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MULTIGRID METHODS FOR SADDLEPOINT PROBLEMS ARISING FROM MORTAR FINITE ELEMENT DISCRETIZATIONS BARBARA I. WOHLMUTH



Abstract. A multigrid algorithm for saddle point problems arising from mortar nite element discretizations is analyzed. Here, we do not require that the constraints at the interface are satis ed in each smoothing step. Using mesh dependent norms for the Lagrange multiplier, suitable approximation and smoothing properties are established. A convergence rate independent of the meshsize is obtained for the W -cycle. Key words. Mortar nite elements, saddle point problems, multigrid methods AMS subject classi cations. 65N22, 65N30, 65N55

1. Introduction. Domain decomposition techniques provide powerful tools for the numerical solution of partial dierential equations. Within the framework of mortar methods [6, 7], the exible coupling of dierent discretization schemes or of nonmatching triangulations can be realized. Adequate weak continuity conditions replace the pointwise continuity at the interfaces. The arising variational problems are either positive de nite nonconforming problems or saddle point problems. Ecient iterative solvers for the discrete nonconforming as well as the discrete saddle point problem are well established, see [1, 2, 3, 10, 11, 12, 15, 16, 17, 18, 19, 22, 23, 28]. Working with the nonconforming variational problem has the drawback that the corresponding nodal basis functions have in general a non-local support. Thus it might be advantageous to work with the unconstraint product space for the numerical realization of the mortar method. Even if the starting point is the positive de nite variational problem, the non-local nodal basis function of the constrained space are in general not explicitly used [19]. Instead one works with the local nodal basis functions of the unconstrained product space and the global mortar projection. An alternative approach is to use the equivalent saddle point problem as starting point [4]. In this case, an inde nite problem has to be solved and standard multigrid methods cannot be applied. Recently, special multigrid techniques for the arising inde nite problems have been established [10, 11, 28]. The characteristic feature of these multigrid methods is the choice of the symmetric smoother for the saddle point problem. It is de ned such that the constraints are satis ed in each smoothing step. As a consequence, each smoothing step requires the exact solution of a modi ed Schur complement system. Using this type of smoother, one works in the positive de nite subspace of the saddle point problem, and the multigrid analysis of the standard positive de nite case carries over. The drawback is that the solution of the modi ed Schur complement system in each smoothing step might be too expensive. In [31], a generalization of this type of smoother is investigated. A dierent approach for the construction of an ecient iterative solver for the saddle point problem is given in [2, 17, 18, 22, 23]. The saddle point problem is solved by a multilevel preconditioned Lanczos iteration. The preconditioner is a block diagonal matrix involving a good preconditioner for the exact Schur complement.  Mathemetisches Institut, Universit at Augsburg, D-86135 Augsburg, Germany. E-mail: [email protected] 1

2

B. I. Wohlmuth

These observations motivate our new approach. In particular, we do not require that the iterates satisfy the weak continuity conditions at the interfaces exactly. As a consequence, we neither need a good preconditioner for the exact Schur complement nor an exact solver for a modi ed Schur complement. In Section 2, we give a brief overview of the mortar method in case of P1 Lagrangian nite elements. We review the de nition of the discrete spaces and the saddle point formulation as well as the a priori estimates. The two basic tools, approximation and smoothing properties, for the convergence analysis of a multigrid method are analyzed in Sections 3-4. In Section 3, the approximation property is formulated. Since we are not working in the positive de nite subspace, we cannot avoid the use of suitable norms for the Lagrange multiplier. Our analysis is based on mesh dependent norms for the Lagrange multiplier. In Section 4, we establish the smoothing property. We follow the lines of [27], and introduce a block diagonal smoother. Finally, the convergence rate for the W -cycle is given. 2. Mortar nite elements. Mortar methods are based on domain decomposition techniques. Within this approach, dierent discretization schemes as well as geometrical nonconforming triangulations can be coupled. The resulting nonconforming method is optimal in the sense that the discretization error is of the same order as the sum of the best approximation errors on the dierent subdomains. Here, we will only brie y review the de nition of a special mortar method. For an overview for more general mortar techniques, we refer to [4, 5, 6, 7, 11]. Let  IR2 be a bounded polygonal domain. We consider a geometrical conforming non-overlapping decomposition into K polygonal subdomains k such that

 =

K [

k=1

 k :

The intersection between the boundaries of two adjacent subdomains l and k , 1  l 6= k  K is called interface ?lk := @ l \ @ k . Furthermore, the union of subdomain boundaries S := [Kk=1 (@ k n @ ) can be decomposed into disjoint open subsets

S=

M [

m=1

m

where for each non-mortar m , 1  m  M , there exists a l(m) and k(m), 1  l(m) < k(m)  K such that m = ?l(m)k(m) . We consider the following elliptic second order boundary problem ?div(aru) + bu = f in ; (2.1) u = 0 on @ ; where a is an uniformly positive function a 2 L1 ( ), f 2 L2 ( ) and 0  b 2 L1 ( ). Each subdomain k , 1  k  K is associated with a quasi-uniform simplicial triangulation Tk;h of meshsize h. For the discretization on the dierent subdomains

k , we use P1 -conforming nite elements satisfying homogeneous boundary conditions on @ k \ @ ; X ( k ; Tk;h ), k = 1; :::; K . Then, the unconstraint product space

Xh :=

K Y

k=1

X ( k ; Tk;h )

3

Multigrid methods for mortar discretizations

satis es no weak continuity condition at the interface and is no suitable space for the discretization of (2.1). In particular, the consistency error would be not bounded by the meshsize h. To de ne an appropriate global discrete space, we have to impose weak continuity conditions at the interfaces of the decomposition. Each non-mortar

m will be associated with a 1D triangulation Sm;h , inherited either from Tl(m);h or Tk(m);h . Then, the local Lagrange multiplier space Mh ( m ; Sm;h ) associated with the non-mortar m is given by

Mh( m ; Sm;h ) := f 2 C ( m ) j je 2 P1 (e); e 2 Sm;h ; je 2 P0 (e); e contains an endpoint of m g; and the global one is de ned as the product space

Mh :=

M Y m=1

Mh ( m ; Sm;h );

see [5, 6, 7]. We refer to [26, 30] for possible modi cations of the Lagrange multiplier space. By means of Mh , a suitable global space Vh for the discretization of (2.1) can be de ned by

Vh := fv 2 Xh j b(v; ) = 0;  2 Mh g; where the bilinear form b(
;
) is given by the following duality pairing

b(v; ) :=

M X m=1

h[v]; i m ; v 2

K Y k=1

H 1 ( k );  2

M  Y m=1

0

H 21 ( m ) :

Here, [
] denotes the jump. The bilinear form b(
;
) restricted on Xh  Mh can be writR P ten as b(v; ) = M m=1 m [v ] ds. By means of Vh , the nonconforming formulation of the mortar method can be given: Find uh 2 Vh such that (2.2)

a(uh ; vh ) = f (vh ); vh 2 Vh ;

R P see [6, 7]. Here, the bilinear form a(
;
) is de ned as a(v; w) := Kk=1 k arv
rw + R Q bv w dx, v; w 2 Kk=1 H 1 ( k ) and f (v) := fv dx, v 2 L2 ( ). The following saddle point problem is equivalent to (2.2): Find (uh ; h ) 2 Xh  Mh such that

(2.3)

a(uh; vh ) + b(vh ; h ) = f (vh ); vh 2 Xh b(uh; h ) = 0;  h 2 Mh :

The Lagrange multiplier h is an approximation of the ux at the interfaces  := a @@un , see [5, 29]. We remark that the sign in the de nition of the jump does not in uence the solution uh and the orientation of n depends on the de nition of the jump. In the rest of this section, we state technical tools which will be necessary for the analysis of the multigrid method. In particular, the coercivity of the bilinear form a(
;
), a discrete inf-sup condition and well-known a priori estimates are given. In the following all constants 0 < c  C < 1 are generic constants depending on the coecients of (2.1), the shape regularity of the triangulation but not on the meshsize.

4

B. I. Wohlmuth

Under the assumption of H 2 -regularity, we have the following O(h) estimate for the discretization error in the broken H 1 -norm and an O(h2 ) estimate for the discretization error in the L2 -norm

jju ? uh jj1  C h jjf jj0; ; jju ? uh jj0;  C h2 jjf jj0;

P Q where jjvjj21 := Kk=1 jjvjj21; k , v 2 Kk=1 H 1 ( k ), see [6, 7, 10]. Furthermore, we have ellipticity of the bilinear form a(
;
) on Y  Y (2.5) a(v; v)  c jjvjj21 ; v 2 Y; (2.4)

and Vh is a subspace of Y

Y := fv 2

K Y k=1

Z

H 1 ( k ) j vj@ = 0; [v] ds = 0; 1  m  M g;

m

see [8]. Our multigrid approach will be based on the saddle point formulation (2.3). It is introduced and analyzed in [5] where a discrete inf-sup condition is given in the H001=2 -norm of the Lagrange multiplier. Here, we work with mesh dependent norms for which a discrete inf-sup condition is established in [29]

jjjj2h?s ;S :=

X X

m e2Sm;h

h2es jjjj20;e ;  2 Mh ; s  0

see also [10, 11]. The notation jj
jjh?s ;S is choosen because the mesh dependent norm represents a kind of discrete H s -dual norm. Then, the following inf-sup condition holds true (2.6) cjj jj 1  sup b(vh ; h ) ;  2 M ; h h? 2 ;S

vh 2Xh vh 6=0

jjvh jj1

h

and we have an a priori estimate for the Lagrange multiplier in the mesh dependent norm jj ? h jjh? 21 ;S  Chjjf jj0; ; (2.7) see [29]. Combining the two a priori estimates (2.4) and (2.7), we get an order h2 a priori estimate for the mesh dependent norm jj(u ? uh ;  ? h )jjh; S , where jj(v; )jj2h; S := jjvjj20; + jjjj2h? 23 ;S ; (v; ) 2 L2( )  L2 (S ):

The corresponding mesh dependent scalar product on L2 ( )  L2 (S ) is denoted by (
;
)h; S X X

h3e (;  )0;e :

m e2Sm;h Finally, the following lemma gives a relation between the weighted L2-norm of the ((v; ); (w;  ))h; S := (v; w)0; + (;  )h? 23 ;S := (v; w)0; +

jump and the nonconformity of an element. A similar result for higher order nite elements can be found in [29].

Multigrid methods for mortar discretizations

5

Lemma 2.1. Let vh 2 Xh satisfy

b(vh ; 2h ) = 0; 2h 2 M2h; where the triangulation Th is obtained by uniform re nement from T2h . Then, the jump of vh can be bounded by X X 1 jj[vh ]jj20;e  C inf1 jjvh ? vjj1 :

m e2Sm;h he

v2H0 ( )

Proof. Here, we will sketch out the proof only for the quasi-uniform case. For more details, we refer to [29]. Using the approximation property of M2h and a trace Theorem, we obtain for each v 2 H01 ( ) and 2h 2 M2h

jj[vh ]jj20;S = ([vh ]; [vh ? v] ? 2h )0;S  C (2h) 21 jj[vh ]jj0;S jj[vh ? v]jj 12 ;S  Ch 12 jj[vh ]jj0;S jjvh ? vjj1 : Here, the norm jj
jj1=2;S is de ned by jjjj212 ;S :=

M X m=1

jjjj221 ; m ;  2

M Y m=1

H 12 ( m );

where jj
jj1=2; m is the standard H 1=2 -norm on m .

3. Approximation property for the saddle point problem. A family of nite element spaces associated with a nested sequence of triangulations Tl of meshsize hl is given by X0  M0  X1  M1 


 Xl  Ml 


 Xn  Mn: Here, we assume that the triangulations are quasi-uniform and that hl?1 = 2hl . The spaces Xl , Ml and Xl  Ml equipped with the norms jj
jj0; , jj
jjh?l 3=2 ;S and jj
jjhl ; S , respectively, are Hilbert spaces. Let T be a linear continuous operator T : H1 ?! H2 . Here, H1 and H2 associated with the norms jj
jjH1 and jj
jjH2 , respectively, are Hilbert spaces. We use the standard operator norm

jjTxjjH2 : jjT jj := xsup 2H jjxjjH1 x6=01 The following lemma is an adaption of Lemma 4.2 in [27] to mortar nite elements. Based on this Lemma a suitable approximation property will be formulated. Lemma 3.1. Let (dl ; l ) 2 Xl  Ml be orthogonal to Xl?1  Ml?1 , i.e. ?




(dl ; l ); (vl?1 ; l?1 ) hl ; S = 0; and (wl ; l ) 2 Xl  Ml be the solution of: Find (wl ; l ) 2 Xl  Ml such that (3.1)

(vl?1 ; l?1 ) 2 Xl?1  Ml?1 ;

a(wl ; vl ) + b(vl ; l ) = (dl ; vl )0; ; b(wl ; l ) = (l ; l )h? 23 ;S ; l

vl 2 X l l 2 Ml :

6

B. I. Wohlmuth

Then, there exists a constant satisfying

jj(wl ; l )jjhl ; S  C h2l jj(dl ; l )jjhl ; S : Proof. The quasi-uniformity of the triangulations yields 3

3

c(jjwl jj0; + hl2 jjl jj0;S )  jj(wl ; l )jjhl ; S  C (jjwl jj0; + hl2 jjl jj0;S ): The estimate for the term in the Lagrange multiplier will be based on the discrete infsup condition (2.6) whereas the bound for jjwl jj0; is based on duality techniques. We start with an estimate for the upper bound of h3l =2 jjl jj0;S . The continuity of a(
;
), the discrete inf-sup condition (2.6) and the orthogonality of dl , (dl ; vl?1 )0; = 0, vl?1 2 Xl?1 , yield an upper bound for jjl jjh?l 1=2 ;S in terms of jjwl jj1

b(vl ; l ) = C inf sup (dl ; vl ? vl?1 )0; ? a(wl ; vl ) jjl jjh? 12 ;S  C vsup vl?1 2Xl?1 vl 2Xl jjvl jj1 2X jjvl jj1 l vl 6=0 vll 6=0l  C ( jjwl jj1 + hl jjdl jj0; ) :

(3.2) Observing that wl 2 Y , applying (2.5) and using (3.2), we nd for wl?1 2 Xl?1

c jjwl jj21  a(wl ; wl ) = (dl ; wl )0; ? b(wl ; l ) = (dl ; wl ? wl?1 )0; ? (l ; l )h? 32 ;S l  7 3  Chl jjdl jj0; + hl2 jjl jj0;S jjwl jj1 + Chl2 jjdl jj0; jjl jj0;S : 3

To obtain an upper bound for jjwl jj1 in terms of hl2 jjl jj0;S and jjdl jj0; , it is sucient to consider the quadratic form

g(s) := s2 ? (a1 + a2 )s ? a1 a2 ; s 2 IR; where a1 := h5l =2 jjl jj0;S  0, a2 := hl jjdl jj0;  0 and  is a positive constant. In case that a1 = a2 = 0, the unique solvability of (3.1) gives jjwl jj1 = 0. For a1 + a2 > 0, an easy calculation shows that g(s) > 0 for s  (1 + )(a1 + a2 ). Thus, we have 



3

jjwl jj1  Chl jjdl jj0; + hl2 jjl jj0;S :

(3.3)

Combining (3.3) with the upper bound for jjl jjh?l 1=2 ;S , we nd (3.4)

3



3



hl2 jjl jj0;S  C h2l jjdl jj0; + hl2 jjl jj0;S :

In the next step, we will focus on an estimate for jjwl jj0; . Let w^ 2 H01 ( ) be the solution of the continuous variational problem: Find w^ 2 H01 ( ) such that

a(w; ^ v) = (wl ; v)0; v 2 H01 ( ): Taking into account that wl , in general, is not contained in H01 ( ), we get jjwl jj20; = a(w; ^ wl ) + b(wl ; ^);

7

Multigrid methods for mortar discretizations

where ^ := arw^n is the ux of w^ at the interfaces S . Using the orthogonalities a(wl ; vl?1 ) + b(vl?1 ; l ) = 0, vl?1 2 Xl?1 , b(wl ; l?1 ) = 0, l?1 2 Ml?1 and observing b(w; ^ l ) = 0, l 2 Ml , we nd for vl?1 2 Xl?1 and l?1 2 Ml?1

jjwl jj20; = a(w^ ? vl?1 ; wl ) + b(wl ; ^) + b(w^ ? vl?1 ; l )  C ( jjw^ ? vl?1 jj1 jjwl jj1 + jj[wl ]jj0;S jj^ ? l?1 jj0;S + jj[w^ ? vl?1 ]jj0;S jjl jj0;S ) : We choose vl?1 2 Vl?1 as a local quasi-projection of w^ such that jjw^ ? vl?1 jj1  Chl3?1 jjw^jj2;

jj[w^ ? vl?1 ]jj0;S  Chl2?1 jjw^jj2; ; see [25], and l?1 2 Ml?1 such that 1

jj^ ? l?1 jj0;S  Chl2?1 jj^jj 12 ;S : Finally, we use the H 2 -regularity, a trace Theorem and Lemma 2.1 to obtain

jjwl jj20;  Chl





jjwl jj1 + jjl jjh? 21 ;S jjwl jj0;

l

which proves together with (3.3) and (3.4) the assertion. Let Al : Xl ?! Xl , Bl : Ml ?! Xl , Bl : Xl ?! Ml be the operators de ned by (Al vl ; wl )0; := a(vl ; wl ); (Bl l ; wl )0; := b(wl ; l ); (Bl wl ; l )h? 23 ;S := b(wl ; l ): l

In particular, the operator Bl satis es (3.5) c 1 jj jj 3  jjB  jj

h2l l h?l 2 ;S

1

l l 0;  C h2 jjl jjh? 32 ;S : l l

The upper bound is obtained by using an inverse estimate and the de nition of Bl jjB  jj = sup (Bl l ; wl )0; = sup b(wl ; l ) l l 0;

jjwl jj0;

wl 2Xl jjwl jj0;

wl 6=0 jj [ w ] jj jj  jj C jj jj 3 : l 0; S l 0; S  wsup  jjwl jj0;

h2l l hl? 2 ;S 2X wl 6=0l wl 2Xl wl 6=0 l

To establish the lower bound, we observe jjBl l jj0;  bjj(wwljj; l ) ; wl 2 Xl ; wl 6= 0: l 0;

Each wl 2 Xl is uniquely de ned by its values at the vertices of the triangulation. De ning wl (p) := l (p) if p is an interior vertex of one of the 1D interface triangulations Sm;l , 1  m  M . For all other vertices p of Tl , we set wl (p) := 0. This special choice yields b(wl ; l )  cjj[wl ]jj0;S jjl jj0;S , we refer to [29] for details. Now, the lower bound in (3.5) follows from jj[wl ]jj0;S  ch?l 1=2 jjwl jj0; . An easy consequence of (3.5) is (3.6)

vinf 2X Bl vll =Bll wl

jjvl jj0;  Ch2l jjBl wl jjh? 23 ;S ; wl 2 Xl : l

8

B. I. Wohlmuth

The self-adjoint non-singular operator Kl : Xl  Ml ?! Xl  Ml associated with the saddle point problem (2.3) is given by

Kl (vl ; l ) := (Al vl + Bl l ; Bl vl ); (vl ; l ) 2 Xl  Ml : The solution (wl ; l ) of the saddlepoint problem (3.1) satis es

(3.7)

Kl (wl ; l ) = (dl ; l ); and thus jjKl?1(dl ; l )jjhl ; S  Ch2l jj(dl ; l )jjhl ; S for (dl ; l ) 2 Xl  Ml being orthogonal on Xl?1  Ml?1 with respect to (
;
)hl ; S . Equivalently, we nd (3.8)

jj(wl ; l )jjhl ; S  Ch2l jjKl (wl ; l )jjhl ; S

for (wl ; l ) 2 Xl  Ml satisfying (Kl (wl ; l ); (vl?1 ; l?1 ))hl ; S = 0, (vl?1 ; l?1 ) 2 Xl?1  Ml?1 .

4. Smoothing property for the saddle point problem. The second basic tool to establish convergence within the multigrid framework is the smoothing property. Using the de nition (3.7) of the operator Kl , the saddle point problem (2.3) on Xl  Ml is equivalent to the following operator equation: Find zl := (ul ; l ) 2 Xl  Ml such that Kl zl = fl ;

where fl := (f~l ; 0) 2 Xl  Ml is de ned by (f~l ; vl )0; := (f; vl )0; , vl 2 Xl . The smoothing operator K^ l : Xl  Ml ?! Xl  Ml is de ned by means of the symmetric positive de nite bilinear forms a^(
;
) and d(
;
) on Xl  Xl and Ml  Ml , respectively (4.1) (K^ l (vl ; l ); (wl ; l ))hl ; S := a^(vl ; wl ) + d(l ; l ); (wl ; l ) 2 Xl  Ml : It has a block diagonal structure K^ l (vl ; l ) = (Al vl ; Dl l ); where the operators A^l : Xl ?! Xl and Dl : Ml ?! Ml are associated with the bilinear forms a^(
;
) and d(
;
), respectively. One smoothing iteration on level l is given by (4.2) zlm := zlm?1 + K^ l?1 Kl K^ l?1(rl ? Kl zlm?1); where rl stands for the right side of the system Kl zl = rl which has to be solved, zl is the exact solution and zlm denotes the iterate in the mth-step. Each smoothing can be easily performed provided that the application of A^?l 1 and Dl?1 is cheap. The following lemma gives the smoothing rate. Lemma 4.1. Let K^ l be de ned as in (4.1), where A^l , jjA^l jj  C=h2l , and Dl , jjDl jj  C=h2l , are self-adjoint positive de nite operators. Under the assumptions that there exists for each wl 2 Xl a wl , 0 < wl < 1 such that (Al wl ; wl )0;  wl (A^l wl ; wl )0; ; (Bl Dl?1 Bl wl ; wl )0;  (1 ? wl )(A^l wl ; wl )0; ;

9

Multigrid methods for mortar discretizations

we get the following smoothing property for the iteration (4.2)

jjKl emljjhl ; S  h2Cpm jje0l jjhl ; S ; m  1;

(4.3)

l m m where el := zl ? zl , 0  m is the iteration error in the mth-smoothing step. Furthermore, if jjA^?l 1 jj  Ch2l , there exists a constant independent of m such that

we have the following stability estimate for the iteration error

jjeml jjhl ; S  C jje0l jjhl ; S ; m  1:

(4.4)

Proof. The iteration error em l is given by eml = (Id ? K^ l?1 Kl K^ l?1 Kl )m e0l ; m  1: Since K^ l is a self-adjoint positive de nite operator and Kl is self-adjoint, there exists a complete set of eigenfunctions zli satisfying

K^ l? 2 Kl K^ l? 2 zli = i zli : 1

1

1 Setting (wli ; il ) := K^ l? 2 zli , we nd for i 6= 0, Al wli + 1 Bl Dl?1 Bl wli = i A^l wli :

i

q

Then, the assumptions on Al and Dl yield ji j  0:5(wli + 2wli + 4(1 ? wli ))  1. Without loss of generality, we can assume that the eigenfunctions are normalized (zli ; zlj )h; S = ij . Then, the norm of Kl em l is bounded by 1

1

1

1

1

1

jjKl emljjhl ; S  jjK^ l2 K^ l? 2 Kl K^ l? 2 (Id ? K^ l? 2 Kl K^ l?1 Kl K^ l? 2 )m K^ l2 e0l jjhl ; S  sup1 js(1 ? s2 )m j jjK^ l jj jje0l jjhl ; S : 1 s2spec (K^ l? 2 Kl K^ l? 2 )

p

Using supt2[0;1] (t(1 ? t2 )m )  C= m and jjK^ l jj  C=h2l , we obtain (4.3). To obtain the stability estimate (4.4), we use the same type of arguments. The assumption on Dl and (3.6) yield an upper bound for jjDl?1=2 jj

jjDl? 2 Bl wl jjh?l 3=2 ;S jjDl? 2 l jjhl?3=2 ;S = sup sup jjBl wl jjh?l 3=2 ;S l 2Ml jjl jjh?3=2 ;S  2M l 6=0 l Bl wll =ll 6=0 ^l12 wl jj0;

jj A  sup inf  Ch?l 1 h2l  Chl :  3=2 ;S l26=M0l Bwlw2lX=ll jjBl wl jjh? l l 1

1 jjDl? 2 jj =

1

l

The last inequality together with the assumption on A?l 1=2 gives jjK^ l?1=2 jj  hl , and thus jjeml jjhl ; S = jj(Id ? K^ l?1Kl K^ l?1 Kl )m1jj jje0l jjhl1; S  sups2[?1;1](1 ? s2 )m jjK^ l2 jj jjK^ l? 2 jj jje0l jjhl ; S  C jje0l jjhl ; S :

10

B. I. Wohlmuth

Remark: Replacing the iteration (4.2) by a conjugate residual algorithm improves the upper bound (4.3). As in the positive de nite case with a suitable Jacobi-type smoothing operator, we obtain jjKl eml jjhl ; S  hC2 m jje0l jjhl ; S ; m  1: l

For more details, we refer to [27]. Combining the approximation property (3.8) and the smoothing property (4.3), we obtain a mesh size independent convergence rate for the two-grid algorithm. The stability estimate (4.4) is necessary in case of the W -cycle analysis. Under the assumptions of Lemma 4.1 the convergence rate of the W -cycle in the jj
jjhl ; S -norm is independent of the number of re nement levels provided that the number of smoothing steps is large enough [21, 27]. Remark: A suitable smoother in the algebraic formulation of the method is given by the following diagonal matrix 



Id 0 0 h2l Id with some constant  > 0. Here, nodal basis functions for the nite elements as well as the Lagrange multiplier are used. Then, the level independent convergence rate of the W -cycle is obtained for the energy norm jj
jjK^ of K^ .

K^ := 

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