Solving the

on the Basis of

Domain Decomp osition Techniques



J Sch ob erl Linz

Abstract

Solving the Signorini Problem on the Basis of Domain Decomp osition

Techniques The nite element discretization of the Signorini Problem leads to a

large scale constrained minimization problem To improve the convergence rate of the

pro jection metho d preconditioning must b e develop ed To b e eective the relative

condition numb er of the system matrix with resp ect to the preconditioning matrix has

to b e small and the applications of the preconditioner as well as the pro jection onto

the of feasible elements have to b e fast computable In this pap er we show how

to construct and analyze such preconditioners on the basis of domain decomp osition

techniques The numerical results obtained for the Signorini problem as well as for

problems conrm the theoretical analysis quite well

AMS Subject Classications T J N F K

Key words contact problem variational domain decomp osition precon

ditioning

Zusammenfassung

o sung des Signorini Problems auf der Basis von Gebietszerlegungs Die Au

metho den Die Finite Elemente Diskretisierung des SignoriniProblems fuhrt zu

einem restringierten Minimierungsproblem mit vielen Freiheitsgraden Zur Verb es

serung der Konvergenzrate des Pro jektionsverfahren mussen Vorkonditionierungs

techniken entwickelt werden Um ezient zu sein mu die relative Konditionszahl

der Systemmatrix in Bezug auf die Vorkonditionierungsmatrix klein sein und die

assige Menge Anwendungen der Vorkonditionierung als auch der Pro jektion in die zul

mussen schnell b erechenbar sein In dieser Arb eit wird die Konstruktion und Analyse

solcher Vorkonditionierer auf Gebietszerlegungsbasis dargestellt Numerische Ergeb

at nisse fur das Signorini Problem und Kontaktprobleme aus der eb enen Elastizit

stimmen mit der theoretischen Analyse gut ub erein



orderung der This research has b een supp orted by the Austrian Science Foundation Fonds zur F

wissenschaftlichen Forschung FWF under pro ject grant PTEC

Intro duction

The contact problem is an imp ortant problem in computational mechanics An elastic b o dy

is deformed due to volume and surface but the b o dy should not p enetrate a given

rigid obstacle This leads to unilateral b oundary conditions called contact conditions We

refer to for mathematical mo deling and analysis

In this pap er we are interested in fast numerical algorithms for solving the nite dimen

sional constrained minimization problem arising from the nite element discretization of

the Signorini problem or elastic contact problem There are clas

sical iterative metho ds like p oint pro jection metho ds and p oint overrelaxation metho ds

These metho ds suer from slow convergence rates on ne meshes Multigrid metho ds

have b een successfully applied to obstacle problems with inequality constraints in the whole

domain by Domain decomp osition metho ds for variational inequalities

have b een investigated in But these domain decomp osition metho ds dif

fer from our metho d based on domain decomp osition preconditioners Boundary element

metho ds have also b een applied for contact problems in

For the sake of simplicity we will consider the Signorini Problem for the Poisson equa

tion to develop and analyze the algorithms but we also provide numerical results for the

contact problem in elasticity

For the Signorini problem the unknown function u is restricted from b elow on the

Signorini part of the b oundary In the classical form the Signorini problem

C

reads as follows

u f in

u on

D

u

on

N

n

u u

u g u g on

C

n n

d

The domain is supp osed to b e b ounded in R d or with a Lipschitzcontinuous

b oundary such that meas and meas

D N C D C

The nite element discretization of the weak form of leads to the nite dimensional

Constrained Minimization Problem CMP

T T

v Av f v Find u K J u inf J v with J v

v K

or to the equivalent

T T

Find u K u Av u f v u v K

The mesh size is denoted by h the total numb er of unknowns is N the numb er of unknowns

on is N The solution u the symmetric and p ositive denite sp d system matrix A

C C

and the righthand side vector f are split into contact b oundary comp onents C and inner

plus Neumann b oundary comp onents I ie

u A A f

C C CI C

u A f

u A A f

I IC I I

N

The of feasible functions K V R is dened by

K fv V v g g

C C

where v g is meant comp onentwise

C C

To solve the nite dimensional constrained minimization problem the pro jection metho d

is applied This metho d is not very p opular in practice b ecause its convergence rate is

very slow on ne meshes unless go o d preconditioning is applied The goal of this pap er is

to present preconditioning techniques satisfying the following prop erties

The relative condition numb er of A with resp ect to C is small

the op eration C v is fast executable

the pro jection P with resp ect to the C energy onto K is fast computable

with the sp d preconditioning matrix C Multilevel preconditioners as well as domain

decomp osition DD preconditioners have b een develop ed for satisfying the rst two re

quirements quite well The basic concept of Dirichlet DD is the approximative decoupling

of the global FE space into inner and extended b oundary subspaces The pro jection in

volves only the b oundary subspace To implement this pro jection the dual problem is

intro duced This enables us to use wellknown b oundary preconditioners To avoid two

cascaded iterations until convergence we analyze also the truncated version by means of

the BramblePasciak transformation intro duced in If the underlying comp onents are

optimal and a uniformly rened mesh is used then the complexity is of optimal order for

the two as well as three dimensional case We will combine our algorithms also with nested

and adaptive concepts and study the b ehavior by means of numerical examples

The rest of the pap er is organized as follows The pro jection algorithm and its approxi

mative version are presented in Section Convergence in the energy norm is proved The

concepts of additive DD preconditioning is shortly rep eated in Section In Section b oth

concepts are combined to develop an optimal preconditioner for the pro jection algorithm

The truncated variant is analyzed in Section Finally numerical results are presented in

Section

The Pro jection Metho d

The nite dimensional CMP can b e solved by the pro jection metho d which reads as

Algorithm Pro jection Metho d

Cho ose an arbitrary initial guess u K

For k do

k k k

f Au u P u C

In Algorithm C is the symmetric and p ositive denite sp d preconditioning matrix We

assume the sp ectral equivalence inequalities

C A C

in the sense of Euclidean inner pro duct with the p ositive sp ectral equivalence constants

and In general these b ounds dep end on the mesh parameter h If they are indep endent

of h and the op eration C v is fast executable ie via O N arithmetical op erations

then we call the preconditioner C asymptotically optimal Asymptotically optimal pre

conditioners can b e constructed by multilevel techniques as well as by multilevel DD

techniques see Section

We assume that the real p ositive relaxation parameter is chosen such that

The op erator P is the pro jection onto K with resp ect to the energy norm induced by the

sp d preconditioning matrix C

P w V P w K kP w w k kv w k v K

C C

k

It is straightforward to show that u converges to its limit u in C energy norm with

convergence factor see eg A convergence rate estimation in Aenergy norm

is necessary for the comp osite algorithm in Section However it was not available from

k

the literature It could not b e shown that u converges monotonically to u in Anorm but

the main theorem of this section provides a monotone decay of the quadratic J

and it estimates the convergence rate

For our application the exact pro jection P is to o exp ensive to compute and so it is

replaced by an approximative pro jection P This leads us to the approximative pro jection

metho d

Algorithm Approximative Pro jection Metho d

Cho ose an arbitrary u K

For k do

k k k

f Au u u C

k k

u P u

The following theorem estimates the convergence rate of the approximative pro jection

metho d which also includes the exact pro jection metho d with

P

Theorem Energy convergence rate estimate

k

Let u b e the sequence generated by Algorithm The relaxation parameter is chosen

in the The approximative pro jection P fullls

k k k k k k

kP u u k ku u k kP u u k

P P

C C C

with Then the estimate

P

k k

J u J u J u

holds for every k N with the convergence rate

P

The error in Aenergy norm is b ounded by

k k

J u J u ku u k

A

Proof First we reduce the approximative pro jection to the case of an exact pro jection

k

We use C A the denition of u and to obtain

k k k k

J u J u ku u k

1

C A

k k k k k

ku u k ku u k J u

C C

k k k k k k k

ku u k kP u u k ku u k J u

P P

C C C

k k k k k k

kP u u k ku u k J u J u

P P

C C

Due to the convexity of K there holds

k k k k

kP u u k kP k u u k

C C

uu

k k

with the pro jection P k u onto the straight line u u This reduces the problem to

uu

k

the plane spanned by u u u From the sketch b elow the next estimates are obvious

  k u  ~k k P[u, u ](u )      K k+1 u    u

u~k

k k k k

kP k u u k ku u k

uu

C C

k k k k k k k k

u u P k u u P k u u P k u u

uu uu uu

C C

k k k k

u u P k u u

uu

C

k k k

u u u u

C

k k k

max min u u u u

C

k

ku u k

C

k

Using the denition of u the variational sp ecication of the solution u and the sp ectral

b ound we get

k T k k k k

f Au u u u u u u

C

T k k

f Au u u ku u k

A

k

k ku u

C

we get Combining and using

k k k k

kP u u k ku u k

C C

n n o o

k k k k k k

max min u u u u u u u u

C C

k k k

u u u u

C

k

Now using the last inequality we continue to estimate the energy functional J u

k k k k k k

u u u u J u J u J u

C P P

k T k k k

f Au u u J u J u

P P

k k k

J u J u ku u k J u

P P

A

k

J u J u

P P

k

J u J u

We use this result to estimate the error in Aenergy norm by

k k T k

ku u k J u J u Au f u u

A

k k

J u J u J u J u

For practical computation we need a computable estimate for the iteration error We

get the same error estimator as for iterative metho ds for linear systems

Corollary

k

The sequence u is generated by Algorithm or by Algorithm Then the iteration error

is b ounded by

k k k T k k

ku u k u u f Au Au

A

with from

Proof From we get

k k

ku u k J u J u

A

and by Theorem we have

k k k

J u J u J u J u

k k T k k

u u f Au Au

that completes the pro of

Domain Decomp osition Preconditioning

In this section we present the approximative additive domain decomp osition DD precon

ditioner intro duced in The domain decomp osition strategy provides us with a splitting

e

of the FE space V into inner V and extended coupling b oundary V subspaces which

I C

is useful for parallel computing In our application the coupling b oundary is replaced by

the contact b oundary By combination with multilevel techniques we gain asymptotically

optimal preconditioners

We recall the splitting of V into the b oundary subspace V and inner subspace V given

C I

in These two subspaces are far away from b eing orthogonal with resp ect to energy

norm To get an idea ab out orthogonality we have to go back to the Sob olev space H

The two subspaces H and the space of the harmonic functions are orthogonal with

resp ect to the H half norm The space V is an approximation to H To approximate

I

the harmonic subspace one takes

e

V fE v v V g

C C C C

T T

where E I E is a computable op erator which approximates the discrete harmonic

C

I

extension of a given b oundary function E is b ounded by c in the sense of

E

kE v k c inf kw k c kv k

C A E A E C S

C

w 2V

w =v

C C

with the BoundarySchurComplement S A A A A

C C CI IC

I

Note the b est extension op erator c would b e the solution of the Dirichlet

E

problem with given b oundary values v But we need a fast executable extension op erator

C

with small constant c

E

On b oth subspaces V and V we need sp d preconditioners C and C resp ectively

I C I C

for which we assume the sp ectral inequalities

T

C and C C E AE C A

C C I I C I I

C I

T

Because E AE is sp ectrally equivalent with constants and c to the Schur complement

E

S we call C Schur complement preconditioner

C C

Using these comp onents we can dene the approximative additive domain decomp osi

tion preconditioner by

T

I E I C

C C

I C

C

I E I C

I I I

I

for which the sp ectral inequalities

C A C

hold with the constants

q q

c c min f max f g and g

C I

C I

E E

If the comp onents E C and C are asymptotically optimal then the preconditioner C

I C

is as well For the inner preconditioner a symmetric multigrid preconditioner

may b e used For D the transformation to the hierarchical basis gives simple Schur

complement preconditioners and extension op erators which are optimal up

to logarithmic factors Optimal comp onents for D and D are constructed by multilevel

techniques see for the Schur complement preconditioner and for the extension

op erator In additional smo othing improves the extension constant c

E

T

All these comp onents have optimal arithmetic complexity ie the op erations E E

and C need O N op erations while the application of C needs O N op erations

C

I C

only The precise analysis of the overall op erator C is given in In it is shown

how a symmetric multiplicative Schwarz preconditioner ts into the framework of additive

Schwarz preconditioners

The Pro jection Metho d with DD Preconditioning

In this section we will apply the DD preconditioner for the pro jection metho d The rst

two requirements namely condition numb ers indep endent of h and fast execution of the

preconditioning op eration are fullled for this preconditioner To construct the pro jection

we use the basis transformation matrix

I

C

T

E I

I I

and express the solution u by u T u Therefore u is the solution of the CMP

inf J v

vK

with K T K and

T T T

v f T J v J T v v T AT v

z

z

A

T

f

T

If C is a preconditioner for A then also C T CT is one for A with the same b ounds

For the DDpreconditioner the transformed C has the blo ck diagonal structure

C

C

C

C

I

Because v can b e chosen arbitrarily in a linear space the set K reduces to K

I

I v

C C

K T K v g K

C

E I v

I I I

Now we apply the pro jection metho d to the transformed system

k k k

f A u u P u C

Using the original quantities A u and f this iteration can b e rewritten as

i h

k k k k T

Au f T P u u T P T u C T

The pro jection involves only b oundary comp onents which are decoupled from the inner

comp onents by the inner pro duct induced by C Therefore it can b e applied just on the

b oundary the inner comp onents stay unchanged Applying the pro jection means to solve

the problem

k k

Find u g ku u k inf kv u k

C C C C

C C C C

C C

v g

C C

This CMP can b e solved by the pro jection metho d with Euclidean inner pro duct and

p ointwise pro jection

k i k i k i

k

u P u C u u

C

C C C

C

In DD we need a fast implementation of the action C v That is ensured by the multi

C

level Schur complement preconditioner mentioned ab ove However for this choice the

op eration C v cannot b e p erformed explicitly By dualizing the b oundary CMP we

C

get the op erator C into the computable direction We rewrite the Kuhn Tucker conditions

C

for the CMP and arrive at the equivalent complementary problem

k

C u p C u

C C C

C

T

u g p p u g

C C

After dening v u g and exchanging primal and dual variables one can write

C

k

C p v g u

C C

T

p v v p

Now this complementary problem can b e rewritten as a CMP in the dual variables as

k T T

g u q C q q min

C C

q

k

After calculating p by the pro jection metho d one gets u by u u C p Now

C C

C C

we can state the whole algorithm develop ed so far

Algorithm Innerouter pro jection algorithm

u

for k

k k

w C f Au

k k k

u u w

p

for i

i i i k

p p P p g u C

C

k k

u u EC p

C

The computational eort is n c n n c with the iteration numb ers n and n for outer

o o o i i o i

and inner iterations and the costs p er iteration step c and c resp ectively The numb er of

o i

outer iterations n is optimal O n is O C which is O h for two as well as three

o i C

dimensional problems The costs in the outer iteration are optimal O N and the costs p er

inner iteration are O N The drawback of Algorithm is the inner iteration which has

C

to b e p erformed until some convergence criterion is fullled To overcome this disadvantage

we lo ok for a b etter initial value p and p erform just a xed numb er of iterations We will

analyze this truncated version in the next section by means of the Augmented Lagrangian

formulation

Augmented Lagrangian Formulation

The dualization of the large CMP gives

T T T T

min A f I g A I q q I q I

C C

C C

N

C

q q R

N N

C

where I is the injection of R into R This functional and its derivative are not

C

fast computable b ecause b oth need the inverse of A Using the Augmented Lagrangian

technique we can change it into something computable Indeed adding some convex

function in v and q the minimum of which in v equals for every xed q we obtain the

CMP

T T T T

inf kAv I q f k q I A I q q I A f I g

1 1

C C C

C A C C

N

C

q 2R : q 0

N

v 2R

where C is some prop erly scaled sp d preconditioner for A such that C A is sp d as

well ie C A The CMP is obviously equivalent to and it can b e handled

easier than as we will see later on

The Bramble Pasciak Transformation

Let us consider now the symmetric but indenite system

u f A I

C

T

p g I

C

arising eg from the weak formulation of the Dirichlet b oundary conditions as

constraints to the energy functional see eg In it is shown how to change

into an sp d system provided that some preconditioner C for A is available such that

A C A

holds with some C may b e one of the DD preconditioners see Section with

a prop er scaling factor Multiplying from the left by the matrices

C I A C

T

I I I I

C

we arrive at the sp d system

A U F

with

AC A A AC I I u AC I f

C

A U F

T T T

I C A I I C I p I C f g

C

C C C

In the third multiplication is not applied explicitly However it is hidden in the inner

pro duct The main result in was the pro of of the sp ectral equivalence of A to the blo ck

diagonal matrix

A C

T

I A I

C

C

with the sp ectral equivalence constants

q

p

and

T T

If we replace the Schur complement I A I by the Schur complement I C I then we

C C

C C

get some additional factors into the sp ectral equivalence constants The following theorem

provides a direct estimation

Theorem

Let C b e a preconditioner to A fullling Then the blo ck diagonal matrix

A C

C

T

I C I

C

C

is sp ectrally equivalent to the blo ck matrix A dened in with sp ectral equivalence

constants

p p

and

Proof First we show that the inequality

A C A C C A

is valid Indeed b ecause the matrices involved are sp d the sp ectral radius C A is

equal to the Rayleigh quotient Therefore we can write

C Au u A C C Au u

AC

max C A max

N N

u2R u2R

A C u u u u

AC

u6=0 u6=0

Au u Au u

max max

N N

u2R u2R

C u u Au u A C u u

u6=0 u6=0

The last estimate follows directly from Further we will use b elow the inequality

u v kuk kv k

which is a consequence of the CauchySchwarz inequality and Youngs inequality ab

a b a b R

Let us now derive the upp er b ound in the sp ectral equivalence inequalities Using

inequalities and we can now estimate

AU U

1

C I p I p A C C Au u A C u I p

C C C

C

C I p I p kI pk A C C Au u kA C uk

1 1

C C C

C C

A C C Au u A C C A C u u C I p I p

C C

A C C Au u A C u u C I p I p

C C

i h

A C u u C I p I p

C C

i h

A C u u C I p I p

C C

p

as the p ositive solution of the equation Now we cho ose

i h

p

that proves the upp er b ound

For the lower b ound we use again the inequalities and with in

advance

AU U

A C C Au u A C C A C u u C I p I p

C C

A C C Au u A C u u C I p I p

C C

i h

A C u u C I p I p

C C

p

Note that we have used the fact for Indeed we nd that

is again the p ositive solution of the equilibration equation

i h

Therefore we obtain the lefthand side of the sp ectral equivalence inequalities

p

AU U CU U

Let us nally reformulate Theorem for an unscaled preconditioner C satisfying the

usual sp ectral equivalence inequalities

C A C

with p ositive sp ectral equivalence constants C A and C A Then

min max

C obviously fullls with Now Theorem the rescaled preconditioner C

immediately shows that the blo ck matrix

AC A A AC I I

C

A

A

T T

I C A I I C I

C

C C

is sp ectrally equivalent to the preconditioner

A C

C

T

I I C

C

C

with the sp ectral equivalence constants

q q

and

Note that and Therefore the relative condition numb er C A

C A C A of the blo ck system is less than four times as worse as the relative

max min

condition numb er C A of A and C ie C A C A The upp er b ound for

provides a simple choice of the relaxation parameter needed in the Richardson iteration

Lemma

T

The Schur complement I C I of the DD preconditioner is exactly C

C

C

C

Proof By multiplication

The Approximative Augmented Pro jection Algorithm

The matrix of the quadratic term of the functional with an unscaled preconditioner C

is exactly the matrix A in The vector of the linear term is

AC I f

F

T

I C f g

C

One can try to use the pro jection algorithm with inner pro duct We apply one

preconditioned Richardson step for the linear system AU F ie

k k

U U C F AU

With the notations

k k k

w C f Au I p

C

u

k k T k

w I w g I u

C

p u C

the Richardson step simplies to

k k k

u u w

u

k

k k

C w

p p

C

p

The restrictions involve only the dual comp onent p and the inner pro duct matrix C decou

ples the primal and the dual comp onents therefore the pro jection involves only the dual

comp onent

k k k k

Find p kp p k inf kq p k

1 1

ex ex

C C

C q C

By Theorem it is enough to have an approximative pro jection fullling

k k k k k k

kp p k kp p k kp p k

1 1 1

P P

ex

C C C

C C C

The approximative pro jection is implemented by applying n steps of the pro jection algo

rithm with Euclidean inner pro duct

k

q p

i i k i

q P q C p q

C

k n

p q

k

It is easily seen that there is no need to compute C w explicitly The condition numb er of

C

p

the Schur complement preconditioning op eration C is O h By applying Theorem

C

to the inner iteration we get the estimate

k k n k k n k k

kp p k kp p k kp p k

1 1 1

i i ex

C C C

C C C

with the convergence rate ch The choice n ch ensures inequality with

i

the hindep endent constant e Theorem proves hindep endent convergence rate

P

of the algorithm which is summarized b elow

Algorithm Approximative Augmented Pro jection Metho d

u g p

for k

k k k

d f Au I p

C

u

k k

w C d

u u

k T k T

w I w g I u

c k

p C u C

for i to n

i+1

i i

k k k k k

n n n

P p C p w C p p

C p C

k k k

u u w

u

To apply the algorithm we have to cho ose some parameters namely and as close

as p ossible fullling the sp ectral equivalence inequalities C A C is calculated

q

I and n is is set as large as p ossible such that C via

C

M ul tig r idl ev el s

prop ortional to the condition numb er of C eg n

C

The iteration error can b e estimated by Corollary which leads us after some simple

calculations to

h i

k k k k k k k

ku u k kp p k c w d N p p w N

1

C

R

A u u p R

C

C

with a constant c only dep ending on and the sp ectral equivalence constants of the pre

conditioner C

The complexity of Algorithm is O n c n n c The costs p er iteration step c and

o o o i i o

c are of optimal order this means O N and O N resp ectively The numb er of inner

i C

iterations n is O h for the two dimensional as well as three dimensional case The

i

numb er of outer iterations n is xed for a given error with This gives the

o

complexity O N h N l n On a regular rened mesh we have N O hN and

C C

therefore the complexity reduces to the optimal term O N l n

The multiplicative version

Algorithm is an approximative additive Schwarz metho d with resp ect to the splitting

N N

C

V R V R

It is well known for linear problems that the multiplicative metho d converges approxima

tively twice as fast as its additive counterpart Table b elow shows a similar b ehavior

for the Signorini problem

With the approximative Pro jection P the multiplicative metho d is given by

k

u

k k

F A p p I C

k

p

k k

p P p

k

u

k k

F A u u I C

k

p

Because A C holds on b oth subspaces V and V the relaxation parameters could b e set

k

to We see that the up date for p is equal to the up date given by the additive version

k

k k k

calculated by one additive step from u and p can The dierence b etween u and u

AS M

b e expressed by

k k k k

u u I C A EC p p

k k

AS M C

p p

Now we can state the multiplicative algorithm which is very similar to the innerouter

pro jection algorithm Algorithm but the inner iteration is now nite

Algorithm Approximative Multiplicative Augmented Pro jection Metho d

u g p

for k

k k k

d f Au I p

C

u

k k

w C d

u u

k T k T

w I w g I u

c k

p C u C

for i to n

i+1

i i

k k k

k k

n n n

P p C p w C p p

C p C

k k k k k

u u w EC p p

C

u

Numerical Results

To give a numerical verication of the theory the algorithms ab ove have b een implemented

and have b een applied to equation with fg f g

D N

fg with source term f and restriction g x

C

The numerical examples have b een carried out within the C nite element co de

FE on a Sun Ultra with MHz We used a sequence of hierarchically rened

triangular meshes for FE discretization The level mesh is drawn in Figure

C A C I C I

C

C

C

level N N

C

C

C

Table Preconditioner and eigenvalue b ounds

additive multiplicative Ritz functional

level N N its times its times J u J u J

C h h

Table Algorithms additive and multiplicative and Ritz functional

As preconditioner C a symmetric multiplicative DD preconditioner with a V multi

grid cycle a multilevel extension with smo othing steps and a b oundary multilevel pre

conditioner for the Schur complement has b een used The sp ectral equivalence constants

and as well as the eigenvalue b ounds and for the matrix C have b een calculated

C C

C

by the Lanczos metho d and are given in Table Based on these calculations we have

l ev el l ev el

and n chosen a priori the b ounds

We applied Algorithms and to reduce the squared error according to the estima

The iteration numb ers and the CPU times are shown in Table tor by a factor of

The iteration numb ers are b ounded by a constant and the CPU times grow prop ortional

to the numb er of unknowns

In our example the FE solution u fullls the restrictions of the continuous problem

h

Therefore the discretization error in energy norm may b e estimated as ku u k

h

J u J u where u is the solution of the continuous problem in weak form Table

h

contains also the minima of the Ritz functional and the dierences to the extrap olated limit

J The estimated error shows the exp ected convergence rate ku u k O h

h

for a solution u H

Next we implemented a nested iterative scheme On level the problem was solved

level N N J u J u J solver timesec total times

C h h

Table Nested iteration

nearly exactly On the rened levels we used the prolongated approximative solution

of the previous level as initial guess and applied just a xed numb er of three steps of the

multiplicative algorithm The prolongation for the primal comp onent u was done naturally

h

by linear interp olation For the dual comp onent p which is a functional in H the

C

L representative in V was calculated and extended naturally By the nested metho d

C h

C

we got nearly the same upp er b ound for the error within the accumulated solver times

and total times including mesh renement assembling and solving shown in Table

Because of the singularities in the solution an adaptive rened mesh has large ad

vantages with resp ect to memory requirements and CPU time For linear problems the

residual error estimator provides one upp er and lower b ound for the discretization er

ror in energy norm Without claiming to compute upp er and lower b ounds to the error we

apply the residual error estimator to the Signorini problem We compute for each element

T the element contribution

X X

h kf k h kn ru k h kn ru k

E E h E E E h

T T T E E

E E T E E E T E

N

h

where h is the longest edge of the triangle T h is the length of the edge E n is the outer

T E E

to the edge and denots the jump accross the edge E T denotes the edges of

E

an the triangle T E contains all inner edges of the triangulation and E contains all edges

h N

on and all edges on on which not b oth no des are in contact All elements T fullling

N C

max are marked for renement Mesh renement is done by regular red

T

sub divison of marked elements plus forming the conforming green Irregular

rened elements are removed b efore further renement takes place We calculated the

P

and compared it to the error b ound J u J Again we error estimator

h

T

T

used a nested solver with three iterations p er level Table shows the results Although

the time complexity is not O N anymore the growing of CPU time is mo derate The

adaptive rened mesh at level is given Figure

As announced in the intro duction we have also numerical results for the plane elasticity

problem For the rst example we set

h i

V v H v on fg

level N N J u J u J N J u J J u J Tsec

C h h h h

Table Adaptive renement

Figure Uniform and adaptive grid with and No des resp ectively

g

level N N J u J u J u total timesec

C h h

E

E

E

E

E

E

E

E

Table Nested iteration for elasticity problem

for the second one we use the circle

h i

p

C V H

For b oth examples we want to nd the eld minimizing the functional

Z Z

i h

T

dx jv j divv f v dx J v

over the set K fv V x v x x g with the linearized strain tensor

T

v v the parameters and the volume f v

ij ji ij

The corresp onding bilinear form is continuous and due to Korns inequalities also el

liptic mo dulo the rigid b o dy motions Therefore a blo ck diagonal preconditioner consisting

of two copies of preconditioners for the Laplace equation can b e used We used two times

the DD preconditioner for the Laplacian

The injection of the b oundary space into the global space is now replaced by the injec

tion in normal direction In the contact zone the normal direction is

We applied the nested multiplicative approximative algorithm with the parameters

and iterations p er level For the rst example we got the discretization

errors shown in Table We observe from Table that the squared error ku u k ch

h

which is the optimal convergence rate for smo oth solutions Therefore an uniformly rened

mesh leading to an assymtotically optimal algorithm can b e used We got similar results

for the second elasticity example Both deformed b o dies are drawn in Figure For the

dual comp onent p the L representative is calculated It is the normal traction which

is also drawn in Figure

Acknowledgements

I would like to thank Prof Ulrich Langer for his inspiration to this work and for his

clear presentations of domain decomp osition techniques I am also very grateful to the

anonymous referees for their valuable comments and suggestions which led to a substantial

improvement of the original pap er

Figure Elasticity problems

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