Domain Decomposition Methods, Hybrid Methods

We have shown in Section 1.3 that duality techniques enable us to obtain alternate variational formulations for some problems. The method that we shall now describe will yield a new family of variational principles that can be more or less grouped under the name of hybrid methods. The common point between the examples that follow is that in all cases the variational principle will depend explicitly, independently of any discretization, on a partition of the domain 0 into subdomains. To make clearer some of the facts that will appear later, we first recall a very classical result.

Example 4.1: A transmission problem.

We consider the very classical case in which a domain 0 is split into two subdomains 01 and O2 by a smooth enough internal boundary S (Figure 1.2).

We consider the case of a

Dirich-let problem with variable coefficient

~ fs ^~

a(x), a(x)

being discontinuous on

Q

1S

Q:

S .

This classically leads to the vari- _ _ _ ational problem [where we want to Figure 1.2 find u E H6(0)]

(4.1)

r ^a1(x)

grad u ·grad

v dx+ r a2(x)gradu.gradvdx

~, ~2

= In

^Iv

^dx,

^{'>Iv E HJ(O).}

Defining

U1 = ^ub,

^and

^U2 = ^{U!02 '}

it is standard to interpret (formally) problem (4.1) in the form

(4.2)

(4.3)

{ -

div(a1(x)

grad

ud = I

in 01 ,

- div(a2(x)

grad

U2)

=

I

in O2 ,

U1!rnao,

=

0, U2!rnao2

=

^0,

where n1 and n2 are obviously the exterior normals to 01 and O2 (respectively) on

S .

Continuity conditions (4.3) are implicitly contained in the variational formulation. An important special case is

a1(x) = a2(x) =

1. We then get the following result. 0

§Io4 Mixed and Hybrid Finite Element Methods Proposition 4.1: Let u be solution of the Dirichlet problem

(404) { -D.u

= f,

ulr = 0.

25

Let the internal boundary S split 0 into 01 and O2 • Then it is equivalent to say that u is solution of the problem

(4.5) { - D.u^l =

f

~n 01 ,

- D.u2

= ^f

^{m O2 ,}

udrnal1,

=

0, u21rnal1,

=

^0,

(4.6)

To show this result we would have to define properly the normal derivatives

oul/onl

^and

OU2/on2

^on S. This would require some regularity on

f,

^for

instance

f

E

L2(0).

0

What we really want to do is to consider a general partition of 0

N

(4.7)

0= URi.

;=1

We now write the classical Dirichlet functional of Example 2.1, in the following apparently strange way.

Example 4.2: A domain decomposition method for Dirichlet problem.

Writing the Dirichlet functional as

(4.8)

and introducing now the functional space

N

(4.9) X(O)

= {vi

v

E

L

²

(O),vIK;

E

Hl(Ki)}

~

II ^Hl(K;),

i=l

we can extend

J(v)

on

X(O).

Moreover

HJ(O)

is a closed subspace of

X(O)

and we may consider

"v

E

HJ(O)"

as a linear constraint on

v

E

X(O).

This constraint states that on eij

=

^oK;

ⁿ

^oKj we must have, in H^{1/ 2 (} eij),

26 Variational Formulations and Finite Element Methods §IA

Ui

=

^{Uj, where Ul}

=

^UIKl" We shall therefore, following a now familiar pro-cedure, impose this constraint through a Lagrange multiplier properly chosen in H-1/ 2( eij) . As we shall see in Chapter III, it will be more convenient to introduce q E

H(

div; Q) and to use as a multiplier the normal trace of q on aKi. This-leads us to the saddle point problem

-(4.10) _vEX(O) inf !EH(dlV,O) sUI?

t

i=1

^{~r

JK;

^I

^{g@dvl2 dx-}J8K; ^[

^{I{.~; v}

^{ds- {} JK;

^Iv

^dX}

for which we have the following optimality conditions: for i = 1, ... , N ,find

U; E Hl(K;) such that,

(4.12)

L

_;=1 N

11{ .

_8K,

^~i

^{Ui ds = 0, 'VI{ E H(div;Q).}

Condition (4.12) expresses continuity of U at interfaces eij and condition

ulr

⁼

O. Condition (4.11) shows that Ui is solution in Ki of a Neumann problem

(4.13) { -6.u;

= I

ⁱⁿ Ki ,

aUi a

^r."

-a

_n; ^{= p. n.} _{- - 1} ^{on 1ii·}

Solving this problem obviously requires [make ^Vi = 1 in (4.11)] a compatibility condition

(4.14)

[ E' ~i

ds

+ [ I

dx =

0,

J8K, JK;

on every subdomain Ki • This condition can also be written (4.15)

From (4.13) we have that the multiplier p·n can be seen as the normal derivative of u . Indeed, when equilibrium is attained, we have on interfaces aui/ ani

=

p. n . = -p. n. = ^{-auj / anj} and Ui = ^{Uj .} A suitable lifting of p in each Ki in

--

order to have div ^-~

p. + I

= 0 can always be done because of (4.14) and (4.15). ^- 0

§I.4 Mixed and Hybrid Finite Element Methods 27 Example 4.3: Dual problem of the domain decomposition method.

We now consider the dual problem of the above saddle point formulations. It will be, as can be expected, very close to the dual problem introduced in Section 1.3 for the Dirichlet problem. Let us first remark that taking the infimum on the constant part of v E XeD) on each K ; leads to the constraint (4.15) on p.

It is therefore possible to suppose div p

+ f =

0, as this can be attained by modifications to p that are internal to I~ (that is, not modifying p . _{- - '} n.) and are transparent to formulation (4.10). Writing

(4.16) _{J8K .} f . 9. . !!:i V ds =

1.

_{K .} ^{div 9. v dx}

⁺ 1

_K; ^{9.' grad v dx,}

one gets from (4.10)

(4.17) . sup inf

t

^{-21 f 19@dvi I2dx_fq. grad Vi dx}.

dlVf+J=o v ;EHl(K;)/lR i =l JK;

JK;-From (4.17) we evidently get, setting p .

-.

= pIK; ,

-(4.18) g@d Ui = P(p.),

-.

where P is the projection operator in (L2(K;))2 on g@d(H1(K;)) . We shall indeed prove in Chapter III that one has

(4.19)

From this we can eliminate v; and write the dual problem (4.20) fEH(dlv;O) _sUI?

-~ t

i =l

J ^r

^{K; IP(9..)12 dx.}

divf+J=o

We are therefore back to a variant of (3.35). Indeed, (3.35) shows that the projection operator P in (4.20) is unnecessary. 0

Remark 4.1: One could obtain a variant of the above dual problem, without constraint (4.15) by using a "least-squares" solution of (4.13) whenever (4.14) does not hold. This could be done, for instance by solving on K ;, in a weak formulation that we shall not describe,

(4.21)

!

^6.2u;

^-a -a

aUi

^a

ⁿⁱ _ni ^{6.u; = =} _{- - .} ^{q . n. 6.f in =}

^{-a af}

^{ni on Ki ,}

aK;,

^on

^aK;,

for which a solution always exists, defined up to an additive constant. Such a procedure could be useful for algorithmic purposes since (4.21) is a local simple problem even if it is a fourth-order problem. 0

28 Variational Formulations and Finite Element Methods §I.4

We shall meet discretization methods, based on such a principle, under the name of dual hybrid methods for the treatment of almost any example considered in this book: Dirichlet problems, elasticity problems, fourth-order problems, etc. 0 Example 4.5: The Hellan-Hermann-lohnson method in elasticity.

This is an example in which a domain decomposition is introduced, not by dualizing a continuity condition but by defining a variational formulation able to bypass this continuity by approximating weak derivatives. We shall first present formal results and delay a precise presentation of the functional framework.

§I.4 Mixed and Hybrid Finite Element Methods 29 (4.28)

These conditions make sense for a space of !!. chosen so that div !!. is well defined, which implies, as we have seen, continuity of!!'n at interfaceS. On the other hand Q can be taken as completely discontinuous on these same interfaces.

What we now try to do is to split continuity conditions between ~n and Q. Let us consider indeed the well-known integration by parts formula,

(4.29) {div g'

In

Q dx

+ ( In

^g: f:(Q) dx

= { JEm

^{O"nn Q' 11 ds}

+ ( JEm

^{O"nt Q'}

t

^ds.

Whenever Q is a smooth [let us say H1(n)] vector, and ^O"nt is continuous, we thus have

N

(4.30)

_I

(f:(Q) : SI dx = -

L {{

^{div g. Q dx}

^{+ {}

^{O"nn Q' 11 dS} ,}

n i=l

JK, J8K,

so that we can rewrite (4.27) and (4.28) in the following form:

(4.31)

~

^{{ !!.D : XD dx}

+

^{2 (A} 1 ) {tr!!. tr 'L-dx J1

In - - +

J1

In

-+ L {{

N ^div

~

^{. 1! dx- { Tnn 1! . 11 dS} = 0,}

V~,

i=l

JK, J8K,

N

(4.32)

L _{;=1 JK,} { {

^{divg. Q dx -}

_J8K.

^{{O"nn Q' 11 dS}}

^{+ {} _In ^t·

^{'!!. dx = 0, VQ.}

Formally this is well defined for!!. chosen with ^Unt continuous at interfaces while 1! . 11 is continuous. Then theterm

(4.33)

therefore depends on the jump of O"nn on oK; and (4.32) can be read as div!!.

+ f

= 0 in the sense of distributions. We shall consider in Chapter VI

a

discretization of problem (4.31) and (4.32) for A

= +00

(that is, tr!!.

=

^0),

i.e., the case of an incompressible material. As we shall see, our main problem will then be to preserve symmetry in the discretized problem.

Up to now we considered a purely formal problem. Giving a good frame-work to (4.31) and (4.32) is a task that requires some care. The presence of traces, appearing explicitly, in the variational formulation leads one to deal with

30 Variational Formulations and Finite Element Methods §I.4 spaces H 1/ 2(oKi ) ^and H- 1/2(oK;) and to subtle considerations about the be-havior of functions in these pathological spaces. Let us define

(4.34) E

=

II(H1(Kd)!

=

^{{g E} (L2(Q))!, consider are known as Galerkin least-squares methods and were introduced by FRANCA-HUGHES [A]. To fix the ideas, let us consider the simplest cases of Examples 3.4 and 3.5, and in particular the saddle point formulations (3.32) and (3.39), respectively. In both cases the Euler equations are given by (3.34). It is clear that we can always add (or substract) the square of one of these equations to the functional without changing the min-max point. For instance, we can

No documento [Springer Series in Computational Mathematics 15] Franco Brezzi, Michel Fortin (auth.), Franco Brezzi, Michel Fortin (eds.) - Mixed and Hybrid Finite Element Methods (1991, Springer-Verlag New York) - libgen.lc (páginas 32-38)