1

. Aachen, EU Regional School Series – February 18, 2013

Introduction to Domain Decomposition Methods

Alfio Quarteroni

Chair of Modeling and Scientific ComputingMathematics Institute of Computational Science and Engineering

Ecole Polytechnique Federale de Lausanne

MOX – Modeling and Scientific ComputingMathematics DepartmentPolitecnico di Milano

2

MOTIVATION

DD can be used in the framework of any discretization method forPDEs (FEM, FV, FD, SEM) to make their algebraic solution moree!cient on parallel computer platforms

DDM allow the reformulation of a boundary-value problem on apartition of the computational domain into subdomains! very convenient framework for the solution of heterogeneous ormultiphysics problems, i.e. those that are governed by di"erentialequations of di"erent kinds in di"erent subregions of thecomputational domain.

3

THE IDEA

The computational domain # where the bvp is set is subdivided into twoor more subdomains in which problems of smaller dimension are to besolved.

Parallel solution algorithms may be used.

There are two ways of subdividing the computational domain:

with disjoint subdomains

with overlapping subdomains

# #1#1

#2#2

$1$2 $

Correspondingly, di"erent DD algorithms will be set up.

4

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...

5

...who wanted to establish the existence of harmonic functions withprescribed boundary values on regions with non-smooth boundaries:

!"#u = f in #

u = g on !#

where

# !#

Schwarz’s idea: split # into two elementary subdomains:

# #1

#2

$1$2

and set up a suitable iterative method passing information betweenthe subproblems.

6

MODERN EXAMPLES OF SUBDIVISIONS

7

REFERENCES

B.F. Smith, P.E. Bjørstad, W.D. Gropp (1996)Domain DecompositionCambridge University Press, Cambridge.

A. Quarteroni and A. Valli (1999)Domain Decomposition Methods for Partial Di!erential EquationsOxford Science Publications, Oxford.

A. Toselli and O.B. Widlund (2005)Domain Decomposition Methods – Algorithms and TheorySpringer-Verlag, Berlin and Heidelberg.

8

CLASSICAL ITERATIVE DD METHODS

9

MODEL PROBLEM

Consider the model problem:

find u : #$ R s.t.

!Lu = f in #u = 0 on !#

L is a generic second order elliptic operator.

The weak formulation reads:

find u % V = H10 (#) : a(u, v) = (f , v) &v % V

where a(·, ·) is the bilinear form associated with L.

10

SCHWARZ METHODS

Consider a decomposition of # with overlap:

# #1

#2

$1$2

Given u(0)2 on $1, for k ' 1:

solve

"#$

#%

Lu(k)1 = f in #1

u(k)1 = u(k!1)2 on $1

u(k)1 = 0 on !#1 \ $1

solve

"###$

###%

Lu(k)2 = f in #2

u(k)2 =

&u(k)1

u(k!1)1

on $2

u(k)2 = 0 on !#2 \ $2.

11

Choice of the trace on $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 #1 and #2, and we

wish the two sequences {u(k)1 } and {u(k)2 } to converge to the restrictionsof the solution u of the original problem:

limk"# u(k)1 = u|!1and limk"# u(k)2 = u|!2

.

The Schwarz method applied to the model problem always converges,with a rate that increases as the measure |$12| of the overlapping region$12 increases.

12

ExampleConsider the model problem

!"u$$(x) = 0 a < x < b,u(a) = u(b) = 0,

where

#1

#2

a b"1"2

The solution is u = 0. We show a few iterations of the method:

a b"1"2

u(0)2

u(1)1

u(1)2u(2)1 u(2)2

Clearly, the method converges with a rate that reduces as the length ofthe interval ("2, "1) gets smaller.

13

Remark

The Schwarz method requires at each iteration the solution of twosubproblems 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.

14

NONOVERLAPPING DECOMPOSITIONWe partition now the domain # in two disjoint subdomains:

# #1

#2

$

The following equivalence result holds.

TheoremThe solution u of the model problem is such that u|!i

= ui for i = 1, 2,where ui is the solution to the problem

!Lui = f in #i

ui = 0 on !#i \ $

with interface conditions

u1 = u2 and!u1!nL

=!u2!nL

on $

!/!nL is the conormal derivative.

15

ExampleThe problem !

"div (µ(u) = f in #u = 0 on !#

is equivalent to "###$

###%

"div (µi(ui ) = fi in #i

u = 0 on !#i

u1 = u2 on $

µ1!u1!n

= µ2!u2!n

on $

RemarkThe proof of the theorem can be done using the weak formulation of theproblem. We refer to Lemma 1.2.1 in [QV99].

16

DIRICHLET-NEUMANN METHOD

Given u(0)2 on $, for k ' 1 solve the problems:

"#$

#%

Lu(k)1 = f in #1

u(k)1 = u(k!1)2 on $

u(k)1 = 0 on !#1 \ $

"###$

###%

Lu(k)2 = f in #2

!u(k)2

!nL=!u(k)1

!nLon $

u(k)2 = 0 on !#2 \ $

The equivalence theorem guarantees that when the two sequences

{u(k)1 } and {u(k)2 } converge, their limit will be necessarily thesolution to the exact problem.

The DN algorithm is therefore consistent.

However, its convergence is not always guaranteed.

17

ExampleLet # = (a, b), " % (a, b), L = "d2/dx2 and f = 0. At every k ' 1 theDN algorithm generates the two subproblems:

"#$

#%

"(u(k)1 )$$ = 0 a < x < "

u(k)1 = u(k!1)2 x = "

u(k)1 = 0 x = a"#$

#%

"(u(k)2 )$$ = 0 " < x < b

(u(k)2 )$ = (u(k)1 )$ x = "

u(k)2 = 0 x = b.

The two sequences converge only if " > (a + b)/2:

a b"

u(0)2

u(1)2

a+b2

u(2)2 a b"

u(0)2

u(1)2

a+b2

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A variant of the DN algorithm can be set up by replacing theDirichlet condition in the first subdomain by

u(k)1 = #u(k!1)2 + (1" #)u(k!1)

1 on $

using a relaxation parameter # > 0.In such a way it is always possible to reduce the error between twosubsequent iterates.

In the previous example, we can easily verify that, by choosing

#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 < #max < 1 such thatthe DN algorithm converges for any possible choice of the relaxationparameter # in the interval (0, #max).

19

NEUMANN-NEUMANN ALGORITHMConsider again a partition of # into two disjoint subdomains and denoteby $ the (unknown) value of the solution u on their interface $:

$ = ui on $ (i = 1, 2)

Consider the following iterative algorithm: for any given $(0) on $, fork ' 0 and i = 1, 2, solve the following problems:

"#$

#%

Lu(k+1)i = f in #i

u(k+1)i = $(k) on $

u(k+1)i = 0 on !#i \ $

"##$

##%

L%(k+1)i = 0 in #i

!%(k+1)i

!n=!u(k+1)

1

!n"!u(k+1)

2

!non $

%(k+1)i = 0 on !#i \ $

with$(k+1) = $(k) " #

'&1%

(k+1)1|" " &2%

(k+1)2|"

(

where # is a positive acceleration parameter, while &1 and &2 are twopositive coe!cients.

20

ROBIN-ROBIN ALGORITHM

For every k ' 0 solve the following problems:

"##$

##%

u(k+1)1 = f in #1

u(k+1)1 = 0 on !#1 \ $

!u(k+1)1

!n+ "1u

(k+1)1 =

!u(k)2

!n+ "1u

(k)2 on $

then"##$

##%

Lu(k+1)2 = f in #2

u(k+1)2 = 0 on !#2 \ $

!u(k+1)2

!n+ "2u

(k+1)2 =

!u(k+1)1

!n+ "2u

(k+1)1 on $

where u(0)2 is assigned and "1, "2 are non-negative accelerationparameters that satisfy "1 + "2 > 0.

Aiming at parallelization, we could use u(k)1 instead of u(k+1)1 .

21

THE STEKLOV-POINCARE INTERFACEEQUATION

22

MULTI-DOMAIN FORMULATION OF POISSONPROBLEM AND INTERFACE CONDITIONS

We consider now the model problem:

!"#u = f in #u = 0 on !#.

For a domain partitioned into two disjoint subdomains, we can write theequivalent multi-domain formulation (ui = u|!i

, i = 1, 2):

"########$

########%

"#u1 = f in #1

u1 = 0 on !#1 \ $"#u2 = f in #2

u2 = 0 on !#2 \ $u1 = u2 on $

!u1!n

=!u2!n

on $.

23

Remark

On the interface $ we have the normal unit vectors n1 and n2:

$

n1#1

n2 #2

There holds: n1 = "n2 on $.

We denote n = n1 so that

!

!n=

!

!n1= "

!

!n2on $.

24

THE STEKLOV-POINCARE OPERATOR

Let $ be the unknown value of the solution u on the interface $:

$ = u|"

Should we know a priori the value $ on $, we could solve the followingtwo independent boundary-value problems with Dirichlet condition on $(i = 1, 2): "

$

%

"#wi = f in #i

wi = 0 on !#i \ $wi = $ on $.

25

With the aim of obtaining the value $ on $, let us split wi as follows

wi = w%i + u0i ,

where w%i and u0i represent the solutions of the following problems

(i = 1, 2): "$

%

"#w%i = f in #i

w%i = 0 on !#i ) !#

w%i = 0 on $

and "$

%

"#u0i = 0 in #i

u0i = 0 on !#i ) !#u0i = $ on $.

26

The functions w%i depend solely on the source data f

! w%i = Gi f

where Gi is a linear continuous operator

u0i depend solely on the value $ on $

! u0i = Hi$

where Hi is the so-called harmonic extension operator of $ on thedomain #i .

We have that ui = w%i + u0i (i = 1, 2)

*!w1

!n=!w2

!non $

*!

!n(w%

1 + u01) =!

!n(w%

2 + u02) on $

*!

!n(G1f + H1$) =

!

!n(G2f + H2$) on $

*

)!H1

!n"!H2

!n

*$ =

)!G2

!n"!G1

!n

*f on $.

27

We have obtained the Steklov-Poincare equation for the unknown $ onthe interface $:

S$ = ' on $

S is the Steklov-Poincare pseudo-di"erential operator:

Sµ =!

!nH1µ"

!

!nH2µ =

2+

i=1

!

!niHiµ

' is a linear functional which depends on f :

' =!

!nG2f "

!

!nG1f = "

2+

i=1

!

!niGi f .

The operator

Si : µ$ Siµ =!

!ni(Hiµ)

,,,,"

i = 1, 2,

is called local Steklov-Poincare operator (Dirichlet-to-Neumann)which operates between the trace space

% = {µ : +v % V s.t. µ = v|"} = H1/200 ($)

and its dual %$.

28

Example

To provide an example of the operator S , we consider a simple 1Dproblem.Let # = (a, b) , R as illustrated below.We split # in two nonoverlapping subdomains. In this case the interface$ reduces to the point " % (a, b) and the Steklov-Poincare operator Sbecomes

S$ =

)dH1

dx"

dH2

dx

*$ =

)1

l1+

1

l2

*$

where l1 = " " a et l2 = b " ".

a b"

l1 l2

$

H1$ H2$

#1 #2

29

EQUIVALENCE BETWEEN THE DD SCHEMESAND CLASSICAL ITERATIVE METHODS

30

The DN method can be reinterpreted as a preconditioned Richardsonmethod for the solution of the Steklov-Poincare interface equation:

PDN($(k) " $(k!1)) = #('" S$(k!1))

The preconditioning operator is PDN = S2 = !(H2µ)/!n2.

The NN method can also be interpreted as a preconditionedRichardson algorithm

PNN($(k) " $(k!1)) = #('" S$(k!1))

with PNN = (&1S!11 + &2S

!12 )!1.

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The Robin-Robin algorithm is equivalent to the following alternatingdirection (ADI) algorithm:

("1I# + S1)µ(k)1 = '+ ("1I# + S2)µ

(k!1)2 ,

("2I# + S2)µ(k)2 = '+ ("2I# + S1)µ

(k!1)1 ,

where I# : %$ %$ here denotes the Riesz isomorphism between theHilbert space % and its dual %$.

At convergence (for a convenient choice of "1 and "2), we haveµ1 = µ2 = $.

32

FINITE ELEMENT APPROXIMATION:MULTI-DOMAIN FORMULATION

33

Consider the Poisson problem:

!"#u = f in #

u = 0 on !#

Its weak formulation reads

find u % V : a(u, v) = F (v) &v % V

where V = H10 (#),

a(v ,w) =

-

!(v ·(w &v ,w % V

and

F (v) =

-

!f v &v % V .

34

Suppose that # is split into two nonoverlapping subdomains and considera uniform triangulation Th of #, conforming on $:

#1

#2

$

35

The Galerkin finite element approximation of the Poisson problem reads:

find uh % Vh : a(uh, vh) = F (vh) &vh % Vh (1)

where

Vh = {vh % C 0(#) : vh|K % Pr r ' 1, &K % Th, vh = 0 on !#}

is the space of finite element functions of degree r with basis {(i}Nh

i=1.

The Galerkin approximation (1) is equivalent to:

find uh % Vh : a(uh,(i ) = F ((i ) &i = 1, . . . ,Nh. (2)

36

We partition the nodes of the triangulation as follows:

{x (1)j , 1 - j - N1} nodes in subdomain #1

{x (2)j , 1 - j - N2} nodes in subdomain #2

{x (")j , 1 - j - N"} nodes on the interface $,

and we split the basis functions accordingly:

((1)j functions associated to the nodes x (1)j

((2)j functions associated to the nodes x (2)j

((")j functions associated to the nodes x (")j on the interface.

37

Example

#1

#2

$

x (1)j

x (2)j

x (")j

38

Problem (2) can be equivalently rewritten as:

find uh % Vh :

"#$

#%

a(uh,((1)i ) = F (((1)

i ) &i = 1, . . . ,N1

a(uh,((2)j ) = F (((2)

j ) &j = 1, . . . ,N2

a(uh,((")k ) = F (((")

k ) &k = 1, . . . ,N".

(3)

We introduce the bilinear form on #i :

ai(v ,w) =

-

!i

(v ·(w &v ,w % V , i = 1, 2,

the space

V ih = {v % H1(#i ) : v = 0 on !#i \ $} (i = 1, 2)

and let u(i)h = uh|!i(i = 1, 2).

Then, problem (3) can be written equivalently as:

find u(1)h % V 1h , u

(2)h % V 2

h s.t."###$

###%

a1(u(1)h ,((1)

i ) = F1(((1)i ) &i = 1, . . . ,N1

a2(u(2)h ,((2)

j ) = F2(((2)j ) &j = 1, . . . ,N2

a1(u(1)h ,((")

k |!1) + a2(u(2)h ,((")

k |!2)

= F1(((")k |!1) + F2((

(")k |!2) &k = 1, . . . ,N".

(4)

39

Remark

Problem (4) corresponds to the finite element approximation of themulti-domain formulation of the Poisson problem:

"#######$

#######%

"#u1 = f in #1

u1 = 0 on !#1 \ $"#u2 = f in #2

u2 = 0 on !#2 \ $u1 = u2 on $!u1!n

=!u2!n

on $.

The condition u1 = u2 on $ is satisfied by definition of u(i)h .

40

We can write:

uh(x) =N1+

j=1

uh(x(1)j )((1)

j (x) +N2+

j=1

uh(x(2)j )((2)

j (x)

+N"+

j=1

uh(x(")j )((")

j (x),

and we substitute this expression in (4) to obtain...

41

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N1%

j=1

uh(x(1)j )a1(!

(1)j ,!(1)

i ) +N"%

j=1

uh(x(")j )a1(!

(")j ,!(1)

i ) = F1(!(1)i ) !i = 1, . . . ,N1

N2%

j=1

uh(x(2)j )a2(!

(2)j ,!(2)

i ) +N"%

j=1

uh(x(")j )a2(!

(")j ,!(2)

i ) = F2(!(2)i ) !i = 1, . . . ,N2

N"%

j=1

uh(x(")j )

&

a1(!(")j ,!(")

i ) + a2(!(")j ,!(")

i )'

+N1%

j=1

uh(x(1)j )a1(!

(1)j ,!(")

i ) +N2%

j=1

uh(x(2)j )a2(!

(2)j ,!(")

i )

= F1(!(")i |!1 ) + F2(!

(")i |!2) !i = 1, . . . ,N".

42

A bit of algebra...

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N1%

j=1

uh(x(1)j )(A11)ij +

N"%

j=1

uh(x(")j )(A1")ij = F1(!

(1)i ) !i = 1, . . . ,N1

N2%

j=1

uh(x(2)j )(A22)ij +

N"%

j=1

uh(x(")j )(A2")ij = F2(!

(2)i ) !i = 1, . . . ,N2

N"%

j=1

uh(x(")j )

&

(A(1)"" )ij + (A(2)

"" )ij'

+N1%

j=1

uh(x(1)j )(A"1)ij +

N2%

j=1

uh(x(2)j )(A"2)ij

= F1(!(")i |!1 ) + F2(!

(")i |!2 ) !i = 1, . . . ,N".

43

A bit of algebra... [continued]!

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N1%

j=1

u1(A11)ij +N"%

j=1

u"(A1")ij = f1 !i = 1, . . . ,N1

N2%

j=1

u2(A22)ij +N"%

j=1

u"(A2")ij = f2 !i = 1, . . . ,N2

N"%

j=1

u"

&

(A(1)"" )ij + (A(2)

"" )ij'

+N1%

j=1

u1(A"1)ij +N2%

j=1

u2(A"2)ij

= f(")1 + f

(")2 !i = 1, . . . ,N".

44

... so that we obtain the algebraic form:"#$

#%

A11u1 + A1"! = f1A22u2 + A2"! = f2

A"1u1 + A"2u2 +'A(1)"" + A(2)

""

(! = f

(")1 + f

(")2

(5)

or

.

/A11 0 A1"

0 A22 A2"

A"1 A"2 A""

0

1

.

/u1

u2

!

0

1 =

.

/f1f2f"

0

1 (6)

where we denoted A"" ='A(1)"" + A(2)

""

(and f" = f

(")1 + f

(")2 .

45

THE SCHUR COMPLEMENT SYSTEM

46

Since $ represents the unknown value of u on $, its finite elementcorrespondent is the vector ! of the values of uh at the interface nodes.

By Gaussian elimination, we can obtain a new reduced system in the soleunknown !:

Matrices A11 and A22 are invertible since they are associated withtwo homogeneous Dirichlet boundary-value problems for the Laplaceoperator:

u1 = A!111 (f1 " A1"!) and u2 = A!1

22 (f2 " A2"!) . (7)

From the third equation we obtain:

2'A(1)"" " A"1A

!111 A1"

(+'A(2)"" " A"2A

!122 A2"

(3! =

f" " A"1A!111 f1 " A"2A

!122 f2.

47

Setting

& = &1 + &2 with &i = A(i)"" " A"iA

!1ii Ai" (i = 1, 2)

and"" = f" " A"1A

!111 f1 " A"2A

!122 f2

we obtain the Schur complement system

&! = ""

& and "" approximate S and '.

& is the so-called Schur complement of A with respect to u1 and u2.

&i are the Schur complements related to the subdomains #i

(i = 1, 2).

48

Remark

After solving the Schur complement system in !, thanks to (7) wecan compute u1 and u2.

This is equivalent to solving two Poisson problems in thesubdomains #1 and #2 with the Dirichlet boundary condition

u(i)h |"= $h (i = 1, 2) on the interface $.

49

PROPERTIES OF THE SCHUR COMPLEMENT !

The Schur complement & inherits some of the properties of A:

if A is singular, so is &;

if A (respectively, Aii ) is symmetric, then & (respectively, &i ) issymmetric too;

if A is positive definite, so is &.

Moreover, concerning the condition number, we have

)(A) . Ch!2

)(&) . Ch!1

50

PRECONDITIONERS FOR THE SCHURCOMPLEMENT SYSTEM

51

The iterative methods that we have illustrated are equivalent topreconditioned Richardson methods for the Schur complement systemwith preconditioners:

for the DN algorithm: Ph = &2

for the ND algorithm: Ph = &1

for the NN algorithm: Ph = (&1&!11 + &2&

!12 )!1

for the RR algorithm: Ph = ("1 + "2)!1("1I + &1)("2I + &2)

52

All these preconditioners are optimal in the sense of the followingdefinition.

DefinitionA preconditioner P is optimal for a matrix A % RN&N if the conditionnumber of P!1A is bounded uniformly with respect to the dimension Nof A.

In particular, we have

)(&!1i &) = O(1) (i = 1, 2)

)((&1&!11 + &2&

!12 )&) = O(1)

for all &1,&2 > 0

53

NONOVERLAPPING METHODS IN THE CASEOF MORE THAN TWO SUBDOMAINS

54

MULTI-DOMAIN FORMULATION FOR M > 2SUBDOMAINS

We generalize now the nonoverlapping methods to the case of a domain# split into M > 2 subdomains:

#i (i = 1, . . . ,M) such that4#i = #

$i = !#i \ !#

$ =4$i .

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1Plotmesh

X − Axis

Y −

Axis

$ik1

$ik4

$ik2

$ik3

#i

#

55

At the di"erential level, we have the equivalent multi-domain formulation:

!"#u = f in #u = 0 on !#

*

"###$

###%

"#ui = f in #i (i = 1, . . . ,M)ui = uk on $ik!ui!ni

=!uk!ni

on $ik

ui = 0 on !#i ) !#

where $ik = !#i ) !#k /= 0.

56

FINITE ELEMENT APPROXIMATION

Considering a conforming finite element approximation, we obtainthe linear system:

)AII AI"

A"I A""

*)uI

!

*=

)fIf"

*(8)

where uI is the vector of unknowns in the internal nodes and ! isthe vector of unknowns on $: ! = u".

The submatrix AII associated to the internal nodes is block-diagonal:

AII =

.

5555/

A11 0 . . . 0

0. . .

......

. . . 00 . . . 0 AMM

0

66661

AI" is a banded matrix (interactions with local interfaces).

57

Remark

On each subdomain #i the matrix

Ai =

)Aii Ai"

A"i A"i"i

*

represents the local sti"ness matrix associated to a Neumann problem.

58

THE SCHUR COMPLEMENT SYSTEM

Since AII is non-singular, from (8) we can derive

uI = A!1II (fI " AI"u") (9)

By eliminating uI from (8), we get

A""! = f" " A"IA!1II (fI " AI"!)

that is 7A"" " A"IA

!1II AI"

8! = f" " A"IA

!1II fI . (10)

Denoting& = A"" " A"IA

!1II AI"

and"" = f" " A"IA

!1II fI

we obtain the Schur complement system in the multi-domain case

&! = ""(11)

59

Example

$1

$2 $3

$4

$c

0 5 10 15 20 25 30

0

5

10

15

20

25

30

nz = 597

Γ1

Γ2

Γ3

Γ4

Γc

On the left, decomposition of ! = (0, 1)" (0, 1) into four square subdomains.

On the right, sparsity pattern of the Schur complement matrix arising from thedecomposition depicted on the left.

RemarkThe local Schur complements are defined as

&i = A"i"i " A"i iA!1ii Ai"i

so that

& = RT"1&1R"1 + . . .RT

"M&MR"M

where R"i is the restriction from the skeleton $ to the boundary of #i .

60

A SIMPLE ALGORITHM

To compute a FE approximation of the solution u of the Poisson problem

!"#u = f in #u = 0 on !#

we can do the following steps:

1 solve the Schur complement system

&! = ""

to compute ! on the whole interface $;

2 solveAIIuI = fI " AI"!

i.e., M independent problems of reduced dimension

AiiuiI = gi (i = 1, . . . ,M)

possibly in parallel.

61

ESTIMATE OF THE CONDITION NUMBER

The following estimate can be proved for the condition number of theSchur complement matrix &:

there exists a constant C > 0, independent of h and H such that

)(&) - CHmax

hH2min

where Hmax and Hmin are the maximal and minimal diameters of thesubdomains, respectively.

62

NUMERICAL RESULTS

We consider the model problem

!"'u = f in # = (0, 1)1 (0, 1)

u = 0 on !#

We decompose # into M square regions #i of characteristic dimension Hsuch that 2Mi=1#i = #:

$1

$2 $3

$4

$c

(Decomposition of ! = (0, 1) " (0, 1) into four square subdomains)

63

Condition number of &:

)(&) H = 1/2 H = 1/4 H = 1/8h=1/8 9.77 14.83 25.27h=1/16 21.49 35.25 58.60h=1/32 44.09 75.10 137.73h=1/64 91.98 155.19 290.43

Notice the dependence on1

hand

1

H.

64

The Schur complement & is dense.

If we use iterative solvers to solve the Schur complement system

&! = ""

it is not convenient from the point of view of memory usageto explicitly compute the elements of &.

We need only to compute the product &x" for any given vector x".

65

For example, in the conjugate gradient method:

a) Choose a starting solution !0

b) Compute r0 = "" " &!0, p0 = r0

For k ' 0 until convergence, Do

c) *k =(pk , rk)

(&pk , pk)d) !k+1 = !k + *kpk

e) rk+1 = rk " *k&pk

f) +k =(&pk , rk+1)

(&pk , pk)g) pk+1 = rk+1 " +kpk

EndFor

66

SCHUR COMPLEMENT MULTIPLICATION BY AVECTOR

We can define the restriction and prolongation operators R"i and RT"i

thatact on the vector of the interface nodal values.

Recalling that

& = &1 + . . .+ &M with &i = A"i"i " A"i iA!1ii Ai"i

we can use the following fully parallel algorithm.

Given x", compute y" = &x" as follows:

a) Set y" = 0

For i = 1, . . . ,M Do in parallel:

b) xi = R"ix"

c) zi = Ai"ixi

d) zi 3 A!1ii zi

e) sum to the global vector y"i 3 A"i"i x" " A"i izi

f) sum to the global vector y" 3 RT"iy"i

EndFor

67

Before using for the first time the Schur complement matrix,an o!-line startup phase is required:

For i = 1, . . . ,M Do in parallel:

a) Built the matrix Ai

b) Reorder Ai as

Ai =

)Aii Ai"i

A"i i A"i"i

*

and extract the submatrices Aii , Ai"i , A"i i and A"i"i

c) Compute the LU or Cholesky factorization of Aii

EndFor

68

SCALABILITY

Definition

A preconditioner Ph of & is said to be scalable if the condition numberof the preconditioned matrix P!1

h & is independent of the number ofsubdomains.

Iterative methods using scalable preconditioners allow henceforth toachieve convergence rates independent of the subdomain number.

Remark

General strategy:

P!1h =

M+

i=1

RT"iP!1i ,h R"i + RT

" P!1H R"

where Pi ,h is a suitable preconditioner for the local Schur complement&i . The last term is the coarse-scale correction.

69

NEUMANN-NEUMANN PRECONDITIONERThe Neumann-Neumann preconditioner for more subdomains reads:

(PNNh )!1 =

M+

i=1

RT"iDi &

%i DiR"i

where &%i is either &!1

i or an approximation of &!1i .

Di is a diagonal matrix of positive weights

Di =

.

5/d1

. . .dn

0

61

dj is the number of subdomains that share the j-th node.

We have the following estimate:

)7(PNN

h )!1&8- CH!2

)1 + log

H

h

*2

70

Remark

The presence of Di and R"i only entails matrix-matrixmultiplications.

Applying &!1i may be done using local inverses: if q is a vector

whose components are the nodal values on $i , then

&!1i q = [0, I ]A!1

i [0, I ]Tq

In particular, the matrix-vector multiplication.

555/

internalnodes

boundary nodes

0

6661

9 :; <A!1i

.

555/

0...0q

0

6661

corresponds to the solution of a Neumann problem in #i :

&"#wi = 0 in #i

!wi

!n= q in $i

71

Example

Condition number of (PNNh )!1&:

)((PNNh )!1&) 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 76.46 194.65h = 1/64 4.53 26.25 105.38 316.54h = 1/128 5.79 31.95 134.02 438.02

Convergence history of the conjugate gradient using PNNh as a

preconditioner and h = 1/32:

0 2 4 6 8 10 12 14 1610−7

10−6

10−5

10−4

10−3

10−2

10−1

100

H=1/4

H=1/8

Conjugate gradient iterations

scal

ed re

sidu

al

Convergence history using PNN

H=1/2

72

BALANCED NEUMANN-NEUMANN PRECONDITIONER

The Neumann-Neumann preconditioner of the Schur complementsystem is not scalable.

A substantial improvement can be achieved by adding a coarse gridcorrection:

(PBNNh )!1& = P0 + (I " P0)

7(PNN

H )!1&8(I " P0)

where P0 = RT

0 &!10 R0& and &0 = R0&R

T

0 , R0 being the restrictionfrom $ onto the coarse level skeleton.

This is called balanced Neumann-Neumann preconditioner.

We can prove that

)7(PBNN

h )!1&8- C

)1 + log

H

h

*2

73

Example

)((PBNNh )!1&) 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 1.29h = 1/64 2.78 2.76 2.08 1.55h = 1/128 3.51 3.67 2.81 2.07

74

CONCLUDING REMARKS

From the numerical results that we have presented, we can conclude withthe following remarks:

even if better conditioned with respect to A, & is ill-conditioned, andtherefore a suitable preconditioner must be applied

the Neumann-Neumann preconditioner can be satisfactory appliedusing a moderate number of subdomains, while for larger M ,)((PNN

h )!1&) > )(&)

the balancing Neumann-Neumann preconditioner is almost optimallyscalable and therefore recommended for partitions with a largenumber of subdomains

75

FETI(Finite Element Tearing & Interconnecting)

METHODS

76

THE MODEL PROBLEM

We consider the variable coe!cient elliptic problem:

&"div(,(u) = f in #,

u = 0 on !#,(12)

where , is piecewise constant, , = ,i % R+ in #i .

We denote by Hi = diam(#i ) the size of the i-th subdomain #i fori = 1, . . . ,M .

77

Domain #, its boundary !#, skeleton $, subdomain #i , and its boundary!#i .

78

THE FINITE ELEMENT SPACES

We introduce:

the space of traces of finite element functions on the boundaries!#i , say Wi = W h(!#i ),

the product space of the previous trace spaces, say W = (Mi=1Wi ,

a subspace of continuous traces across the skeleton $, say =W ,W .

We denote by #ih the nodes in #i , by !#ih the nodes on !#i , by !#h

the nodes on !#, and by $h the nodes on the skeleton $.

79

Finite element degrees of freedom on the boundary (!#h), on theskeleton ($h), in the subdomain (#ih), and on its boundary (!#ih).

80

THE SCALING COUNTING FUNCTIONS

Let us introduce the following scaling counting functions:

-i(x) =

"###$

###%

1 x % !#ih ) (!#h \ $h),'>

j'Nx,!j (x)

(/,!i (x) x % !#ih ) $h,

0 elsewhere,

&x % $h 2 !#h, where: " % [1/2,+4) and Nx is the set of indices of thesubregions having x on their boundaries.

Then we set:

-†i (x) (= pseudo inverses) =

&-!1i (x) if -i(x) /= 0,

0 if -i(x) = 0.

81

THE FINITE ELEMENT APPROXIMATION

Based on the finite element approximation of the model problem (12), letus consider the local Schur complements:

&i = A(i)"" " A",!iA

!1!i ,!i

A!i ,", i = 1, . . . M ,

for which:& = RT

"1&1R"1 + . . .+ RT

"M&MR"M

,

where R"i is a restriction operator, that is a rectangular matrix of zerosand ones that map values on $ onto those on $i , i = 1, . . . ,M . Thematrices &i are positive semi-definite.

We indicate the interface nodal values on !#i as ui , and we setu = (u1, . . . , uM), the local load vectors on !#i as "i and we set"$ = ("1, . . . ,"M). Finally, we set:

&$ = diag(&1, . . . ,&M),

a block diagonal matrix.

82

THE REDUCED FEM PROBLEM

The original FEM problem, when reduced to the interface $, reads :

"$

%Find u %W s.t. J(u) =

1

25&$u, u6 " 5"$, u6 $ min,

B"u = 0.

(13)

The reduced problem (13) admits a unique solution i"Ker{&$} ) Ker{B"} = 0, that is if &$ is invertible on Ker(B").

The matrix B" is not unique, so that we should impose continuity when ubelongs to more than one subdomain; B" is made of {0,"1, 1} since itenforces interfaces continuity constraints at interfaces nodes.

RemarkWe are using the same notation W to denote the finite element spacetrace and that of their nodal values at points of $h.

83

THE CHOICE OF THE MATRIX B$

In 2D, there is a little choice on how to write the constraint of continuityat 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 vertexnode, e.g. one common to 4 subdomains, a minimum set of threeconstraints can be chosen in many di"erent ways to assure continuity atthe node in question.

Continuity constraints enforced by 3 (non-redundant) conditions on theleft, by 6 (redundant) conditions on the right.

84

THE REDUCED FEM PROBLEM WITH LAGRANGEMULTIPLIERS

We reformulate the reduced problem (13) by introducing suitableLagrange multipliers:

&Find (u,!) %W 1 U s.t. &$u+ BT

" ! = "$,

B"u = 0.(14)

Because of the inf-sup (LBB) condition, the component ! of the solutionto (14) is unique up to an additive vector of Ker(BT

" ), so we chooseU = range(B").

85

Let the matrix R = diag7R (1), . . . ,R (M)

8be made of null-space elements

of &$. (E.g. R (i) corresponds to the rigid body motions of #i , in case oflinear elasticity operator.)

R is a full column rank matrix.

The solution of the first equation of the reduced problem (14) exists i""$ "BT

" ! % range(&$), a limitation that will be resolved by introducinga suitable projection operator P . Then,

u = &†$("$ " BT

" !) + R# if "$ " BT" ! 7 Ker(&$),

where # is an arbitrary vector and &†$ a pseudoinverse of &$.

86

Substituting u into the second equation of the reduced problem (14)yields:

B"&†$B

T" ! = B"&

†$"$ + B"R#. (15)

Let us set F = B"&†$B

T" and d = B"&

†$"$.

Then choose PT to be a suitable projection matrix, e.g.PT = I " G(GTG)!1GT , with G = B"R .

We obtain: &PTF! = PTd,

GT! = e (= RT"$).

87

ONE-LEVEL FETI METHOD

The original one-level FETI method is a CG method in the space Vapplied to:

PTF! = PTd , ! % !0 + V ,

with an initial !0 s.t. GT!0 = e. Here

V = {! % U : 5!,B"z6 = 0, z % Ker(&$)}

is the so-called space of admissible increments, Ker(GT ) = range(P)and:

V $ = {µ % U : 5µ,B"z6Q = 0, z % Ker(&$)} = range(PT ).

88

FETI PRECONDITIONERS

The original, most basic FETI preconditioner is:

P!1h = B"&$BT

" =>M

i=1 B(i)&iB(i)T .

It is called a Dirichlet preconditioner since its application to a givenvector involves the solution of M independent Dirichlet problems,one in every subdomain.

The coarse space in FETI consists of the nullspace on eachsubstructure.

89

In case B" has full row rank (i.e. the constraints are linearlyindependent and there are no redundant Lagrange multipliers), abetter preconditioner can be defined as follows:

?P!1h = (B"D!1BT

" )!1B"D!1&$D!1BT" (B"D!1BT

" )!1,

where D is a block diagonal matrix with positive entries

D = diag'D(1), . . . ,D(M)

(with D(i) a diagonal matrix whose

elements are -†i (x) corresponding to the point x % !#ih (recall thescaling counting functions).

90

RemarkSince B"D!1BT

" is block-diagonal, its inverse can be easilycomputed by inverting small blocks whose size is nx , the number ofLagrange multipliers used to enforce continuity at point x .

The matrix D, that operates on elements of the product space W ,can be regarded as a scaling from the right of B" by D!1/2. Withthis choice:

K2(P ?P!1h PTF ) - C (1 + log(H/h))2,

where K2(·) is the spectral condition number and C is a constantindependent of h, H , " and the values of the ,i .

91

FETI-DP(Dual Primal FETI)

METHODS

92

OVERVIEW OF THE FETI-DP METHODS

The FETI-DP method is a domain decomposition method thatenforces equality of the solution at subdomains interfaces byLagrange multipliers except at subdomains corners, which remainprimal variables.

The first mathematical analysis of the method was provided byMandel and Tezaur (2001).

The method was further improved by enforcing the equality ofaverages across the edges or faces on subdomain interfaces.

This is important for parallel scalability.

93

FINITE ELEMENT SPACES

We consider a 2D case for simplicity.

We introduce a space @W s.t. =W , @W ,W for which we have continuityof the primal variables at subdomain vertices.

We write @W as:

@W = =W% 8@W$,

where:=W% , =W is the space of continuous interface functions that vanishat all nodal points of $h except at the subdomain vertices. =W% isgiven in terms of the vertex variables and the averages of the valuesover the individual edges of the set of interface nodes $h.

@W$ is the direct sum of local subspaces @W$,i : @W$ =AM

i=1@W$,i ,

where @W$,i ,Wi consists of local functions on !#i that vanish atthe vertices of #i and have zero average on each individual edge.

94

FINITE ELEMENT SPACES

According to this space splitting:

the continuous degrees of freedom associated with the subdomainvertices and with the subspace =W% are called primal (();

those (that are potentially discontinuous across $) that are

associated with the subspaces @W$,i and with the interior of thesubdomain edges are called dual (').

The subspace =W%, together with the interior subspace, defines thesubsystem which is fully assembled, factored, and stored in eachiteration step.

95

THE SCHUR COMPLEMENT

All unknowns of the subspace =W% as well as the interior variables areeliminated to obtain a new Schur complement B&$.

We consider the following procedure.

Let BA denote the sti"ness matrix obtained by restrictingdiag (A1, . . . ,AM) from

AMi=1W

h(#i ) to @W h(#) (these spaces now referto subdomains, not their boundaries).

Then BA is no longer block diagonal because of the coupling that nowexists between subdomains that share a common vertex.

96

According to the previous space decomposition, BA can be split as:

BA =

C

DAII AI% AI$

ATI% A%% A%$

ATI$ AT

%$ A$$

E

F .

The subscript I refers to the internal degrees of freedom of thesubdomains, ( to those associated to the subdomains vertices, and ' tothose of the interior of the subdomains edges.

The matrices AII and A$$ are block diagonal (one block per subdomain).

Any non-zero entry of AI$ represents a coupling between degrees offreedom associated with the same subdomain.

97

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 verticesonly.

98

Upon eliminating the variables of the I and ( sets, a Schur complementassociated with the variables of the ' sets (interior and edges) isobtained as follows:

B& = A$$ "GATI$A

T%$

H I AII AI%

ATI% A%%

J!1 IAI$

A%$

J.

Correspondingly we obtain a reduced right hand side "$.

99

THE REDUCED FEM PROBLEM

By indicating with u$ % @W$ the vector of degrees of freedom associatedwith the edges, similarly to what done in (13) for FETI, the finiteelement problem can be reformulated as a reduced minimization problemwith constraints given by the requirement of continuity across all of $:

"$

%Find u$ % @W$ : J(u$) =

1

25B&u$, u$6 " 5"$, u$6 $ min,

B$u$ = 0.

(16)The matrix B$ is made of {0,"1, 1} as it was for B".

RemarkThe constraints associated with the vertex nodes are dropped since theyare assigned to the primal set.

RemarkSince all the constraints refer to edge points, no distinction needs to bemade between redundant and non-redundant constraints and Lagrangemultipliers.

100

THE REDUCED FEM PROBLEM WITH LANGRANGEMULTIPLIERS

A saddle point formulation of (16), similar to (14), can be obtained byintroducing a set of Lagrange multipliers ! % V = range(B$).

Indeed, since BA is s.p.d., so is B&: by eliminating the subvectors u$ weobtain the reduced system:

F$! = d$, (17)

where F$ = B$B&!1BT

$ and d$ = B$B&!1"$.

RemarkOnce ! is found, u$ = B&!1("$ " BT

$!) % @W$, while the interiorvariables uI and the vertex variables u% are obtained by back-solving thesystem associated with BA.

101

A FETI-DP PRECONDITIONER

A preconditioner for F is introduced as done for FETI (in case ofnon-redundant Lagrange multipliers):

P!1$ = (B$D

!1$ BT

$ )!1B$D!1$ S$$D

!1$ BT

$ (B$D!1$ BT

$)!1.

D$ is a diagonal scaling matrix with blocks D(i)$ : each of their

diagonal elements corresponds to a Lagrange multiplier that enforcescontinuity between the nodal values of some wi %Wi and wj %Wj

at some point x % $h and it is given by -†j (x);

S$$ = diag (&1,$$, . . . ,&M,$$) with &i ,$$ being the restriction of

the local Schur complement &i to @W$,i ,Wi .

102

RemarkWhen using the conjugate gradient method for the preconditionedsystem:

P!1$ F$! = P!1

$ d$,

in contrast with one level FETI methods we can use an arbitrary initialguess !0.

We have a condition number that scales polylogarithmically, that is:

K2(P!1$ F$) - C (1 + log(H/h))2,

where C is independent of h,H , " and the values of the ,i .

103

COMPARISON OF FETI AND FETI-DP METHODS

FETI-DP algorithms do not require the characterization of thekernels of local Neumann problems (as required by one-level FETImethods), because the enforcement of the additional constraints ineach iteration always makes the local problems nonsingular and atthe same time provides an underlying coarse global problem.

FETI-DP methods do not require the introduction of a scalingmatrix Q, which enters in the construction of a coarse solver forone-level FETI algorithms.

One-level FETI methods are projected conjugate gradient algorithmsthat cannot start from an arbitrary initial guess. In contrast,FETI-DP methods are standard preconditioned conjugate algorithmsand can therefore employ an arbitrary initial guess !0.

104

BDDC(Balancing Domain Decomposition

with Constraints)METHODS

105

A QUICK VIEW OF BDDC METHODS

The BDDC method was introduced by Dohrmann (2003) as asimpler primal alternative to the FETI-DP domain decompositionmethod.

The name BDDC was coined by Mandel and Dohrmann because itcan be understood as further development of the balancing domaindecomposition method.

BDDC is used as a preconditioner for the conjugate gradient method.

106

A specific version of BDDC is characterized by the choice of coarsedegrees of freedom, which can be values at the corners of thesubdomains, or averages over the edges of the interface between thesubdomains.

One application of the BDDC preconditioner combines the solutionof local problems on each subdomain with the solution of a globalcoarse problem with the coarse degrees of freedom as the unknowns.

The local problems on di"erent subdomains are completelyindependent of each other, so the method is suitable for parallelcomputing.

107

GENERAL FORM OF BDDC PRECONDITIONERS

A BDDC preconditioner reads:

P!1BDDC = BRT

D"B&!1BRD",

where BR" : =W $ @W is a restriction matrix, BRD" is a scaled variant of BR"

with scale factor -†i (featuring the same sparsity pattern of BR").

This scaling is chosen in such a way that BR"BRTD" is a projection (then it

coincides with its square).

108

ALGEBRAIC FORM OF SCHWARZ ITERATIVEMETHODS

109

ALGEBRAIC FORM OF SCHWARZ METHODS FOR FEDISCRETIZATION

Let Au = f be the system associated to the finite elementapproximation of the Poisson problem.We still assume that # is decomposed in two overlappingsubdomains #1 and #2.We denote by Nh the total number of interior nodes of #, and by n1and n2 the interior nodes of #1 and #2: Nh - n1 + n2The sti"ness matrix A contains two submatrices, say A1 and A2,which correspond to the local sti"ness matrices associated to theDirichlet problems in #1 and #2:

A

A1

A2

Nh

n1

n2

110

MULTIPLICATIVE SCHWARZ METHODConsider the multiplicative method: given u(0)2 on $1, for k ' 1:

"#$

#%

Lu(k)1 = f in #1

u(k)1 = u(k!1)2 on $1

u(k)1 = 0 on !#1 \ $1

*

7A!1 A"1

8Ku(k)!1

u(k)"1

L= f1

u(k)"1

= u(k!1)!2|"1

"#$

#%

Lu(k)2 = f in #2

u(k)2 = u(k)1 on $2u(k)2 = 0 on !#2 \ $2

*

7A!2 A"2

8Ku(k)!2

u(k)"2

L= f2

u(k)"2

= u(k)!1|"2

#1

#2

$1$2

111

We find:

A!1u(k)!1

= f1 " A"1u(k!1)"1

A!2u(k)!2

= f2 " A"2u(k)"2

or, equivalently, by adding and subtracting u(k!1)!i

:

u(k)!1

= u(k!1)!1

+ A!1!1

(f1 " A!1u(k!1)!1

" A"1u(k!1)"1

)

u(k)!2

= u(k!1)!2

+ A!1!2

(f2 " A!2u(k!1)!2

" A"2u(k)"2

)

Remark

This method can be seen as a block Gauss-Seidel method to solve thesystem )

A!1 A"1

A"2 A!2

*)u!1

u!2

*=

)f1f2

*

112

If there is no direct coupling between the matrices assembled on oppositesides of $1 and $2:

A!2\!1= 0 and A!1\!2

= 0

#1

#2

$1$2

then, formally, we can write

A"1 = A(!\!1)\(!2\!1)= A(!\!1)

so that

A"1u(k!1)"1

= A(!\!1)u(k!1)(!\!1)

Proceeding in a similar way also for the second equation, we finallyobtain:

u(k)!1

= u(k!1)!1

+ A!1!1

(f1 " A!1u(k!1)!1

" A(!\!1)u(k!1)(!\!1)

)

u(k)!2

= u(k!1)!2

+ A!1!2

(f2 " A!2u(k!1)!2

" A(!\!2)u(k)(!\!2)

)

113

In compact form we can write

u(k+12 ) = u(k) +

)A!1!1

00 0

*7f " Au(k)

8

u(k+1) = u(k+12 ) +

)0 00 A!1

!2

*'f " Au(k+1

2 )(

Let Ri be the (rectangular) restriction matrix that returns the vectorof coe!cients in the interior of #i :

u!i = Riu = (I 0)

)u!i

u!\!1

*

Then, we obtain:

u(k+12 ) = u(k) + RT

1 (R1ART1 )!1R1

7f " Au(k)

8

u(k+1) = u(k+12 ) + RT

2 (R2ART2 )!1R2

'f " Au(k+1

2 )(

114

Define, for i = 1, 2

Qi = RTi (RiART

i )!1Ri

We get

u(k+12 ) = u(k) + Q1

'f " Au(k)

(

u(k+1) = u(k+12 ) + Q2

'f " Au(k+1

2 )(

and

u(k+1) = u(k)+Q1

'f " Au(k)

(+Q2

2f " A

'u(k) + Q1

'f " Au(k)

((3

or, equivalently,

u(k+1) = u(k) + (Q1 + Q2 " Q2AQ1)7f " Au(k)

8

115

ADDITIVE SCHWARZ METHOD

Proceeding as before, we can prove that the additive Schwarzmethod can be seen as a block Jacobi method to solve the system

)A!1 A"1

A"2 A!2

*)u!1

u!2

*=

)f1f2

*

One iteration of the additive Schwarz method can be written as

u(k+1) = u(k) + (Q1 + Q2)(f " Au(k))

In the case of a M ' 2 overlapping subdomains {#i} we have:

u(k+1) = u(k) +

KM+

i=1

Qi

L

(f " Au(k))

116

THE SCHWARZ METHOD AS A PRECONDITIONER

Multiplicative Schwarz method: can be interpreted as aRichardson iterative procedure for Au = f with the multiplicativeSchwarz preconditioner

P!1ms = Q1 + Q2 " Q2AQ1

Pms is not symmetric and can be used within GMRES or BiCGStabiterations.

Additive Schwarz method: corresponds to Richardson iterationsfor Au = f with the additive Schwarz preconditioner

Pas =

KM+

i=1

Qi

L!1

117

The convergence rate for the additive Schwarz method is slowerthan 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 Jacobiand Gauss-Seidel methods)

Pas is not optimal as the condition number blows up if the size ofthe subdomains reduces:

)(P!1as A) - C

1

-H

where C is a constant independent of h, H , and -, - being a linearmeasure of the overlapping region(s) (h ! - ! H).

118

ALGORITHMIC ASPECTS & NUMERICAL RESULTS

Create overlapping subdomains

a) Triangulate the computational domain # ! Th.b) Partition the mesh into M nonoverlapping subdomains {#i}Mi=1.c) Extend every subdomain #i by adding all the strips

of finite elements of Th within distance - from #i .

#5

#1

#2

#3

#4

#5

#6

#7

#8

#9

Start-up phase for the application of Pas

a) On each subdomain #i “build” Ri and RTi .

b) Build the matrix A corresponding to the FE discretization on Th.c) On each #i form the local submatrices A!i = RiART

i .d) On each #i prepare the code to solve a linear system with A!i :

compute a factorization (LU, ILU, Cholesky...) of A!i .

119

One-level additive Schwarz preconditioner Pas

Given a vector r, compute z = P!1as r as follows

a) set z = 0;For i = 1, . . . ,M , do in parallel

b) restrict the residual over #i \ !#i : ri = Ri r;c) solve A!i zi = ri ;d) add to the global residual: z3 z+ RT

i zi .EndFor

Example

)(P!1as 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 54.52 104.85 207.67h = 1/64 63.98 109.22 210.07 416.09h = 1/128 127.99 218.48 420.04 832.57

120

The main limitation of Schwarz methods is to propagate information onlyamong neighboring subdomains, since only local solves are involved bythe application of (Pas)!1.

"$ We have to introduce a “coarse” global problem over the wholedomain to guarantee a mechanism of global communication among allsubdomains.

121

TWO-LEVEL SCHWARZ PRECONDITIONERS

As for the Neumann-Neumann method, we can introduce acoarse-grid mechanism that allows for a sudden information di"usionon the whole domain #:

consider the subdomains as macro-elements of a new coarse grid TH

build a corresponding sti"ness matrix AH

The matrix

Q0 = RTH A!1

H RH

represents the coarse level correction for the two-level preconditioner,with RH the restriction operator from the fine to the coarse grid.

The two-level preconditioner Pcas is defined as:

P!1cas =

M+

i=0

Qi

122

We can prove that there exists a constant C > 0, independent of both hand H such that

)(P!1casA) - C

)1 +

H

-

*

If - is a fraction of H (generous overlap), the preconditioner Pcas isscalable and optimal.

If - . h, the preconditioner Pcas is neither scalable or optimal.

Iterations on the original finite element system using Pcas convergewith a rate independent of h and H (and therefore of the number ofsubdomains)

Thanks to the additive structure of the preconditioner, thepreconditioning step is fully parallel as it involves the solution ofindependent systems, one per each local matrix A!i .

123

ALGORITHMIC ASPECTS & NUMERICAL RESULTS

Start-up phase to apply Pcas

a) Do the start-up phase for Pas .b) Define a coarse space triangulation T0 with elements of order H

and n0 dofs:

c) Build the restriction operator RH % Rn0&Nh , RH(i , j) = )i (xj).)i is the basis function associated to the coarse grid node iand xj are the coordinates of the fine grid node j .

d) Construct the coarse matrix AH , either as

AH(i , j) =

-

!()i ·()j or AH = RHART

H

124

Two-levels additive Schwarz preconditioner Pcas

Given a vector r, compute z = P!1cas r as follows

a) set z = 0;For i = 1, . . . ,M , do in parallel

b) restrict the residual over #i \ !#i : ri = Ri r;c) solve A!i zi = ri ;d) add to the global residual: z3 z+ RT

i zi .EndFor

e) Compute the coarse-grid contribution zH = A!1H (RHr);

f) Add to the global residual z3 z+ RTH zH .

Example

)(P!1casA) H = 1/4 H = 1/8 H = 1/16

h = 1/32 7.03 4.94 -h = 1/64 12.73 7.59 4.98h = 1/128 23.62 13.17 7.66h = 1/256 45.33 24.34 13.28

If H/- = constant, this two-level preconditioner is either optimal andscalable.

125

Practical indications:

For decompositions with a small number of subdomains, the singlelevel Schwarz preconditioner Pas is very e!cient.

When the number M of subdomains gets large, using two-levelpreconditioners becomes crucial.

When generating a coarse grid is di!cult, other algebraictechniques, like aggregation, can be adopted to define AH , and

)(P!1aggreA) - C

)1 +

H

-

*

P!1aggreA H = 1/4 H = 1/8 H = 1/16

h = 1/16 13.37 8.87 -h = 1/32 26.93 17.71 9.82h = 1/64 54.33 35.21 19.70h = 1/128 109.39 70.22 39.07

126

REFERENCES

127

References

Barry Smith, Petter Bjorstad, William Gropp. DomainDecomposition, Parallel Multilevel Methods for Elliptic PartialDi!erential Equations. Cambridge University Press, 1996.

Alfio Quarteroni, Alberto Valli. Domain Decomposition Methods forPartial Di!erential Equations. Oxford Science Publications, 1999.

Barbara I. Wholmuth. Discretization Methods and Iterative SolversBased on Domain Decomposition. Springer, 2001.

Andrea Toselli, Olof Widlund. Domain Decomposition Methods –Algorithms and Theory. Springer, 2004.

Clemens Pechstein. Finite and Boundary Element Tearing andInterconnecting Solvers for Multiscale problems. Springer, 2013.

Alfio Quarteroni. Numerical Models for Di!erential Problems.Springer, 2012.

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