Advection Reaction Diffusion-Reaction Diffusion coupling

−η˜²− ²

p2(D+k_min)²(λ+1)(λD+k_min) (λ−1)⁴p

π(D+kmin) qD+kmin

1−λ −η˜²h¹², max_{kmin≤k≤π/h}|ρ(k,p^∗₂,q₂^∗)| ∼λ−²

p2λ(1+λ)p

(kmin+D) pπ(1−λ) h¹².

(2.1.23)

Proof. Guided by numerical experiments, forλ≥1 we make the ansatzp∼Cp+Ah¹², q∼Qh⁻¹+Bh⁻¹², and ˆk=Ckh⁻¹². First of all considering the equation∂kρ(ek,p,q)=0, we find setting to zero the first non zero termC_k =p

Cp·Q. Inserting this into (2.1.19) and comparing the two leading terms, we get the result. Similarly forλ<1, we make the ansatzp∼C_ph⁻¹+Ah⁻¹²,q∼Q+Bh¹² and ˆk=C_kh⁻¹² and we getC_k =

q C_pp

Q²+η˜². Substituting and matching the leading order terms we obtain the result.

If we set ˜η²=0, thenD =k_minand we recover the results of [78]. Note that in contrast to the one sided case, the convergence factor does not deteriorate to 1 ash→0, but it is bounded either by ¹_λ ifλ≥1 or byλifλ<1, so we obtain a non-overlapping OSM that converges independently of the mesh sizeh. We emphasize that the heterogeneity makes the method faster instead of presenting a difficulty. A heuristic explanation is that the heterogeneity tends to decouple the problems, making them less dependent one from the other. In contrast with other domain decomposition methods, OSMs can be tuned according to the physics and therefore they can benefit from this decoupling.

2.2 Advection Reaction Diffusion-Reaction Diffusion coupling

In this Section, we consider the domain decomposition described at the beginning of Sec-tion 2.1. InΩ1we have a reaction diffusion equation, while inΩ2we have an advection reaction diffusion equation. We allow the reaction and diffusion coefficients to be differ-ent among the subdomains. The OSM reads

(η²₁−ν1∆)uⁿ₁ = f, in Ω1,

(ν1∂x+S1)(u₁ⁿ)(0,·) = (ν2∂x−a·(1, 0)^>+S1)(uⁿ⁻¹₂ )(0,·), (η²₂+a· ∇ −ν2∆)uⁿ₂ = f, in Ω2,

(ν2∂x−a·(1, 0)^>−S₂)(u₂ⁿ)(0,·) = (ν1∂x−S₂)(uⁿ₁⁻¹)(0,·),

(2.2.1)

wherea=(a1,a2)^>. The additional term in the transmission conditions arises from the conservation of the flux in divergence form, see Chapter 6 in [139]. We first suppose a₂=0. Then we can solve the error equations in the subdomains through separation of variables and we obtaineⁿ_i =P

2ν2 . Insertinge1,e2into the transmission conditions we get ν1

The convergence factor is therefore given by ρ(k,σ1,σ2)=ν2λ−(k)−a1+σ1(k) dependence onk, the convergence factor becomes

ρ(k,σ1,σ2)=ν2

We can define two optimal operatorsS^opt_j associated to the eigenvaluesσ^opt₁ (k) :=ν2

pk²+δ²+

a1

2 andσ^opt₂ (k) :=ν1

q

k²+η˜²₁which lead to convergence in just two iterations.

2.2.1 Zeroth order single sided optimized transmission conditions

Following the strategy of the previous section, we chooseσ1(k),σ2(k) so that they coin-cide with the optimal choice for the frequencyk=p, i.e. σ1(k)=ν2 Theorem 2.2.1. The unique optimized Robin parameter p^∗solving the min-max problem

minp∈R max

kmin≤k≤kmax

|ρ(k,p)|,

is given by the unique root of the non linear equation

|ρ(p^∗,k_min)| = |ρ(p^∗,k_max)|.

Proof. The proof is very similar to the proof of Theorem 2.1.3, therefore we just sketch the main steps. We start observing thatρ(k,p) has only one zero located atk=p and ρ(k,p)>0 ∀k,p. Thus we may neglect the absolute value. Analysing the derivative with respect to p, we find

sign

µ∂ρ(k,p)

∂p

¶

= −sign(k−p).

This implies that^∂ρ_∂^(k,p)_p >0 ifk<pand ^∂ρ_∂^(k,p)_p <0 ifk>p. We conclude thatpmust lie in the interval [kmin,kmax]. Similarly the derivative with respect toksatisfies^∂ρ_∂^(k_k^,p)<0 if k<pand ^∂ρ(k,p)_∂k >0 ifk>p. Hence, the local maxima with respect tokare located at the boundary pointsk=k_minandk=k_max. Repeating the final argument of Theorem 2.1.3 we get the result.

Since a closed form formula is again not available, we study the asymptotic behaviour for the optimal parameterp^∗when taking finer and finer meshes.

Theorem 2.2.2. If the physical parameters are fixed, kmax=^π_h and h is small enough, then the optimized Robin parameter p^∗satisfies

p^∗∼Ca·h⁻¹²,Ca= r

ν2(λ+1)π³ 2q

k_min² +η˜²₁λν2+2q

k_min² +δ²ν2−a1

´ p2ν2(λ+1) . Furthermore the asymptotic convergence factor is

kminmax≤k≤π/h|ρ(k,p^∗)| ∼1−h¹²

µCa(λ+1)² λπ

¶ .

Proof. We insert the ansatzp=Ca·h^−αinto the equation (2.1.12). Expanding for smallh, we get that

ρ(p,k_min)∼1−h^α

µC_a(λ+1)² λπ

¶ . On the other hand,

ρ(p,kmax)∼1+h^−α+1





 1 2

(λ+1)³

−2q

k_min² +η˜²₁λν2−2q

k_min² δ²ν2+a1

´ Caν2λ





. Comparing the first two terms we get the result.

2.2.2 Zeroth order two sided optimized transmission conditions

In this paragraph we generalize the previous transmission conditions, introducing an-other degree of freedomq. The operatorsSjare such that their eigenvalues are

σ1(k)=ν2

q

q²+δ²+a1

2 , σ2(k)=ν1

q

p²+η˜²₁, and the convergence factor becomes

ρ(k,p)= q

k²+η˜²₁− q

p²+η˜²₁

λ1

³p

k²+δ²+₂^a_ν¹₂´ +q

p²+η˜²₁·

pk²+δ²−p q²+δ² λq

k²+η˜²₁+³p

q²+δ²+₂^a_ν¹₂´.

In order to prove a similar result as in Theorem 2.1.9, we suppose that ˜η1=0, i.e. only dif-fusion is present inΩ1, anda1>0, i.e. the advection flux is pointing into the subdomain Ω2.

Theorem 2.2.3. There are two pairs of parameters(p^∗₁,q₁^∗)and(p₂^∗,q₂^∗)such that we ob-tain equioscillation between all the three local maxima located at the boundary extrema k_min,k_maxand at the interior pointk,e

|ρ(kmin,p^∗_j,q^∗_j)| = |ρ(kmax,p^∗_j,q^∗_j)| = |ρ(k,e p^∗_j,q^∗_j)| j=1, 2.

The optimal pair of parameters is the one which realizes the

(p^∗_j,qmin^∗_j),j=1,2|ρ(kmin,p^∗_j,q^∗_j)|.

Proof. Similarly to the proof of Theorem 2.1.9, we observe that the function admits two zeros, one located atk=p, the other atk=qdue to the choice of the transmission oper-ators. Computing the derivatives with respect to p and q we get

sign(∂|ρ|

∂p )= −sign(ρ)·sign(k−q)= −sign(k−p), sign(∂|ρ|

∂q )= −sign(ρ)·sign(k−p)= −sign(k−q).

We conclude that, at the optimum, bothp andq lie in [kmin,kmax], i.e. the function at the optimum has two zeros in the interval. Now we study the behaviour with respect to k. Computing the derivative with respect tok, we find that the potential local maxima are given by the roots of

pδ²+k²−p δ²+q² k(λk+p

q²+δ²+₂^a_ν¹₂)= p−k pk²+δ²³

pλ+p

k²+δ²+_2ν^a1₂

´.

With some algebraic manipulations, we find that a sufficient condition such that

(pλ+p

k²+δ²+a1/(2ν2))

has a monotonic behaviour with respect tokis thata1>0. Lettingp,qin [kmin,kmax], we have that the local maxima of the function are located atkmin,kmax,ke. Moreover we have

∂|ρ|

∂p |k=kmin>0, ∂|ρ|

∂q |k=kmin>0,

∂|ρ|

∂p |k=k_max<0, ∂|ρ|

∂q |k=k_max<0, (2.2.3)

∂|ρ|

∂p |_k=ek<0, ∂|ρ|

∂q |_k=ke>0.

We can thus repeat the same arguments as in the proof of Theorem 2.1.9 since all steps are now exclusively based on the sign of the partial derivatives with respect to the parameters, see (2.2.3), and the result follows.

Theorem 2.2.4. Let D := q

k_min² +δ². If the physical parametersη˜²₂,ν1,ν2,a₁ are fixed, kmax=^π_h and h goes to zero, the optimized two-sided Robin parameters are forλ≥1,

p₁^∗∼P1h⁻¹+E1h⁻¹²,q₁^∗∼Q1−F1h¹², max

k_min≤k≤^π_h|ρ(k,p^∗₁,q₁^∗)| ∼λ−^E_(P¹₁^π_λ+π^(λ+1)₎2h¹², with

P1:=π(λ−1)

2λ , Q1:= v u u t

D+k_min+_2ν^a1_2λ 1−¹_λ −δ²,

E₁:=

(2(P₁q

δ²+Q²₁+C²_h)(λ+1)ν2+P₁a₁)(λP₁+π)² 2λ²P₁ν2C_hπ(λ+1) , F₁:=

(2(P₁

qδ²+Q₁²+C_h²)(λ+1)ν2+P₁a₁)(2ν2(λk_min+

qδ²+Q₁²)+a₁)²

qδ²+Q²₁ 4λ²P1ν²₂C_hQ1(2ν2(λkmin+D)+a1) ,

C_h:= r

P₁(2

qδ²+Q²₁ν2(λ+1)+a₁) p2ν2(λ+1) . and forλ<1,

p^∗₂∼P2−E2h¹²,q₂^∗∼Q2h⁻¹+F2h⁻¹², max

k_min≤k≤^π_h|ρ(k,p^∗₂,q₂^∗)| ∼λ−^F₍²_λπ+^λπ(1_Q^+λ)

2)² h¹².

with

P₂:=

D+k_min+₂^a_ν¹₂

1−λ , Q₂:=π(λ−1)

2 ,

E₂:=((λ+1)(D_h²+P2Q2)ν2+^a1₂^Q2)(2ν2(λP2+D)+a1)² 2ν²₂D_hQ₂(2k_minλν2+2ν2D+a₁) , F₂:=

pλ+1q

(D+kmin)(λ+1)+₂^a_ν¹₂p

π(3λ−1)²

p2(1−λ²) ,

D_h:=

pQ₂(2P₂ν2(λ+1)+a₁) p2ν2(λ+1) .

Proof. The proof follows the same steps as in the proof of Theorem 2.1.10.

2.2.3 Advection tangential to the interface

In the previous section we restricted our study to the case of advection normal to the in-terface. Here we consider the other relevant physical case, namely advection tangential to the interface, so thata1=0 anda26=0 in (2.2.1). For homogeneous problems, this case has been studied through Fourier transform in unbouded domains, see for instance [64].

However, in [95] we have shown that for homogeneous problems with tangential advec-tion this procedure does not yield efficient optimized parameters. The reason behind this failure lies in the separation of variables technique which applied to the error equation,

(η²₁−ν1∆)eⁿ₁ = 0, in Ω1,

(ν1∂x+S1)(eⁿ₁)(0,·) = (ν2∂x+S1)(eⁿ⁻¹₂ )(0,·), (η²₂+a₂∂y−ν2∆)eⁿ₂ = 0, in Ω2,

(ν2∂x−S2)(eⁿ₂)(0,·) = (ν1∂x−S2)(eⁿ₁⁻¹)(0,·),

(2.2.4)

leads to eⁿ₁ =X

k∈V

ˆ

e₁ⁿ(0,k)e^λ1(k)xsin(k y) and e₂ⁿ= X

k∈V

ˆ

eⁿ₂(0,k)e^−λ2(k)xe

a2y

2ν2sin(k y), (2.2.5)

whereλ1(k)=p

k²+η˜1,λ2(k)=

p4ν²₂k²+4ν²₂η˜22+a₂²

2ν2 with ˜η²_j:=^η

2 j

νj. Since the functionsψk(y) := sin(k y) andφk(y) :=e

a2y

2ν2 sin(k y) are not orthogonal, it is not possible to get a recurrence relation which expresses ˆeⁿ_j(0,k) only as a function of ˆeⁿ_j⁻²(0,k) for eachk and j =1, 2.

Nevertheless, we propose a more general approach. First let us define two scalar prod-ucts, the classicalL²scalar product and the weighted scalar product

〈f,g〉 =2 L Z

Γf g d y, 〈f,g〉w=2 L Z

Γf g e⁻

a2y ν2 d y.

It follows that〈ψk,ψj〉 =δk,jand〈φk,φj〉w=δk,j. SettingS₁:=ν2λ2(p)IandS₂:=ν1λ1(q)I forp,q∈Rand inserting the expansions (2.2.5) into the transmission conditions of (2.2.4)

we obtain

We truncate the expansions fori,l>N, since higher frequencies are not represented by the numerical grid, and we project the first equation ontoψkwith respect the scalar prod-uct〈·,·〉and the second one ontoφjwith respect to the weighted scalar product〈·,·〉w,

eˆⁿ₁(0,k)(ν1λ1(k)+ν2λ2(p)) = Since for two given matricesA,Bthe spectral radius satisfiesρ(AB)=ρ(B A), we conclude thatρ( ˜D⁻¹₁ V D1D˜⁻¹₂ W D2)=ρ( ˜D⁻¹₂ W D2D˜₁⁻¹V D1) and therefore, in order to accelerate the method, we are interested in the minimization problem

p,q∈Rminρ(( ˜D⁻¹₁ V D1D˜⁻¹₂ W D2)(p,q)). (2.2.10) Remark2.2.5. Problem (2.2.10) does not have a closed formula solution. However in the next subsection we show its efficiency by solving numerically the minimization problem.

For these cases where the theoretical analysis falls short without providing any good es-timates, in subsection 2.5 we discuss a cheap and reliable numerical procedure to find optimized transmission conditions.

Remark2.2.6. Equation (2.2.10) is a straight generalization of the min-max problem (2.1.8).

Indeed, assuming that the functionsψkandφjare orthogonal, the matricesVandW are the identity matrix. Therefore equation (2.2.9) simplifies toeⁿ₁ =D¯eⁿ₁⁻²andeⁿ₂ =De¯ ⁿ₂⁻², where the diagonal matrix ¯Dsatisfies ( ¯D)_k,k=^ν_ν²₁^λ_λ²₁^(k)−ν_(k)+ν²₂^λ_λ²₂^(p)_(p)^ν_ν¹₂^λ_λ¹₂^(k)−ν_(k)+ν¹₁^λ_λ¹₁^(q)_(q). Since the eigen-values of a diagonal matrix are its diagonal entries we get that ifW =V =I,

p,q∈Rminρ(( ˜D₁⁻¹V D1D˜⁻¹₂ W D2)(p,q))=min

Remark2.2.7. The case of an arbitrary advection, i.e.a16=0 anda26=0 has been recently treated in [95] for homogeneous problems. Considering a heterogeneous problem with advection fieldsaj=(a_1j,a_2j)^>in domainΩj,j=1, 2, a separation of variables approach would lead to non orthogonal functionsψk(y)=e

a21y

2ν1 sin(k y) andφk(y)=e

a22y

2ν2 sin(k y) unless ₂^a_ν²¹

1 = ₂^a_ν²²₂, and thus it is not possible to obtain a recurrence relation as shown in (2.2.5). However the approach developed in this section can be readily applied. The sub-domain solutions are

e₁ⁿ(x,y)= X

k∈V

eˆⁿ_1,ke

a21y

2ν1 sin(k y)e^λ1(k)x, eⁿ₂(x,y)= X

k∈V

eˆⁿ_2,ke

a22y

2ν2 sin(k y)e^−λ2(k)x,

withλ1(k)=^a11+

p4ν²1k²+4ν²1η˜12+a²₁₁+a²₂₁

2ν1 andλ2(k)=^−a12+

p4ν²2k²+4ν²2η˜22+a₁₂²+a²₂₂

2ν2 . DefiningS1= ν2λ2(p)+a12,S2=ν1λ1(p)−a11, the two scalar products〈f,g〉w1 =_L²R

Γf g e⁻

a21y

ν1 d y and

〈f,g〉w2=²_LR

Γf g e⁻

a22y

ν2 d yand repeating the calculations (2.2.6)-(2.2.8), one finds the re-currence relation (2.2.9), withV_k,l := 〈ψk,φl〉w1,W_j,i:= 〈φj,ψi〉w2 and the diagonal ma-trices (D1)l,l :=(−ν2λ2(l)+ν2λ2(p)), ( ˜D1)k,k:=(ν1λ1(k)+ν2λ2(p)−a11+a12), (D2)i,i:=

(ν1λ1(i)−ν1λ1(q)), ( ˜D2)j,j:=(−ν2λ2(j)−ν1λ1(q)−a12+a11).

Dans le document Domain decomposition methods for multiphysics problems (Page 70-77)