Abstract Schwarz framework

In this section we compress in a nutshell, the extensive literature on the so called Ad-ditive Schwarz preconditioners. For more details, we refer the interested reader to the monographs [151, Chapters 2,3], [16, Chapter 7], [143, Chapter 5], [61, Chapter 5] and the survey article [159].

Let V be a finite dimensional vector space, V⁰ its dual, and A :V →V⁰ a symmetric positive definite operator. We can think of A as the operator which satisfies 〈Au,v〉 = a(u,v),∀u,v∈V, where〈·,·〉denotes the duality betweenV andV⁰anda(·,·) is the sym-metric and coercive bilinear form defined in (1.1.2). We introduce a set of auxiliary spaces Vj, 0≤ j ≤N, the operatorsBj:Vj →V_j⁰ and the interpolation operatorsR^>_j :Vj→V. We suppose thatV=PN

j=0R^>_jV_j, i.e. every elementv∈V can be written as sum of terms vj∈Vj, and this decomposition does not need to be unique. We should think ofBj as approximations ofAon the smaller spaceVj. Then, an abstract additive Schwarz precon-ditioner forAcan be defined as

B=

N

X

j=0

R^>_jB⁻¹_j Rj, B:V⁰→V. (1.2.7)

It can be proven thatBis a symmetric positive definite operator, see [16, Theorem 7.1.11], andB Ais symmetric positive definite with respect the scalar product〈B⁻¹·,·〉. There is a great freedom in the choice of the auxiliary spacesV_j and of the operatorsB_j. For in-stance, the auxiliary spaces can be defined through a domain decomposition, or from several discretizations ofΩwith different mesh sizes. The operatorsBj can be defined as the restriction ofAonto the subspacesV_j, or they can be constructed through differ-ent scalar products. Such freedom results in a wide variety of differdiffer-ent preconditioners.

We briefly introduce three of them and we focus more on the two-level additive Schwarz preconditioner.

1.2.1.1 Hierarchical Basis preconditioner

In the Hierarchical basis preconditioner [161] one considers a set of meshT1,T2, ...,TN, such thatTj is obtained by regular subdivision ofTj−1, withhj=2^N−jhN, whereh_k:= max_T∈TkdiamT fork=1, ...,N is the maximum diameter of a triangle on the meshTk. The unknowns on each meshTj form a spaceWj. The auxiliary spacesVj are then de-fined as subspaces ofWj,

Vj:={v∈Wj:v(p)=0 for all vertices p ofTj−1} .

If we assumeW1=V1, then it holdsWj=Wj−1+Vj, and thusWN=V1⊕ · · · ⊕VN. On each auxiliary spaceV_j, the operatorsB^HB_j are defined as

〈B^HB_j v1,v2〉 = X

p∈Vj\Vj−1

v1(p)v2(p), ∀v1,v2∈Vj,

whereVJdenotes the set of vertices ofTj. The extension operatorsR^>_j are straightforward injections operators betweenVj andVN. The Hierarchical basis preconditioner is then equal toB^HB:=PN

j=0R^>_j ³ B^HB_j ´−1

Rj. 1.2.1.2 BPX preconditioner

The BPX preconditioner has been introduced by Bramble, Pasciak and Xu in [14]. The auxiliary spacesVj, 0≤j≤N, are defined as the finite element spaces on the meshesTj

and the operatorsB^BPX_j are defined as

〈B^BPX_j v1,v2〉 = X

p∈Vj

v1(p)v2(p), ∀v1,v2∈Vj.

The extension operatorsR^>_j are straightforward injections operators betweenVjandVN. The global BPX preconditioner is thenB^BPX:=PN

j=0R^>_j ³

B^BPX_j ´₋1

Rj. 1.2.1.3 Two-Level additive Schwarz preconditioner

In the two-level additive Schwarz preconditioner, we consider a domainΩ discretized with a meshThand a corresponding finite element spaceV. We divideΩinto a collection

of open subsetsΩ⁰₁,Ω⁰₂, ...,Ω⁰_N, whose diameters have orderH. The boundaries ofΩ⁰_j are aligned with the fine meshTh for every j. We assume there exist nonnegativeC^∞(Ω) functionsθj, j=1, ...,N (their existence can be proven under suitable assumptions on the subdomainsΩ⁰_j, see [151, Lemma 3.4]) which satisfy

θj=0 onΩ\Ω⁰_j,

N

X

j=1

θj=1 onΩ,

There exists a positive constantδsuch thatk∇θjkL^∞(Ω)≤C/δ.

(1.2.8)

These hypotheses imply thatn Ω⁰_jo

j form an overlapping decomposition ofΩ, andδis a measure of the overlap. On each subdomain, we introduce the spaceVj ={ v∈V : v=0 onΩ\Ω⁰_j} and an operatorB^AD_j :Vj→V_j⁰such that〈B^AD_j uj,vj〉 =bj(uj,vj), where bj:Vj×Vj→Ris a coercive and symmetric bilinear form which approximatesa(·,·) on the subspaceVj. One can choose the bilinear formsbj(·,·) to be ‘exact local solvers’, which means defining them through the ‘exact’ bilinear forma(·,·)

bj(uj,vj)=a(R^>_juj,R^>_j vj), ∀uj,vj∈Vj. (1.2.9) In this case we would haveB^AD_j =RjAR^>_j.

Besides the subdomain spacesVj, we further consider a coarse discretization ofΩ, re-sulting in the coarse meshT0and the coarse finite element spaceV₀. Then the two-level additive Schwarz preconditioner is

P_ad=B^ADA, where B^AD:=

N

X

j=0

R^>_j ³ B^AD_j ´−1

Rj. (1.2.10)

We remark thatB^ADcorresponds exactly to the AS preconditionerM⁻¹_ASdefined in (1.2.6), if one uses exact solvers on each subspaceVj and neglects the coarse solver indexed by j=0.

It is interesting to observe that the preconditioned operatorP_adcan be written as a sum of projection operators. To see this, for j=0, ...,N, we introduce the operatorsP_j:V → R^>_jVj⊂V, Pj:=R^>_jPej, wherePej:V→Vjare defined by

bj(Peju,vi)=a(u,R^>_jvj), ∀vj∈Vj. (1.2.11) If we use exact solvers, it follows immediately that

a(Pju,R^>_jvj)=a(R^>_jPeju,R^>_jvj)=bj(Peju,R^>_jvj)=a(u,R^>_jvj), ∀vj∈Vj. The following Lemma holds.

Lemma 1.2.1(Lemma 2.1 in [151]). The operators Pj are self adjoint with respect to the scalar product defined by a(·,·)and positive semi-definite. Moreover it holds

P_j=R^>_j(B^AD_j )⁻¹R_jA, 0≤j≤N, and if b_j(·,·)satisfy(1.2.9), then P_jis a projection and thus P²_j=P_j.

Proof. We first show the equalityPj =R^>_j(B^AD_j )⁻¹RjA. To do so, we consider the matrix form of (1.2.11),

v^>_i B^AD_j Pejuj=v^>_jRjAuj, ∀uj,vj∈Vj.

Since it holds for everyuj,vj ∈Vj we can writeB^AD_j Pej =RjA. We assume thatbj(·,·) is coercive, thusB^AD_j is invertible and using the definition ofPj the result follows. To prove thatPj is self adjoint, it is sufficient to show that

a(P_ju,v)=v^>AP_ju=v^>A(R^>_j(B^AD_j )⁻¹R_jAu)=(R^>_j(B^AD_j )⁻¹R_jAv)^>Au=a(u,P_jv).

The positive definiteness ofPifollows from the positive definiteness ofB^AD_j , indeed, a(Pju,u)=u^>APju=u^>AR^>_j(B^AD_j )⁻¹RjAu=w^>_j(B^AD_j )⁻¹wj≥0,

wherewj:=RjAu. Finally, in case we are using exact solvers, we haveB^AD_j =RjAR^>_j, and thus

P²_j=R^>_j(RjAR^>_j)⁻¹RjAR^>_j(RjAR^>_j)⁻¹A=R^>_j(RjAR^>_j)⁻¹A=Pj.

We close this subsection observing that the preconditioned operatorPad can be written as

Pad=B^ADA=

N

X

j=0

R^>_j(RjAR^>_j)⁻¹RjA=

N

X

j=0

Pj.

1.2.1.4 Convergence theory

As discussed in the introduction, the convergence of the conjugate gradient method de-pends on the condition number of the matrix A. As the preconditioned systemB A is self-adjoint with respect to the scalar product defined bya(·,·), see Lemma (1.2.1), the largest and smallest eigenvalue ofB Aare given by the Rayleigh quotient formula as

λmax(B A) :=sup

u∈V

a(B Au,u)

a(u,u) , λmin(B A) := inf

u∈V

a(B Au,u) a(u,u) .

There is a well-developed theory to find estimates for the maximum and minimum eigen-value for additive Schwarz preconditioners and it is called abstract Schwarz framework.

This theory relies on three assumptions. For every new preconditioner, if one verifies these three assumptions, then a general result permits to find estimates for the extreme eigenvalues and thus for the condition number of the preconditioned system. These three assumptions are:

Assumption1.2.2 (Stable decomposition). There exists a constantC0such that everyu∈V admits a decompositionu=PN

j=0R^>_juj, for someuj∈Vj, such that

N

X

j=0

〈Bjuj,uj〉 ≤C²₀〈Au,u〉. (1.2.12)

Assumption 1.2.2 permits to find a lower bound forλmin(B A). To have a robust precondi-tioner, it is essential that the constantC₀does not depend strongly on some parameters of the problem such as the mesh size on the finest grid or the size/number of the subdo-mains. In this perspective, the choice of the coarse spaceV0plays the key role.

Assumption1.2.3 (Strengthened Cauchy-Schwarz Inequality). There exist constants 0≤

²i j≤1, 1≤i,j≤N, such that

|a(R_i^>ui,R^>_juj)| ≤²i ja(R_i^>ui,R^>_i ui)¹²a(R^>_juj,R^>_j uj)¹², (1.2.13) forui∈Vianduj∈Vj. We will denote the spectral radius ofE=©

²i jª

byρ(E).

The quantityρ(E) will be involved in the upper bound ofλmax(B A).

Assumption1.2.4 (Local stability). There exists a constantω>0, such that,

a(R_i^>ui,R^>_i ui)≤ω〈Biui,ui〉, ui∈Vi, 0≤i≤N. (1.2.14) Assumption 1.2.4 guarantees that the bilinear forms induced byB_iare coercive. If all these assumptions are verified it is possible to prove the following theorem.

Theorem 1.2.5(Theorem 2.7 in [151]). Let Assumptions(1.2.2)-(1.2.3)-(1.2.4)be satisfied.

Then the condition number of the additive Schwarz operator satisfies

κ(B A)≤C₀²ω(ρ(E)+1). (1.2.15)

Of course, different methods will lead to different values of these parameters. We con-clude this section reporting some classical bounds for the condition numbers of the HB, BPX and two-level AS preconditioners,

κ(B^HBA)≤C1(1+ |lnhN|²), κ(B^BPXA)≤C2, κ(B^ADA)≤C3

µ 1+H

δ

¶ ,

wherehN is the finest mesh size. For more details, we refer the reader to [16, Chapter 7]

and [159] for BPX and HB preconditioners, and Chapter 2 and 3 in the dedicated mono-graph [151] for the two-level AS preconditioner.

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