Introduction to Domain Decomposition Methods. Alfio Quarteroni

. Aachen, EU Regional School Series February 18, 2013 Introduction to Domain Decomposition Methods Alfio Quarteroni Chair of Modeling and Scientific Computing Mathematics Institute of Computational Science and Engineering École Polytechnique Fédérale de Lausanne MOX Modeling and Scientific Computing Mathematics Department Politecnico di Milano 1

MOTIVATION DD can be used in the framework of any discretization method for PDEs (FEM, FV, FD, SEM) to make their algebraic solution more efficient on parallel computer platforms DDM allow the reformulation of a boundary-value problem on a partition of the computational domain into subdomains very convenient framework for the solution of heterogeneous or multiphysics problems, i.e. those that are governed by differential equations of different kinds in different subregions of the computational domain. 2

THE IDEA The computational domain Ω where the bvp is set is subdivided into two or more subdomains in which problems of smaller dimension are to be solved. Parallel solution algorithms may be used. There are two ways of subdividing the computational domain: with disjoint subdomains with overlapping subdomains Ω 2 Ω 2 Ω 1 Γ 2 Ω Ω 1 Γ Γ 1 Correspondingly, different DD algorithms will be set up. 3

AN HISTORICAL REMARK The classical logo of DDM is [http://www.ddm.org/] Why this symbol? What does it mean? We have to go back to a work of 1869 by Hermann Schwarz... 4

...who wanted to establish the existence of harmonic functions with prescribed boundary values on regions with non-smooth boundaries: { u = f in Ω where u = g on Ω Ω Ω Schwarz s idea: split Ω into two elementary subdomains: Ω 2 Ω Ω 1 Γ 2 Γ 1 and set up a suitable iterative method passing information between the subproblems. 5

MODERN EXAMPLES OF SUBDIVISIONS 6

REFERENCES B.F. Smith, P.E. Bjørstad, W.D. Gropp (1996) Domain Decomposition Cambridge University Press, Cambridge. A. Quarteroni and A. Valli (1999) Domain Decomposition Methods for Partial Differential Equations Oxford Science Publications, Oxford. A. Toselli and O.B. Widlund (2005) Domain Decomposition Methods Algorithms and Theory Springer-Verlag, Berlin and Heidelberg. 7

CLASSICAL ITERATIVE DD METHODS 8

MODEL PROBLEM Consider the model problem: find u :Ω R s.t. { Lu = f in Ω u =0 on Ω L is a generic second order elliptic operator. The weak formulation reads: find u V = H0 1 (Ω) : a(u, v) =(f, v) v V where a(, ) isthebilinearformassociatedwithl. 9

SCHWARZ METHODS Consider a decomposition of Ω with overlap: Ω 2 Ω Ω 1 Γ 2 Γ 1 Given u (0) 2 on Γ 1,fork 1: solve solve Lu (k) 1 = f in Ω 1 u (k) 1 = u (k 1) 2 on Γ 1 u (k) 1 =0 on Ω 1 \ Γ 1 Lu (k) 2 = { f in Ω 2 u (k) 1 u (k 1) on Γ 2 1 u (k) 2 = u (k) 2 =0 on Ω 2 \ Γ 2. 10

Choice of the trace on Γ 2 : if u (k) 1 multiplicative Schwarz method if u (k 1) 1 additive Schwarz method We have two elliptic bvp with Dirichlet conditions in Ω 1 and Ω 2,andwe wish the two sequences {u (k) 1 } and {u(k) 2 } to converge to the restrictions of the solution u of the original problem: lim k u (k) 1 = u Ω1 and lim k u (k) 2 = u Ω2. The Schwarz method applied to the model problem always converges, with a rate that increases as the measure Γ 12 of the overlapping region Γ 12 increases. 11

Example Consider the model problem { u (x) =0 a < x < b, u(a) =u(b) =0, where Ω 2 a γ 2 γ 1 b Ω 1 The solution is u =0. Weshowafewiterationsofthemethod: u (1) 1 u (2) 1 γ 2 a b Clearly, the method converges with a rate that reduces as the length of the interval (γ 2,γ 1 )getssmaller. γ 1 u (0) 2 u (1) 2 u (2) 2 12

Remark The Schwarz method requires at each iteration the solution of two subproblems of the same kind as those of the original problem. The boundary conditions of the initial problem remain unchanged. Dirichlet-type boundary conditions are imposed across the interfaces. 13

NONOVERLAPPING DECOMPOSITION We partition now the domain Ω in two disjoint subdomains: Ω Ω 1 Ω 2 Γ The following equivalence result holds. Theorem The solution u of the model problem is such that u Ωi = u i for i =1, 2, where u i is the solution to the problem { Lui = f in Ω i u i =0 on Ω i \ Γ with interface conditions u 1 = u 2 and u 1 n L = u 2 n L on Γ / n L is the conormal derivative. 14

Example The problem { div (µ u) =f in Ω u =0 on Ω is equivalent to div (µ i u i )=f i u =0 u 1 = u 2 u 1 µ 1 n = µ u 2 2 n in Ω i on Ω i on Γ on Γ Remark The proof of the theorem can be done using the weak formulation ofthe problem. We refer to Lemma 1.2.1 in [QV99]. 15

DIRICHLET-NEUMANN METHOD Given u (0) 2 on Γ, for k 1solvetheproblems: Lu (k) 1 = f in Ω 1 u (k) 1 = u (k 1) 2 on Γ u (k) 1 =0 on Ω 1 \ Γ Lu (k) 2 = f in Ω 2 u (k) 2 = u(k) 1 n L on Γ n L u (k) 2 =0 on Ω 2 \ Γ The equivalence theorem guarantees that when the two sequences {u (k) 1 } and {u(k) 2 } converge, their limit will be necessarily the solution to the exact problem. The DN algorithm is therefore consistent. However, its convergence is not always guaranteed. 16

Example Let Ω = (a, b), γ (a, b), L = d 2 /dx 2 and f =0. Ateveryk 1the DN algorithm generates the two subproblems: (u (k) 1 ) =0 a < x <γ u (k) 1 = u (k 1) 2 x = γ u (k) 1 =0 x = a (u (k) 2 ) =0 γ<x < b (u (k) 2 ) =(u (k) 1 ) x = γ u (k) 2 =0 x = b. The two sequences converge only if γ>(a + b)/2: u (0) 2 u (0) 2 2 a γ a+b b u (2) 2 a a+b 2 γ b u (1) 2 u (1) 2 17

AvariantoftheDNalgorithmcanbesetupbyreplacingthe Dirichlet condition in the first subdomain by u (k) 1 = θu (k 1) 2 +(1 θ)u (k 1) 1 on Γ using a relaxation parameter θ>0. In such a way it is always possible to reduce the error between two subsequent iterates. In the previous example, we can easily verify that, by choosing θ opt = u (k 1) 1 u (k 1) 2 u (k 1) 1 the algorithm converges to the exact solution in a single iteration. More in general, there exists a suitable value 0 <θ max < 1suchthat the DN algorithm converges for any possible choice of the relaxation parameter θ in the interval (0,θ max ). 18

NEUMANN-NEUMANN ALGORITHM Consider again a partition of Ω into two disjoint subdomains and denote by λ the (unknown) value of the solution u on their interface Γ: λ = u i on Γ (i =1, 2) Consider the following iterative algorithm: for any given λ (0) on Γ, for k 0andi =1, 2, solve the following problems: with Lu (k+1) i = f in Ω i u (k+1) i = λ (k) on Γ u (k+1) i =0 on Ω i \ Γ Lψ (k+1) i =0 inω i ψ (k+1) i = u(k+1) 1 u(k+1) 2 n n n on Γ i =0 on Ω i \ Γ ψ (k+1) λ (k+1) = λ (k) θ ( ) σ 1 ψ (k+1) 1 Γ σ 2 ψ (k+1) 2 Γ where θ is a positive acceleration parameter, while σ 1 and σ 2 are two positive coefficients. 19

ROBIN-ROBIN ALGORITHM For every k 0solvethefollowingproblems: then u (k+1) 1 = f in Ω 1 u (k+1) 1 =0 on Ω 1 \ Γ u (k+1) 1 n + γ 1 u (k+1) 1 = u(k) 2 n + γ 1u (k) 2 on Γ Lu (k+1) 2 = f in Ω 2 u (k+1) 2 =0 on Ω 2 \ Γ u (k+1) 2 n + γ 2 u (k+1) 2 = u(k+1) 1 n + γ 2 u (k+1) 1 on Γ where u (0) 2 is assigned and γ 1, γ 2 are non-negative acceleration parameters that satisfy γ 1 + γ 2 > 0. Aiming at parallelization, we could use u (k) 1 instead of u (k+1) 1. 20

THE STEKLOV-POINCARÉ INTERFACE EQUATION 21

MULTI-DOMAIN FORMULATION OF POISSON PROBLEM AND INTERFACE CONDITIONS We consider now the model problem: { u = f in Ω u =0 on Ω. For a domain partitioned into two disjoint subdomains, wecanwritethe equivalent multi-domain formulation (u i = u Ωi, i =1, 2): u 1 = f in Ω 1 u 1 =0 on Ω 1 \ Γ u 2 = f in Ω 2 u 2 =0 on Ω 2 \ Γ u 1 = u 2 on Γ u 1 n = u 2 n on Γ. 22

Remark On the interface Γ we have the normal unit vectors n 1 and n 2 : Ω1 Γ n 1 n 2 Ω 2 There holds: n 1 = n 2 on Γ. We denote n = n 1 so that n = n 1 = n 2 on Γ. 23

THE STEKLOV-POINCARÉ OPERATOR Let λ be the unknown value of the solution u on the interface Γ: λ = u Γ Should we know a priori the value λ on Γ, wecouldsolvethefollowing two independent boundary-value problems with Dirichlet condition on Γ (i =1, 2): w i = f in Ω i w i =0 on Ω i \ Γ w i = λ on Γ. 24

With the aim of obtaining the value λ on Γ, let us split w i as follows w i = w i + u 0 i, where wi and ui 0 represent the solutions of the following problems (i =1, 2): wi = f in Ω i wi =0 on Ω i Ω wi =0 on Γ and ui 0 = 0 in Ω i ui 0 =0 on Ω i Ω ui 0 = λ on Γ. 25

The functions w i depend solely on the source data f w i = G i f where G i is a linear continuous operator u 0 i depend solely on the value λ on Γ u 0 i = H i λ where H i is the so-called harmonic extension operator of λ on the domain Ω i. We have that u i = wi + ui 0 (i =1, 2) w 1 n = w 2 n on Γ n (w 1 + u0 1 )= n (w 2 + u0 2 ) on Γ n (G 1f + H 1 λ)= n (G 2f + H 2 λ) on Γ ( H1 n H ) ( 2 G2 λ = n n G ) 1 f on Γ. n 26

We have obtained the Steklov-Poincaré equation for the unknown λ on the interface Γ: Sλ = χ on Γ S is the Steklov-Poincaré pseudo-differential operator: Sµ = n H 1µ n H 2µ = χ is a linear functional which depends on f : The operator χ = n G 2f n G 1f = S i : µ S i µ = 2 i=1 2 i=1 n i H i µ n i G i f. (H i µ) n i i =1, 2, Γ is called local Steklov-Poincaré operator (Dirichlet-to-Neumann) which operates between the trace space and its dual Λ. Λ={µ : v V s.t. µ = v Γ } = H 1/2 00 (Γ) 27

Example To provide an example of the operator S, weconsiderasimple1d problem. Let Ω = (a, b) R as illustrated below. We split Ω in two nonoverlapping subdomains. In this case the interface Γreducestothepointγ (a, b) andthesteklov-poincaréoperators becomes ( dh1 Sλ = dx dh ) ( 2 1 λ = + 1 ) λ dx l 1 l 2 where l 1 = γ a et l 2 = b γ. λ H 1 λ H 2 λ a l 1 l 2 Ω 1 γ Ω 2 b 28

EQUIVALENCE BETWEEN THE DD SCHEMES AND CLASSICAL ITERATIVE METHODS 29

The DN method can be reinterpreted as a preconditioned Richardson method for the solution of the Steklov-Poincaré interface equation: P DN (λ (k) λ (k 1) )=θ(χ Sλ (k 1) ) The preconditioning operator is P DN = S 2 = (H 2 µ)/ n 2. The NN method can also be interpreted as a preconditioned Richardson algorithm P NN (λ (k) λ (k 1) )=θ(χ Sλ (k 1) ) with P NN =(σ 1 S 1 1 + σ 2 S 1 2 ) 1. 30

The Robin-Robin algorithm is equivalent to the following alternating direction (ADI) algorithm: (γ 1 I Λ + S 1 )µ (k) 1 = χ +(γ 1 I Λ + S 2 )µ (k 1) 2, (γ 2 I Λ + S 2 )µ (k) 2 = χ +(γ 2 I Λ + S 1 )µ (k 1) 1, where I Λ :Λ Λ here denotes the Riesz isomorphism between the Hilbert space Λ and its dual Λ. At convergence (for a convenient choice of γ 1 and γ 2 ), we have µ 1 = µ 2 = λ. 31

FINITE ELEMENT APPROXIMATION: MULTI-DOMAIN FORMULATION 32

Consider the Poisson problem: { u = f in Ω u =0 on Ω Its weak formulation reads find u V : a(u, v) =F (v) v V where V = H0 1(Ω), a(v, w) = v w v, w V and F (v) = fv v V. Ω Ω 33

Suppose that Ω is split into two nonoverlapping subdomains and consider a uniform triangulation T h of Ω, conforming on Γ: Ω 1 Γ Ω 2 34

The Galerkin finite element approximation of the Poisson problem reads: where find u h V h : a(u h, v h )=F (v h ) v h V h (1) V h = {v h C 0 (Ω) : v h K P r r 1, K T h, v h =0on Ω} is the space of finite element functions of degree r with basis {ϕ i } N h i=1. The Galerkin approximation (1) is equivalent to: find u h V h : a(u h,ϕ i )=F (ϕ i ) i =1,...,N h. (2) 35

We partition the nodes of the triangulation as follows: {x (1) j, 1 j N 1 } nodes in subdomain Ω 1 {x (2) j, 1 j N 2 } nodes in subdomain Ω 2 {x (Γ) j, 1 j N Γ } nodes on the interface Γ, and we split the basis functions accordingly: ϕ (1) j functions associated to the nodes x (1) j ϕ (2) j functions associated to the nodes x (2) j ϕ (Γ) j functions associated to the nodes x (Γ) j on the interface. 36

Example Ω 1 x (1) j Γ x (Γ) j Ω 2 x (2) j 37

Problem (2) can be equivalently rewritten as: find u h V h : a(u h,ϕ (1) i )=F (ϕ (1) i ) i =1,...,N 1 a(u h,ϕ (2) j )=F(ϕ (2) j ) j =1,...,N 2 a(u h,ϕ (Γ) k )=F(ϕ(Γ) k ) k =1,...,N Γ. We introduce the bilinear form on Ω i : a i (v, w) = v w v, w V, i =1, 2, Ω i the space V i h = {v H 1 (Ω i ):v =0on Ω i \ Γ} (i =1, 2) and let u (i) h = u h Ωi (i =1, 2). Then, problem (3) can be written equivalently as: find u (1) h V 1 h, u(2) h V 2 h s.t. a 1 (u (1) h,ϕ(1) i )=F 1 (ϕ (1) i ) i =1,...,N 1 a 2 (u (2) h,ϕ(2) j )=F 2 (ϕ (2) j ) j =1,...,N 2 a 1 (u (1) h,ϕ(γ) k Ω 1 )+a 2 (u (2) h,ϕ(γ) k Ω 2 ) = F 1 (ϕ (Γ) k Ω 1 )+F 2 (ϕ (Γ) k Ω 2 ) k =1,...,N Γ. (3) (4) 38

Remark Problem (4) corresponds to the finite element approximation of the multi-domain formulation of the Poisson problem: u 1 = f in Ω 1 u 1 =0 on Ω 1 \ Γ u 2 = f in Ω 2 u 2 =0 on Ω 2 \ Γ u 1 = u 2 on Γ u 1 n = u 2 on Γ. n The condition u 1 = u 2 on Γ is satisfied by definition of u (i) h. 39

We can write: u h (x) = N 1 j=1 u h (x (1) j )ϕ (1) j N Γ + u h (x (Γ) j j=1 N 2 (x)+ u h (x (2) )ϕ (Γ) j (x), j=1 j )ϕ (2) j (x) and we substitute this expression in (4) to obtain... 40

N 1 j=1 N 2 j=1 N Γ j=1 u h (x (1) j )a 1(ϕ (1) j,ϕ (1) u h (x (2) j )a 2(ϕ (2) j,ϕ (2) [ u h (x (Γ) j ) a 1(ϕ (Γ) j N Γ i )+ j=1 N Γ i )+,ϕ (Γ) i N 1 + u h (x (1) j )a 1(ϕ (1) j j=1 = F 1(ϕ (Γ) i j=1 u h (x (Γ) j )a 1(ϕ (Γ) j,ϕ (1) i )=F 1(ϕ (1) i ) i =1,...,N 1 u h (x (Γ) j )a 2(ϕ (Γ) j,ϕ (2) i )=F 2(ϕ (2) i ) i =1,...,N 2 ] )+a 2(ϕ (Γ) j,ϕ (Γ) i ) N 2 i )+ u h (x (2) j )a 2(ϕ (2) j,ϕ (Γ) i ),ϕ (Γ) j=1 Ω1 )+F 2(ϕ (Γ) i Ω2 ) i =1,...,N Γ. 41

A bit of algebra... N 1 j=1 N 2 j=1 N Γ j=1 N Γ u h (x (1) j )(A 11) ij + j=1 N Γ u h (x (2) j )(A 22) ij + [ u h (x (Γ) j ) j=1 j=1 (A (1) ΓΓ )ij + (A(2) u h (x (Γ) j )(A 1Γ ) ij = F 1(ϕ (1) i ) i =1,...,N 1 u h (x (Γ) j )(A 2Γ ) ij = F 2(ϕ (2) i ) i =1,...,N 2 ΓΓ )ij ] N 1 N 2 + u h (x (1) j )(A Γ1 ) ij + u h (x (2) j )(A Γ2 ) ij = F 1(ϕ (Γ) i j=1 Ω1 )+F 2(ϕ (Γ) i Ω2 ) i =1,...,N Γ. 42

A bit of algebra... [continued] N 1 N Γ u 1(A 11) ij + u Γ (A 1Γ ) ij = f 1 i =1,...,N 1 j=1 j=1 N 2 j=1 N Γ j=1 N Γ u 2(A 22) ij + u Γ (A 2Γ ) ij = f 2 i =1,...,N 2 u Γ [ j=1 ] (A (1) ΓΓ )ij +(A(2) ΓΓ )ij N 1 N 2 + u 1(A Γ1 ) ij + u 2(A Γ2 ) ij j=1 j=1 = f (Γ) 1 + f (Γ) 2 i =1,...,N Γ. 43

... so that we obtain the algebraic form: A 11 u 1 + A 1Γ λ = f 1 A 22 u 2 + A 2Γ λ = f( 2 ) A Γ1 u 1 + A Γ2 u 2 + A (1) ΓΓ + A(2) ΓΓ λ = f (Γ) 1 + f (Γ) 2 (5) or A 11 0 A 1Γ 0 A 22 A 2Γ A Γ1 A Γ2 A ΓΓ where we denoted A ΓΓ = u 1 u 2 λ = f 1 f 2 f Γ ( ) A (1) ΓΓ + A(2) ΓΓ and f Γ = f (Γ) 1 + f (Γ) 2. (6) 44

THE SCHUR COMPLEMENT SYSTEM 45

Since λ represents the unknown value of u on Γ, its finite element correspondent is the vector λ of the values of u h at the interface nodes. By Gaussian elimination, we can obtain a new reduced system in the sole unknown λ: Matrices A 11 and A 22 are invertible since they are associated with two homogeneous Dirichlet boundary-value problems for the Laplace operator: u 1 = A 1 11 (f 1 A 1Γ λ) and u 2 = A 1 22 (f 2 A 2Γ λ). (7) From the third equation we obtain: [( ) ( )] A (1) ΓΓ A Γ1A 1 11 A 1Γ + A (2) ΓΓ A Γ2A 1 22 A 2Γ λ = f Γ A Γ1 A 1 11 f 1 A Γ2 A 1 22 f 2. 46

Setting and Σ=Σ 1 +Σ 2 with Σ i = A (i) ΓΓ A ΓiA 1 ii A iγ (i =1, 2) χ Γ = f Γ A Γ1 A 1 11 f 1 A Γ2 A 1 22 f 2 we obtain the Schur complement system Σλ = χ Γ Σ and χ Γ approximate S and χ. Σistheso-calledSchur complement of A with respect to u 1 and u 2. Σ i are the Schur complements related to the subdomains Ω i (i =1, 2). 47

Remark After solving the Schur complement system in λ, thanks to(7)we can compute u 1 and u 2. This is equivalent to solving two Poisson problems in the subdomains Ω 1 and Ω 2 with the Dirichlet boundary condition u (i) h Γ = λ h (i =1, 2) on the interface Γ. 48

PROPERTIES OF THE SCHUR COMPLEMENT Σ The Schur complement Σ inherits some of the properties of A: if A is singular, so is Σ; if A (respectively, A ii )issymmetric,thenσ(respectively,σ i )is symmetric too; if A is positive definite, so is Σ. Moreover, concerning the condition number, wehave κ(a) Ch 2 κ(σ) Ch 1 49

PRECONDITIONERS FOR THE SCHUR COMPLEMENT SYSTEM 50

The iterative methods that we have illustrated are equivalent to preconditioned Richardson methods for the Schur complement system with preconditioners: for the DN algorithm: P h =Σ 2 for the ND algorithm: P h =Σ 1 for the NN algorithm: P h =(σ 1 Σ 1 1 + σ 2 Σ 1 2 ) 1 for the RR algorithm: P h =(γ 1 + γ 2 ) 1 (γ 1 I +Σ 1 )(γ 2 I +Σ 2 ) 51

All these preconditioners are optimal in the sense of the following definition. Definition ApreconditionerPisoptimal for a matrix A R N N if the condition number of P 1 Aisboundeduniformly with respect to the dimension N of A. In particular, we have κ(σ 1 i Σ) = O(1) (i =1, 2) κ((σ 1 Σ 1 1 + σ 2 Σ 1 2 )Σ) = O(1) for all σ 1,σ 2 > 0 52

NONOVERLAPPING METHODS IN THE CASE OF MORE THAN TWO SUBDOMAINS 53

MULTI-DOMAIN FORMULATION FOR M > 2 SUBDOMAINS We generalize now the nonoverlapping methods to the case of a domain ΩsplitintoM > 2subdomains: 1 Ω i (i =1,...,M) suchthat Ω i = Ω Γ i = Ω i \ Ω Γ= Γ i. Plotmesh Ω 0.8 0.6 0.4 0.2 Γ ik1 Γ ik2 Y Axis 0 0.2 0.4 0.6 Γ ik4 Ω i Γ ik3 0.8 1 1 0.5 0 0.5 1 X Axis 54

At the differential level, we have the equivalent multi-domain formulation: { u = f in Ω u =0 on Ω u i = f in Ω i (i =1,...,M) u i = u k on Γ ik u i = u k n i n i on Γ ik u i =0 on Ω i Ω where Γ ik = Ω i Ω k. 55

FINITE ELEMENT APPROXIMATION Considering a conforming finite element approximation, weobtain the linear system: ( )( ) ( ) AII A I Γ ui fi = (8) A ΓI A ΓΓ λ f Γ where u I is the vector of unknowns in the internal nodes and λ is the vector of unknowns on Γ: λ = u Γ. The submatrix A II associated to the internal nodes is block-diagonal: A 11 0... 0. A II = 0....... 0 0... 0 A MM A I Γ is a banded matrix (interactions with local interfaces). 56

Remark On each subdomain Ω i the matrix ( ) Aii A A i = iγ A Γi A Γi Γ i represents the local stiffness matrix associated to a Neumann problem. 57

THE SCHUR COMPLEMENT SYSTEM Since A II is non-singular, from (8) we can derive By eliminating u I from (8), we get that is Denoting and u I = A 1 II (f I A I Γ u Γ ) (9) A ΓΓ λ = f Γ A ΓI A 1 II (f I A I Γ λ) ( AΓΓ A ΓI A 1 ) II A I Γ λ = fγ A ΓI A 1 II f I. (10) Σ=A ΓΓ A ΓI A 1 II A I Γ χ Γ = f Γ A ΓI A 1 II f I we obtain the Schur complement system in the multi-domain case Σλ = χ Γ (11) 58

Example 0 Γ 1 Γ 4 Γ 2 Γ 3 5 10 15 Γ 2 Γ c Γ 1 Γ c 20 25 Γ 3 Γ 4 30 0 5 10 15 20 25 30 nz = 597 On the left, decomposition of Ω = (0, 1) (0, 1) into four square subdomains. On the right, sparsity pattern of the Schur complement matrix arisingfromthe decomposition depicted on the left. Remark The local Schur complements are defined as so that Σ i = A Γi Γ i A Γi ia 1 ii A iγi Σ=R T Γ 1 Σ 1 R Γ1 +...R T Γ M Σ M R ΓM where R Γi is the restriction from the skeleton Γ to the boundary of Ω i. 59

ASIMPLEALGORITHM To compute a FE approximation of the solution u of the Poisson problem { u = f in Ω u =0 on Ω we can do the following steps: 1 solve the Schur complement system Σλ = χ Γ to compute λ on the whole interface Γ; 2 solve A II u I = f I A I Γ λ i.e., M independent problems of reduced dimension A ii u i I = g i (i =1,...,M) possibly in parallel. 60

ESTIMATE OF THE CONDITION NUMBER The following estimate can be proved for the condition number of the Schur complement matrix Σ: there exists a constant C > 0, independent of h and H such that κ(σ) C H max hh 2 min where H max and H min are the maximal and minimal diameters of the subdomains, respectively. 61

NUMERICAL RESULTS We consider the model problem { u = f in Ω = (0, 1) (0, 1) u =0 on Ω We decompose Ω into M square regions Ω i of characteristic dimension H such that M i=1 Ω i = Ω: Γ 4 Γ 2 Γ 3 Γ c Γ 1 (Decomposition of Ω = (0, 1) (0, 1) into four square subdomains) 62

Condition number of Σ: κ(σ) H =1/2 H =1/4 H =1/8 h=1/8 9.77 14.83 25.27 h=1/16 21.49 35.25 58.60 h=1/32 44.09 75.10 137.73 h=1/64 91.98 155.19 290.43 Notice the dependence on 1 h and 1 H. 63

The Schur complement Σ is dense. If we use iterative solvers to solve the Schur complement system Σλ = χ Γ it is not convenient from the point of view of memory usage to explicitly compute the elements of Σ. We need only to compute the product Σx Γ for any given vector x Γ. 64

For example, in the conjugate gradient method: a) Choose a starting solution λ 0 b) Compute r 0 = χ Γ Σλ 0, p 0 = r 0 For k 0untilconvergence,Do c) α k = (p k, r k ) (Σp k, p k ) d) λ k+1 = λ k + α k p k e) r k+1 = r k α k Σp k f) β k = (Σp k, r k+1 ) (Σp k, p k ) g) p k+1 = r k+1 β k p k EndFor 65

SCHUR COMPLEMENT MULTIPLICATION BY A VECTOR We can define the restriction and prolongation operators R Γi and R T Γ i that act on the vector of the interface nodal values. Recalling that Σ=Σ 1 +...+Σ M with Σ i = A Γi Γ i A Γi ia 1 ii A iγi we can use the following fully parallel algorithm. Given x Γ,computey Γ =Σx Γ as follows: a) Set y Γ = 0 For i =1,...,M Do in parallel: b) x i = R Γi x Γ c) z i = A iγi x i d) z i A 1 ii z i e) sum to the global vector y Γi A Γi Γ i x Γ A Γi iz i f) sum to the global vector y Γ RΓ T i y Γi EndFor 66

Before using for the first time the Schur complement matrix, an off-line startup phase is required: For i =1,...,M Do in parallel: a) Built the matrix A i b) Reorder A i as ( ) Aii A A i = iγi A Γi i A Γi Γ i and extract the submatrices A ii, A iγi, A Γi i and A Γi Γ i c) Compute the LU or Cholesky factorization of A ii EndFor 67

SCALABILITY Definition A preconditioner P h of Σ is said to be scalable if the condition number of the preconditioned matrix P 1 h Σ is independent of the number of subdomains. Iterative methods using scalable preconditioners allow henceforth to achieve convergence rates independent of the subdomain number. Remark General strategy: P 1 h = M RΓ T i P 1 i,h R Γ i + RΓ T P 1 H i=1 where P i,h is a suitable preconditioner for the local Schur complement Σ i.thelasttermisthecoarse-scalecorrection. R Γ 68

NEUMANN-NEUMANN PRECONDITIONER The Neumann-Neumann preconditioner for more subdomains reads: where Σ i is either Σ 1 i (P NN h ) 1 = M RΓ T i D i Σ i D i R Γi i=1 or an approximation of Σ 1 i. D i is a diagonal matrix of positive weights d 1 D i =... d n d j is the number of subdomains that share the j-th node. We have the following estimate: κ ( (Ph NN ) 1 Σ ) ( CH 2 1+log H ) 2 h 69

Remark The presence of D i and R Γi only entails matrix-matrix multiplications. Applying Σ 1 i may be done using local inverses: if q is a vector whose components are the nodal values on Γ i,then Σ 1 i q =[0, I ]A 1 [0, I ] T q In particular, the matrix-vector multiplication internal nodes } boundary nodes {{ } A 1 i i 0.. 0 q corresponds to the solution of a Neumann problem in Ω i : { wi =0 inω i w i n = q in Γ i 70

Example Condition number of (Ph NN ) 1 Σ: κ((ph NN ) 1 Σ) H =1/2 H =1/4 H =1/8 H =1/16 h =1/16 2.55 15.20 47.60 - h =1/32 3.45 20.67 76.46 194.65 h =1/64 4.53 26.25 105.38 316.54 h =1/128 5.79 31.95 134.02 438.02 Convergence history of the conjugate gradient using Ph NN preconditioner and h = 1/32: as a 10 0 Convergence history using P NN 10 1 10 2 scaled residual 10 3 10 4 H=1/4 H=1/8 10 5 H=1/2 10 6 10 7 0 2 4 6 8 10 12 14 16 Conjugate gradient iterations 71

BALANCED NEUMANN-NEUMANN PRECONDITIONER The Neumann-Neumann preconditioner of the Schur complement system is not scalable. Asubstantialimprovementcanbeachievedbyaddingacoarsegrid correction: (Ph BNN ) 1 Σ=P 0 +(I P 0 ) ( (PH NN ) 1 Σ ) (I P 0 ) where P 0 = R T 0 Σ 1 0 R 0ΣandΣ 0 = R 0 ΣR T 0, R 0 being the restriction from Γ onto the coarse level skeleton. This is called balanced Neumann-Neumann preconditioner. We can prove that κ ( (Ph BNN ) 1 Σ ) ( C 1+log H ) 2 h 72

Example κ((ph BNN ) 1 Σ) H =1/2 H =1/4 H =1/8 H =1/16 h =1/16 1.67 1.48 1.27 - h =1/32 2.17 2.03 1.47 1.29 h =1/64 2.78 2.76 2.08 1.55 h =1/128 3.51 3.67 2.81 2.07 73

CONCLUDING REMARKS From the numerical results that we have presented, we can conclude with the following remarks: even if better conditioned with respect to A, Σisill-conditioned,and therefore a suitable preconditioner must be applied the Neumann-Neumann preconditioner can be satisfactory applied using a moderate number of subdomains, while for larger M, κ((ph NN ) 1 Σ) >κ(σ) the balancing Neumann-Neumann preconditioner is almost optimally scalable and therefore recommended for partitions with a large number of subdomains 74

FETI (Finite Element Tearing & Interconnecting) METHODS 75

THE MODEL PROBLEM We consider the variable coefficient elliptic problem: { div(ρ u) = f in Ω, u =0 on Ω, (12) where ρ is piecewise constant, ρ = ρ i R + in Ω i. We denote by H i = diam(ω i )thesizeofthei-th subdomain Ω i for i =1,...,M. 76

Domain Ω, its boundary Ω, skeleton Γ, subdomain Ω i,anditsboundary Ω i. 77

THE FINITE ELEMENT SPACES We introduce: the space of traces of finite element functions on the boundaries Ω i,sayw i = W h ( Ω i ), the product space of the previous trace spaces, say W =Π M i=1 W i, a subspace of continuous traces across the skeleton Γ, say Ŵ W. We denote by Ω ih the nodes in Ω i,by Ω ih the nodes on Ω i,by Ω h the nodes on Ω, and by Γ h the nodes on the skeleton Γ. 78

Finite element degrees of freedom on the boundary ( Ω h ), on the skeleton (Γ h ), in the subdomain (Ω ih ), and on its boundary ( Ω ih ). 79

THE SCALING COUNTING FUNCTIONS Let us introduce the following scaling counting functions: 1 x Ω ih ( Ω h \ Γ h ), ( ) δ i (x) = ργ j Nx j (x) /ρ γ i (x) x Ω ih Γ h, 0 elsewhere, x Γ h Ω h,where:γ [1/2, + ) andn x is the set of indices of the subregions having x on their boundaries. Then we set: δ i (x) (= pseudo inverses) = { δ 1 i (x) if δ i (x) 0, 0 if δ i (x) =0. 80

THE FINITE ELEMENT APPROXIMATION Based on the finite element approximation of the model problem (12), let us consider the local Schur complements: for which: Σ i = A (i) ΓΓ A Γ,Ω i A 1 Ω i,ω i A Ωi,Γ, i =1,... M, Σ=R T Γ 1 Σ 1 R Γ1 +...+ R T Γ M Σ M R ΓM, where R Γi is a restriction operator, that is a rectangular matrix of zeros and ones that map values on Γ onto those on Γ i, i =1,...,M. The matrices Σ i are positive semi-definite. We indicate the interface nodal values on Ω i as u i,andweset u =(u 1,...,u M ), the local load vectors on Ω i as χ i and we set χ =(χ 1,...,χ M ). Finally, we set: ablockdiagonalmatrix. Σ = diag(σ 1,...,Σ M ), 81

THE REDUCED FEM PROBLEM The original FEM problem, when reduced to the interface Γ,reads: Find u W s.t. J(u) = 1 2 Σ u, u χ, u min, B Γ u = 0. (13) The reduced problem (13) admits a unique solution iff Ker{Σ } Ker{B Γ } =0,thatisifΣ is invertible on Ker(B Γ ). The matrix B Γ is not unique, sothatweshouldimposecontinuity when u belongs to more than one subdomain; B Γ is made of {0, 1, 1} since it enforces interfaces continuity constraints at interfaces nodes. Remark We are using the same notation W to denote the finite element space trace and that of their nodal values at points of Γ h. 82

THE CHOICE OF THE MATRIX B Γ In 2D, there is a little choice on how to write the constraint of continuity at a point sitting on an edge, there are many options for a vertex point. For the edge node we only need to choose the sign, whereas for a vertex node, e.g. one common to 4 subdomains, a minimum set of three constraints can be chosen in many different ways to assure continuity at the node in question. Continuity constraints enforced by 3(non-redundant)conditions on the left, by 6(redundant)conditionson the right. 83

THE REDUCED FEM PROBLEM WITH LAGRANGE MULTIPLIERS We reformulate the reduced problem (13) by introducing suitable Lagrange multipliers: { Find (u, λ) W U s.t. Σ u + B T Γ λ = χ, B Γ u = 0. (14) Because of the inf-sup (LBB) condition, thecomponentλ of the solution to (14) is unique up to an additive vector of Ker(BΓ T ), so we choose U = range(b Γ ). 84

Let the matrix R = diag ( R (1),...,R (M)) be made of null-space elements of Σ. (E.g. R (i) corresponds to the rigid body motions of Ω i,incaseof linear elasticity operator.) R is a full column rank matrix. The solution of the first equation of the reduced problem (14) exists iff χ B T Γ λ range(σ ), a limitation that will be resolved by introducing asuitableprojectionoperatorp. Then, u =Σ (χ BT Γ λ)+rα if χ BT Γ λ Ker(Σ ), where α is an arbitrary vector and Σ apseudoinverseofσ. 85

Substituting u into the second equation of the reduced problem (14) yields: B Γ Σ BT Γ λ = B ΓΣ χ + B Γ Rα. (15) Let us set F = B Γ Σ BT Γ and d = B ΓΣ χ. Then choose P T to be a suitable projection matrix, e.g. P T = I G(G T G) 1 G T,withG = B Γ R. We obtain: { P T F λ = P T d, G T λ = e (= R T χ ). 86

ONE-LEVEL FETI METHOD The original one-level FETI method is a CG method in the space V applied to: P T F λ = P T d, λ λ 0 + V, with an initial λ 0 s.t. G T λ 0 = e. Here V = {λ U : λ, B Γ z =0, z Ker(Σ )} is the so-called space of admissible increments, Ker(G T )=range(p) and: V = {µ U : µ, B Γ z Q =0, z Ker(Σ )} = range(p T ). 87

FETI PRECONDITIONERS The original, most basic FETI preconditioner is: P 1 h = B Γ Σ B T Γ = M i=1 B(i) Σ i B (i)t. It is called a Dirichlet preconditioner since its application to a given vector involves the solution of M independent Dirichlet problems, one in every subdomain. The coarse space in FETI consists of the nullspace on each substructure. 88

In case B Γ has full row rank (i.e. the constraints are linearly independent and there are no redundant Lagrange multipliers), a better preconditioner can be defined as follows: P 1 h =(B Γ D 1 B T Γ ) 1 B Γ D 1 Σ D 1 B T Γ (B ΓD 1 B T Γ ) 1, where D is ( a block diagonal matrix with positive entries D = diag D (1),...,D (M)) with D (i) adiagonalmatrixwhose elements are δ i (x) correspondingtothepointx Ω ih (recall the scaling counting functions). 89

Remark Since B Γ D 1 BΓ T is block-diagonal, itsinversecanbeeasily computed by inverting small blocks whose size is n x,thenumberof Lagrange multipliers used to enforce continuity at point x. The matrix D, thatoperatesonelementsoftheproductspacew, can be regarded as a scaling from the right of B Γ by D 1/2.With this choice: 1 K 2 (P P h PT F ) C(1 + log(h/h)) 2, where K 2 ( ) isthespectral condition number and C is a constant independent of h, H, γ and the values of the ρ i. 90

FETI-DP (Dual Primal FETI) METHODS 91

OVERVIEW OF THE FETI-DP METHODS The FETI-DP method is a domain decomposition method that enforces equality of the solution at subdomains interfaces by Lagrange multipliers except at subdomains corners, which remain primal variables. The first mathematical analysis of the method was provided by Mandel and Tezaur (2001). The method was further improved by enforcing the equality of averages across the edges or faces on subdomain interfaces. This is important for parallel scalability. 92

FINITE ELEMENT SPACES We consider a 2D case for simplicity. We introduce a space W s.t. Ŵ W W for which we have continuity of the primal variables at subdomain vertices. We write W as: where: W = ŴΠ W, Ŵ Π Ŵ is the space of continuous interface functions that vanish at all nodal points of Γ h except at the subdomain vertices. ŴΠ is given in terms of the vertex variables and the averages of the values over the individual edges of the set of interface nodes Γ h. W is the direct sum of local subspaces W,i : W = M i=1 W,i, where W,i W i consists of local functions on Ω i that vanish at the vertices of Ω i and have zero average on each individual edge. 93

FINITE ELEMENT SPACES According to this space splitting: the continuous degrees of freedom associated with the subdomain vertices and with the subspace ŴΠ are called primal (Π); those (that are potentially discontinuous across Γ) that are associated with the subspaces W,i and with the interior of the subdomain edges are called dual ( ). The subspace ŴΠ, togetherwiththeinteriorsubspace,definesthe subsystem which is fully assembled, factored, and stored in each iteration step. 94

THE SCHUR COMPLEMENT All unknowns of the subspace ŴΠ as well as the interior variables are eliminated to obtain a new Schur complement Σ. We consider the following procedure. Let à denote the stiffness matrix obtained by restricting diag (A 1,...,A M )from M i=1 W h (Ω i )to W h (Ω) (these spaces now refer to subdomains, not their boundaries). Then à is no longer block diagonal because of the coupling that now exists between subdomains that share a common vertex. 95

According to the previous space decomposition, Ã can be split as: A II A I Π A I Ã = A T I Π A ΠΠ A Π. A T I AT Π A The subscript I refers to the internal degrees of freedom of the subdomains, Π to those associated to the subdomains vertices, and to those of the interior of the subdomains edges. The matrices A II and A are block diagonal (one block per subdomain). Any non-zero entry of A I represents a coupling between degrees of freedom associated with the same subdomain. 96

Degrees of freedom of the space W for one-level FETI (left) and those of the space W for one-level FETI-DP (right) in the case of primal vertices only. 97

Upon eliminating the variables of the I and Π sets, a Schur complement associated with the variables of the sets (interior and edges) is obtained as follows: Σ =A [ A T I A T Π ] [ ] 1 [ ] A II A I Π AI A T. I Π A ΠΠ A Π Correspondingly we obtain a reduced right hand side χ. 98

THE REDUCED FEM PROBLEM By indicating with u W the vector of degrees of freedom associated with the edges, similarly to what done in (13) for FETI, the finite element problem can be reformulated as a reduced minimization problem with constraints given by the requirement of continuity across all of Γ: Find u W : J(u ) = 1 2 Σu, u χ, u min, B u = 0. The matrix B is made of {0, 1, 1} as it was for B Γ. Remark The constraints associated with the vertex nodes are dropped since they are assigned to the primal set. Remark Since all the constraints refer to edge points, no distinction needs to be made between redundant and non-redundant constraints and Lagrange multipliers. (16) 99

THE REDUCED FEM PROBLEM WITH LANGRANGE MULTIPLIERS A saddle point formulation of (16), similar to (14), can be obtained by introducing a set of Lagrange multipliers λ V = range(b ). Indeed, since à is s.p.d., so is Σ: by eliminating the subvectors u we obtain the reduced system: F λ = d, (17) where F = B Σ 1 B T and d = B Σ 1 χ. Remark Once λ is found, u = Σ 1 ( χ B T λ) W,whiletheinterior variables u I and the vertex variables u Π are obtained by back-solving the system associated with Ã. 100

AFETI-DPPRECONDITIONER A preconditioner for F is introduced as done for FETI (in case of non-redundant Lagrange multipliers): P 1 =(B D 1 BT ) 1 B D 1 S D 1 BT (B D 1 BT ) 1. D is a diagonal scaling matrix with blocks D (i) :eachoftheir diagonal elements corresponds to a Lagrange multiplier that enforces continuity between the nodal values of some w i W i and w j W j at some point x Γ h and it is given by δ j (x); S = diag (Σ 1,,...,Σ M, ) with Σ i, being the restriction of the local Schur complement Σ i to W,i W i. 101

Remark When using the conjugate gradient method for the preconditioned system: P 1 F λ = P 1 d, in contrast with one level FETI methods we can use an arbitrary initial guess λ 0. We have a condition number that scales polylogarithmically, that is: K 2 (P 1 F ) C(1 + log(h/h)) 2, where C is independent of h, H,γ and the values of the ρ i. 102

COMPARISON OF FETI AND FETI-DP METHODS FETI-DP algorithms do not require the characterization of the kernels of local Neumann problems (as required by one-level FETI methods), because the enforcement of the additional constraints in each iteration always makes the local problems nonsingular and at the same time provides an underlying coarse global problem. FETI-DP methods do not require the introduction of a scaling matrix Q, whichentersintheconstructionofacoarsesolverfor one-level FETI algorithms. One-level FETI methods are projected conjugate gradient algorithms that cannot start from an arbitrary initial guess. In contrast, FETI-DP methods are standard preconditioned conjugate algorithms and can therefore employ an arbitrary initial guess λ 0. 103

BDDC (Balancing Domain Decomposition with Constraints) METHODS 104

AQUICKVIEWOFBDDCMETHODS The BDDC method was introduced by Dohrmann (2003) as a simpler primal alternative to the FETI-DP domain decomposition method. The name BDDC was coined by Mandel and Dohrmann because it can be understood as further development of the balancing domain decomposition method. BDDC is used as a preconditioner for the conjugate gradient method. 105

AspecificversionofBDDCischaracterizedbythechoiceofcoarse degrees of freedom, whichcanbevaluesatthecornersofthe subdomains, or averages over the edges of the interface between the subdomains. One application of the BDDC preconditioner combines the solution of local problems on each subdomain with the solution of a global coarse problem with the coarse degrees of freedom as the unknowns. The local problems on different subdomains are completely independent of each other, so the method is suitable for parallel computing. 106

GENERAL FORM OF BDDC PRECONDITIONERS ABDDCpreconditionerreads: P 1 BDDC = R T DΓ Σ 1 RDΓ, where R Γ : Ŵ W is a restriction matrix, R DΓ is a scaled variant of R Γ with scale factor δ i (featuring the same sparsity pattern of R Γ ). This scaling is chosen in such a way that R Γ R T DΓ is a projection (then it coincides with its square). 107

ALGEBRAIC FORM OF SCHWARZ ITERATIVE METHODS 108

ALGEBRAIC FORM OF SCHWARZ METHODS FOR FE DISCRETIZATION Let Au = f be the system associated to the finite element approximation of the Poisson problem. We still assume that Ω is decomposed in two overlapping subdomains Ω 1 and Ω 2. We denote by N h the total number of interior nodes of Ω, and by n 1 and n 2 the interior nodes of Ω 1 and Ω 2 : N h n 1 + n 2 The stiffness matrix A contains two submatrices, saya 1 and A 2, which correspond to the local stiffness matrices associated to the Dirichlet problems in Ω 1 and Ω 2 : n 1 A 1 A N h A 2 n 2 109

MULTIPLICATIVE SCHWARZ METHOD Consider the multiplicative method: givenu (0) 2 on Γ 1,fork 1: Lu (k) 1 = f in Ω 1 u (k) 1 = u (k 1) 2 on Γ 1 u (k) 1 =0 on Ω 1 \ Γ 1 ( ) ( ) u (k) AΩ1 A Ω 1 Γ1 u (k) = f 1 Γ 1 u (k) Γ 1 = u (k 1) Ω 2 Γ 1 Lu (k) 2 = f in Ω 2 u (k) 2 = u (k) 1 on Γ 2 u (k) 2 =0 on Ω 2 \ Γ 2 ( ) ( ) u (k) AΩ2 A Ω 2 Γ2 u (k) = f 2 Γ 2 u (k) Γ 2 = u (k) Ω 1 Γ 2 Ω 2 Ω 1 Γ 2 Γ 1 110

We find: A Ω1 u (k) Ω 1 = f 1 A Γ1 u (k 1) Γ 1 A Ω2 u (k) Ω 2 = f 2 A Γ2 u (k) Γ 2 or, equivalently, by adding and subtracting u (k 1) Ω i : u (k) Ω 1 u (k) Ω 2 = u (k 1) Ω 1 = u (k 1) Ω 2 + A 1 Ω 1 (f 1 A Ω1 u (k 1) Ω 1 A Γ1 u (k 1) Γ 1 ) + A 1 Ω 2 (f 2 A Ω2 u (k 1) Ω 2 A Γ2 u (k) Γ 2 ) Remark This method can be seen as a block Gauss-Seidel method to solve the system ( )( ) ( ) AΩ1 A Γ1 uω1 f1 = A Ω2 u Ω2 f 2 A Γ2 111

If there is no direct coupling between the matrices assembled on opposite sides of Γ 1 and Γ 2 : A Ω2\Ω 1 =0 and A Ω1\Ω 2 =0 Ω 2 Ω 1 Γ 2 Γ 1 then, formally, we can write so that A Γ1 = A (Ω\Ω1)\(Ω 2\Ω 1) = A (Ω\Ω 1) A Γ1 u (k 1) Γ 1 = A (Ω\Ω1)u (k 1) (Ω\Ω 1) Proceeding in a similar way also for the second equation, we finally obtain: u (k) Ω 1 u (k) Ω 2 = u (k 1) Ω 1 = u (k 1) Ω 2 + A 1 Ω 1 (f 1 A Ω1 u (k 1) Ω 1 A (Ω\Ω1)u (k 1) (Ω\Ω ) 1) + A 1 Ω 2 (f 2 A Ω2 u (k 1) Ω 2 A (Ω\Ω2)u (k) (Ω\Ω ) 2) 112

In compact form we can write ( ) u (k+ 1 2 A ) = u (k) 1 + Ω 1 0 (f ) Au (k) 0 0 ( ) 0 0 ( u (k+1) = u (k+ 1 2 ) + 0 A 1 f Au (k+ 1 Ω 2 Let R i be the (rectangular) restriction matrix that returns the vector of coefficients in the interior of Ω i : ( ) uωi u Ωi = R i u =(I 0) u Ω\Ω1 Then, we obtain: u (k+ 1 2 ) = u (k) + R T 1 (R 1AR T 1 ) 1 R 1 ( f Au (k) ) 2 ) ) u (k+1) = u (k+ 1 2 ) + R T 2 (R 2AR T 2 ) 1 R 2 ( f Au (k+ 1 2 ) ) 113

Define, for i =1, 2 Q i = R T i (R i AR T i ) 1 R i We get u (k+ 1 2 ) = u (k) + Q 1 ( f Au (k)) u (k+1) = u (k+ 1 2 ) + Q 2 ( f Au (k+ 1 2 )) and ( u (k+1) = u (k) +Q 1 f Au (k)) +Q 2 [f A (u (k) + Q 1 (f Au (k)))] or, equivalently, u (k+1) = u (k) +(Q 1 + Q 2 Q 2 AQ 1 ) ( f Au (k)) 114

ADDITIVE SCHWARZ METHOD Proceeding as before, we can prove that the additive Schwarz method can be seen as a block Jacobi method to solve the system ( )( ) ( ) AΩ1 A Γ1 uω1 f1 = A Ω2 u Ω2 f 2 A Γ2 One iteration of the additive Schwarz method can be written as u (k+1) = u (k) +(Q 1 + Q 2 )(f Au (k) ) In the case of a M 2overlappingsubdomains{Ω i } we have: ( M ) u (k+1) = u (k) + Q i (f Au (k) ) i=1 115

THE SCHWARZ METHOD AS A PRECONDITIONER Multiplicative Schwarz method: can be interpreted as a Richardson iterative procedure for Au = f with the multiplicative Schwarz preconditioner P 1 ms = Q 1 + Q 2 Q 2 AQ 1 P ms is not symmetric and can be used within GMRES or BiCGStab iterations. Additive Schwarz method: corresponds to Richardson iterations for Au = f with the additive Schwarz preconditioner ( M ) 1 P as = Q i i=1 116

The convergence rate for the additive Schwarz method is slower than for the multiplicative Schwarz method. In the case of two subdomains, there is approximately a factor 1 2. (This is similar to the classical convergence results for the Jacobi and Gauss-Seidel methods) P as is not optimal as the condition number blows up if the size of the subdomains reduces: κ(p 1 as A) C 1 δh where C is a constant independent of h, H, andδ, δ being a linear measure of the overlapping region(s) (h δ H). 117

ALGORITHMIC ASPECTS & NUMERICAL RESULTS Create overlapping subdomains a) Triangulate the computational domain Ω T h. b) Partition the mesh into M nonoverlapping subdomains {ˆΩ i } M i=1. c) Extend every subdomain ˆΩ i by adding all the strips of finite elements of T h within distance δ from ˆΩ i. ˆΩ 1 ˆΩ 4 ˆΩ 7 ˆΩ 2 ˆΩ 5 ˆΩ 8 Ω 5 ˆΩ 3 ˆΩ 6 ˆΩ 9 Start-up phase for the application of P as a) On each subdomain Ω i build R i and Ri T. b) Build the matrix A corresponding to the FE discretization on T h. c) On each Ω i form the local submatrices A Ωi = R i ARi T. d) On each Ω i prepare the code to solve a linear system with A Ωi : compute a factorization (LU, ILU, Cholesky...) of A Ωi. 118

One-level additive Schwarz preconditioner P as Given a vector r, computez = Pas 1 r as follows a) set z = 0; For i =1,...,M, doinparallel b) restrict the residual over Ω i \ Ω i : r i = R i r; c) solve A Ωi z i = r i ; d) add to the global residual: z z + Ri T z i. EndFor Example κ(pas 1 A) H =1/2 H =1/4 H =1/8 H =1/16 h =1/16 15.95 27.09 52.08 - h =1/32 31.69 54.52 104.85 207.67 h =1/64 63.98 109.22 210.07 416.09 h =1/128 127.99 218.48 420.04 832.57 119

The main limitation of Schwarz methods is to propagate information only among neighboring subdomains, sinceonly local solves are involved by the application of (P as ) 1. We have to introduce a coarse global problem over the whole domain to guarantee a mechanism of global communication among all subdomains. 120

TWO-LEVEL SCHWARZ PRECONDITIONERS As for the Neumann-Neumann method, we can introduce a coarse-grid mechanism that allows for a sudden information diffusion on the whole domain Ω: consider the subdomains as macro-elements of a new coarse grid T H build a corresponding stiffness matrix A H The matrix Q 0 = RH T A 1 H R H represents the coarse level correction for the two-level preconditioner, with R H the restriction operator from the fine to the coarse grid. The two-level preconditioner P cas is defined as: P 1 cas = M i=0 Q i 121

We can prove that there exists a constant C > 0, independent of both h and H such that ( κ(pcas 1 A) C 1+ H ) δ If δ is a fraction of H (generous overlap), the preconditioner P cas is scalable and optimal. If δ h, thepreconditioner P cas is neither scalable or optimal. Iterations on the original finite element system using P cas converge with a rate independent of h and H (and therefore of the number of subdomains) Thanks to the additive structure of the preconditioner, the preconditioning step is fully parallel as it involves the solution of independent systems, one per each local matrix A Ωi. 122

ALGORITHMIC ASPECTS & NUMERICAL RESULTS Start-up phase to apply P cas a) Do the start-up phase for P as. b) Define a coarse space triangulation T 0 with elements of order H and n 0 dofs: c) Build the restriction operator R H R n0 N h, R H (i, j) =Φ i (x j ). Φ i is the basis function associated to the coarse grid node i and x j are the coordinates of the fine grid node j. d) Construct the coarse matrix A H,eitheras A H (i, j) = Φ i Φ j or A H = R H ARH T Ω 123

Two-levels additive Schwarz preconditioner P cas Given a vector r, computez = Pcas 1 r as follows a) set z = 0; For i =1,...,M, doinparallel b) restrict the residual over Ω i \ Ω i : r i = R i r; c) solve A Ωi z i = r i ; d) add to the global residual: z z + Ri T z i. EndFor e) Compute the coarse-grid contribution z H = A 1 H (R Hr); f) Add to the global residual z z + RH T z H. Example κ(pcas 1 A) H =1/4 H =1/8 H =1/16 h =1/32 7.03 4.94 - h =1/64 12.73 7.59 4.98 h =1/128 23.62 13.17 7.66 h =1/256 45.33 24.34 13.28 If H/δ = constant, thistwo-levelpreconditioneriseitheroptimaland scalable. 124

Practical indications: For decompositions with a small number of subdomains, the single level Schwarz preconditioner P as is very efficient. When the number M of subdomains gets large, using two-level preconditioners becomes crucial. When generating a coarse grid is difficult, other algebraic techniques, likeaggregation, canbeadoptedtodefinea H,and ( κ(paggre 1 A) C 1+ H ) δ Paggre 1 A H =1/4 H =1/8 H =1/16 h =1/16 13.37 8.87 - h =1/32 26.93 17.71 9.82 h =1/64 54.33 35.21 19.70 h =1/128 109.39 70.22 39.07 125

REFERENCES 126

References Barry Smith, Petter Bjorstad, William Gropp. Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations. CambridgeUniversityPress,1996. Alfio Quarteroni, Alberto Valli. Domain Decomposition Methods for Partial Differential Equations. OxfordSciencePublications,1999. Barbara I. Wholmuth. Discretization Methods and Iterative Solvers Based on Domain Decomposition. Springer,2001. Andrea Toselli, Olof Widlund. Domain Decomposition Methods Algorithms and Theory. Springer,2004. Clemens Pechstein. Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale problems. Springer,2013. Alfio Quarteroni. Numerical Models for Differential Problems. Springer, 2012. Official page of Domain Decomposition Methods www.ddm.org 127