ω1

To study the iteration, we use the Fourier expansioneⁿ_j(x,y)=P

k∈K vⁿ_j(x,k) sin(k y) with K :=n_π

Lb,^2π

bL , . . .o

. Inserting this expansion into (1.4.2) and (1.4.3), the Fourier coefficients vⁿ_j satisfy solution to (1.4.4) is given by

vⁿ_j(x,k)=Rⁿ₋⁻¹(aj)

where

We are ready to prove the scalability of the OSM in the overlapping case.

Theorem 1.4.1. Recall(1.4.7)and defineϕ(k,δ,p) := |g3−pg1|+|g4−pg2|. The overlapping OSM(1.4.2)is scalable, in the sense that

ρ(T_2D^O (k,δ,p))≤ kT_2D^O(k,δ,p)k∞≤max

k max{ϕ(k,δ,p),kTe₁(k,δ,p)k∞}<1, independently of N for every p≥0.

Proof. Because of the structure ofT_2D^O , the normkT_2D^O k∞is given by

By computing the derivative ofϕwith respect topwe find

∂ϕ

∂p =− 2ke^2δk+2kL−2ke^2δk

k²e^4δk+2ke^4δk+p²e^4δke^2kL+2k²e^2δk−2e^2δkp²e^kL+(p−k)² forp<k,

∂ϕ

∂p = 2ke^2δk+2kL−2ke^2δk

k²e^4δk +2ke^4δk+p²e^4δke^2kL+2p²e^2δk −2e^2δkk²e^kL+(k−p)² forp>k.

Analyzing the signs of these derivatives, we see thatϕ(k,δ,p) is strictly decreasing for p<kand it is strictly increasing forp>k, thus it reaches a minimum forp=k. Therefore the maximum ofϕ(k,δ,p) with respect to the variable p is obtained for p=0 and for p→ +∞:

ϕ(k,δ,p)≤max©

ϕ(k,δ, 0) , lim_p

→∞ϕ(k,δ,p)ª . Forp=0,δ>0 andL>0 we have

ϕ(k,δ,p=0)=e^2δk−e^−2δk+e^kL−e^−kL e^kL+2δk−e^−kL−2δk

= sinh(2δk)+sinh(kL)

sinh(kL) cosh(2δk)+sinh(2δk) cosh(kL)<1, and, under the same conditions,

plim→∞ϕ(k,δ,p)= sinh(2δk)+sinh(kL)

sinh(kL) cosh(2δk)+sinh(2δk) cosh(kL)=ϕ(k,δ, 0)<1.

Hence, it holds thatϕ(k,δ,p)≤ϕ(k,δ, 0)<1. We now focus onkTe1k∞andkTe2k∞. Notice thatkTe₁k∞= kTe₂k∞and

kTe1k∞=

¯

¯

¯

¯

¯

(k+p)e^−kL+(k−p)e^kL (k+p)e^k(L+2δ)+(k−p)e^−k(L+2δ)

¯

¯

¯

¯

¯

=

¯

¯

¯

¯

kcosh(kL)−psinh(kL) kcosh(k(L+2δ))+psinh(k(L+2δ))

¯

¯

¯

¯<1.

In order to get a bound independently ofk, we observe that lim

k→∞ϕ(k,δ,p)= lim

k→∞kTe1k∞= 0 if δ>0. Therefore defining ¯ρ(δ) :=max_kmax{ϕ(k,δ,p),kTe₁(k,δ,p)k∞}, we see that kT_2D^O k∞=max{ϕ,kTe1k,kTe2k}<ρ¯(δ)<1, for everyδ,p>0.

For the case without overlap, we need a further argument because forδ=0 bothρ(T_2D^O ) andkT_2D^Ok∞are less than one for any finite frequencyk, but tend to one ask→ ∞. One can therefore construct a situation where the method would not be scalable as follows:

suppose we haveN subdomains, and on the j-th subdomain we choose as initial guess e⁰_j the j-th frequencye⁰_j =eˆ⁰_jsin(j^π

Lby). Then the convergence of the method is deter-mined by the frequency which maximizesρ(T_2D^O (k)). When the number of subdomains N becomes large, this maximum is attained for the largest frequencykN =N ^π

L yb since

ρ(T_2D^O(k))→1 ask→ ∞. Thus, every time we add a subdomain to the chain with a new initial condition on the interfaceN+1 according to our rule, the convergence rate of the method deteriorates fromρ(T_2D^O (N^π

Lb)) toρ(T_2D^O ((N+1)^π

Lb)) and the scalability property is lost. Theorem 1.4.2 gives however a sufficient condition such that the OSM is weakly scal-able also without overlap, and to see this we introduce the vectoreⁿwitheⁿ_k = krⁿ(k)k∞

whererⁿ(k) contains the Robin traces at the interfaces of thek-th Fourier mode.

Theorem 1.4.2. Given a toleranceTol, and supposing there exists ak that does not depend˜ on N such that e⁰_k<Tolfor every k>k, then the OSM without overlap,˜ δ=0, and p>0is weakly scalable.

Proof. Suppose that the initial guess satisfieske⁰k∞>Tol, since otherwise there is noth-ing to prove. Then, due to the hypothesis, we have that maxπ

L≤k≤k˜e⁰_k>Tol. We now show that the method contracts with aρindependent of the number of subdomains up to the toleranceTol, and therefore we have scalability. Indeed, for everyksuch that^π_L≤k≤k˜

eⁿ_k= krⁿ(k)k∞≤ kT_2D^O(k)k∞krⁿ⁻¹(k)k∞≤ kT_2D^O ( ¯k)k∞krⁿ⁻¹(k)k∞= kT_2D^O ( ¯k)k∞e_kⁿ⁻¹, wherekT_2D^O( ¯k)k∞=maxπ

L≤k≤k˜kT_2D^O (k)k∞<1 becausekT_2D^O (k)k∞is strictly less than 1 for every finitek. Now fork>k˜,

eⁿ_k = krⁿ(k)k∞≤ kT_2D^O(k)k∞krⁿ⁻¹(k)k∞≤ krⁿ⁻¹(k)k∞=eⁿ_k⁻¹,

sincekT_2D^O (k)k∞≤1. Therefore we observe that the method does not increase the error for the frequenciesk>k˜while it contracts for the other frequencies with a contraction factor of at least ¯ρ= kT_2D^O ( ¯k)k∞<1. Hence, as long askeⁿk∞>Tol^{, we have}keⁿk∞≤ρ¯ⁿke⁰k∞

with ¯ρindependent ofN.

The technical assumption in Theorem 1.4.2 on the frequency content of the initial error is not restrictive, since in a numerical implementation we have a maximum frequency kmax which can be represented by the grid. Choosing ˜k=kmax, the hypothesis of The-orem 1.4.2 is verified. Note also that without overlap,δ=0, we have thatkT_2D^Ok∞=1 forp=0 orp→ ∞. Therefore we can not conclude that the method is scalable in these two cases. Forp=0, the OSM exchanges only partial derivatives information on the in-terface. Forp→ ∞, we obtain the classical Schwarz algorithm and it is well known that without overlap (δ=0), the method does not converge. We finally show the behaviour of p7→ kT_2D^O (k,δ,p)k∞for a fixed pair (δ,k) in Figure 1.4. According to the proof of Theorem 1.4.1, the minimum of the functionp7→ϕ(k,δ,p) is located atp=k. Even though it is a minimum forϕ(k,δ,p) and not necessarily forkT_2D^O (k,δ,p)k∞orρ(T_2D^O ), we might de-duce from Figure 1.4 that in order to eliminate thek-th frequency, a good choice would be to setp:=kin the OSM. For the Laplace equation, it has been shown for two subdomains that settingp:=kleads to a vanishing convergence factorρ(k) for the frequencyk[74].

In the case of many subdomains, a similar result has not been proved yet, but Figure 1.4 indicates that it might hold as well.

0 20 40 60 80 100 p

0 0.005

0.01 0.015 0.02

kTO 2D)k

Figure 1.4: Infinity norm of the iteration matrixT_2D^O as a function ofpforL=1,Lb=1,δ= 0.1,k=20,N=50.

a₀ a₁ ^{· · ·} a_j−1 a_j ^{· · ·} a_N−1 a_N x

Ω1 · · · Ωj · · · ΩN

Lb

L L L

Figure 1.5: Nonoverlapping domain decomposition in two dimensions. Notice thataj= j L.

1.4.1.2 Scalability analysis for the Dirichlet-Neumann method

We now consider a two dimensional problem decomposed into nonoverlapping subdo-mains as shown in Figure 1.5. The error form of the parallel Dirichlet Neumann method (PDNM) is given by

−∆eⁿ_j =0 inΩj, eⁿ_j(·, 0)=0,eⁿ_j(·,bL)=0,

eⁿ_j(aj,·)=(1−θ)eⁿ⁻¹_j (aj,·)+θeⁿ⁻¹_j+1(aj,·),

∂xeⁿ_j(aj−1,·)=(1−µ)∂xeⁿ⁻¹_j (aj−1,·)+µ∂xeⁿ⁻¹_j−1(aj−1,·), forj=2, . . . ,N−1, and

−∆eⁿ₁=0 inΩ1, e₁ⁿ(·, 0)=0,eⁿ₁(·,bL)=0, e₁ⁿ(a₀,·)=0,

e₁ⁿ(a1,·)=(1−θ)eⁿ⁻¹₁ (a1,·)+θe₂ⁿ⁻¹(a1,·),

and

where (1.4.10) may be rewritten as

eⁿ=T_2D^{D N}eⁿ⁻¹, infinity-norm is not suitable to bound the spectral radius and conclude convergence and scal-ability. Nevertheless in Theorem 1.4.3, we prove scalability of the PDNM under certain assumptions on the parametersµ,θand using similarity arguments..

Theorem 1.4.3. Denote by kminthe minimum frequency and defineα(x) :=1/ cosh(x). If θ=µ, then

minL)², which implies that the PDNM is convergent and scalable.

We show in Figure 1.6 the function ¯ρ(µ) for the caseLb=1, that isk_min=π. The proof of Theorem 1.4.3 relies on the following lemma.

Lemma 1.4.4. Letα(x) :=1/ cosh(x). Then for any x∈(0,∞)such thatcosh(x)>2it holds

0 0.2 0.4 0.6 0.8 1

We are now ready to prove Theorem 1.4.3.

Proof. Ifµ=θ, the matrixT_2D^{D N}has the structure

whereB,eB,b B∈R^2×2. We introduce an invertible block diagonal matrix

Notice that

T_off^>T_off=diag Ã

0, 0,4µ²

γ²₂ , . . . ,4µ² γ²₂ , 0, 0

! , and henceq

ρ(T_off^>T_off)=²_γ^µ₂. Now, we focus on the termρ(T_diag^> T_diag). The block diagonal structure ofT_diag^> T_diagallows us to write

ρ(T_diag^> T_diag)= q

max©

ρ(Ce^>Ce),ρ(Cb^>Cb),ρ(C^>C)ª

. (1.4.13)

The evaluation of the spectral radiiρ(Ce^>Ce),ρ(Cb^>Cb), andρ(C^>C) leads to the analysis of cumbersome formulas, and we thus bound instead the spectral radii by the correspond-ing infinity-norms. To do so, settcorrespond-ingde1:=γ1andde2:=kγ2, we obtain

ρ(Ce^>C)e =ρ(GeBe^>Ge⁻¹Ge⁻¹BeG)e ≤ kGeBe^>Ge⁻¹Ge⁻¹BeGke ∞=2µ²−2µ+1.

Next, we setdb₁:=γ2anddb₂:=kγ1and get

ρ(C^>C)=ρ(GBb ^>Gb⁻¹Gb⁻¹BG)b ≤ kGBe ^>Ge⁻¹Ge⁻¹BGek∞

=max (

1−µ, 1−µ+µ²(e^−kL−e^kL)⁴ (e^−kL+e^kL)⁴

)

≤1−µ+µ²,

where the fact that^(e_(e⁻_−kL^kL−ekL)4

+e^kL)⁴ ≤1 for anykis used. Now, a direct calculation shows that 2µ²−2µ+1≤2µ²−2µ+1+ 4µ(1−µ)

(e^kminL+e^−kminL)²≤1−µ+µ², for anyµ∈(0, 1). Therefore, we obtain

kT_diagk2=ρ(T_diag^> T_diag)≤ q

1−µ+µ². Recalling (1.4.12) and (1.4.13), we conclude that

ρ(T_2D^{D N})≤ kTdiagk2+ kToffk2≤ q

1−µ+µ²+2µ γ2

≤ q

1−µ+µ²+ 2µ

(e^kminL+e^−kminL)=: ¯ρ(µ),

which is the first statement of the theorem. The second part follows now from Lemma 1.4.4 by observing that if ¯ρ(µ)<1, thenρ(T_2D^{D N})≤ρ¯(µ)<1 where ¯ρ(µ) is independent of N.

1.4.1.3 Scalability analysis for the Neumann-Neumann method

Finally we study the convergence of the Neumann-Neumann method (NNM). For our model problem, the error form for the NNM is the following: first solve

−∆eⁿ_j =0 inΩj, eⁿ_j(·, 0)=0,eⁿ_j(·,L)=0, eⁿ_j(aj−1,·)=Dⁿ_j−1,eⁿ_j(aj,·)=Dⁿ_j, forj=2, . . . ,N−1 and

−∆eⁿ₁=0 inΩ1, eⁿ₁(·, 0)=0,eⁿ₁(·,L)=0, eⁿ₁(a0,·)=0,eⁿ₁(a1,·)=D₁ⁿ,

−∆eⁿ_N=0 inΩN, eⁿ_N(·, 0)=0,eⁿ_N(·,L)=0, eⁿ_N(aN−1,·)=Dⁿ_N−1,eⁿ_N(aN,·)=0, then solve

−∆ψⁿ_j =0 inΩj,

∂xψⁿ_j(·, 0)=0,ψⁿ_j(·,L)=0,

∂xψⁿ_j(a_j−1,·)=∂xeⁿ_j(a_j−1,·)−∂xeⁿ_j−1(a_j−1,·),

∂xψⁿ_j(a_j,·)=∂xeⁿ_j(a_j,·)−∂xeⁿ_j+1(a_j,·), forj=2, . . . ,N−1 and

−∆ψⁿ₁=0 inΩ1,

ψⁿ₁(·, 0)=0,ψⁿ₁(·,L)=0,ψⁿ₁(a₀,·)=0,

∂xψⁿ1(a1,·)=∂xe₁ⁿ(a1,·)−∂xeⁿ₂(a1,·),

and −∆ψⁿ_N=0 inΩN,

ψⁿN(·, 0)=0,ψⁿN(·,L)=0,ψⁿN(aN,·)=0,

∂xψⁿN(a_N−1,·)=∂xeⁿ_N(a_N−1,·)−∂xeⁿ_N−1(a_N−1,·), and finally set

Dⁿ_j⁺¹:=Dⁿ_j −ϑ(ψⁿ_j+1(a_j,·)+ψⁿ_j(a_j,·)), (1.4.14) forj=1,· · ·,N−1, whereϑ>0. We expand botheⁿ_j andψⁿ_j in Fourier series ,

eⁿ_j(x,y)= X∞ m=1

vⁿ_j(x,k) sin(k y), ψⁿ_j(x,y)= X∞ m=1

wⁿ_j(x,k) sin(k y), wherek∈K. The Fourier coefficientsvⁿ_j(x,k) andwⁿ_j(x,k) solve the problems

k²vⁿ_j −∂xxvⁿ_j =0 in (a_j−1,a_j), vⁿ_j(a_j−1,k)=Dⁿ_j−1,

vⁿ_j(aj,k)=Dⁿj,

k²wⁿ_j −∂xxwⁿ_j =0 in (a_j−1,a_j),

∂xwⁿ_j(a_j−1,k)=∂xvⁿ_j(a_j−1,k)−∂xvⁿ_j−1(a_j−1,k),

∂xwⁿ_j(aj,k)=∂xvⁿ_j(aj,k)−∂xvⁿ_j+1(aj,k),

forj=2, . . . ,N−1, and which is used to solve the problems inwⁿ_j, and we obtain

wⁿ_j(x,k)= 1

We definee = D₁,D₂,· · ·,D_N−1 , and write equations (1.4.15)-(1.4.16) aseⁿ⁺¹=T_2D e , where the iteration matrixT_2D^{N N}is given by

T_2D^{N N}=

Proof. The infinity-norm ofT_2D^{N N}is given by kT_2D^{N N}k∞=max©

1. Therefore, we get kT_2D^{N N}k∞=max holds, meaning thatγ1>2, and since the mapk7→γ1=2 sinh(kL) is strictly increasing in k, it suffices thatγ1>2 is satisfied for justk=^π

Lb. Hence the condition becomes sinh(kL)>

1 or equivalentlykL>arcsinh(1)=ln(1+p

2), which concludes the proof.

1.4.1.4 Numerical results

We close this subsection with a numerical experiment. We start with a random initial guess and we apply the different methods to solve (1.4.1) withf =g=0. We set the ge-ometric parameters equal tobL=L=1 and we discretize each subdomain square with N_h=100 interior unknowns. For the overlapping OSM we chooseδ=10h, whereh=_Nh¹₊₁ and we setp=π. For the PDNM, we setθ=λ=¹₂ while for the PNNMθ=¹₄. In Table 1.1,

N 10 20 30 40 50

OSM 13 13 13 13 13

PDNM 70 70 70 70 70

PNNM 6 6 6 6 6

Table 1.1: Number of iterations to reach convergence as the number of subdomainsN increases.

we report the number of iterations to reach convergence with a tolerance of Tol :=10⁻¹² for the different methods as the numberN of the subdomains increases. We can observe that every method requires a constant number of iterations to reach convergence. There-fore, these numerical experiments are in agreement with the theoretical results presented in this subsection. According to Table 1.1, it seems that the PNNM is the fastest method.

However, we remark that each iteration of the PNNM requires two subdomain solves, so its cost is comparable with the OSM. Moreover, the PNNM is extremely sensitive on the choice ofθ, see [26].

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