Numerische Mathematik

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1 Numer. Math. (1999) 84: Digita Object Identifier (DOI) /s Numerische Mathematik c Springer-Verag 1999 Mutigrid methods for a parameter dependent probem in prima variabes Joachim Schöber Johannes-Keper-Universität, Institut für Mathematik, Atenberger Strasse 69, A-4040 Linz, Austria Received June 17, 1998 / Revised version received October 26, 1998 / Pubished onine September 7, 1999 Summary. In this paper we consider mutigrid methods for the parameter dependent probem of neary incompressibe materias. We construct and anayze mutieve-projection agorithms, which can be appied to the mixed as we as to the equivaent, non-conforming finite eement scheme in prima variabes. For proper norms, we prove that the smoothing property and the approximation property hod with constants that are independent of the sma parameter. Thus we obtain robust and optima convergence rates for the W- cyce and the variabe V-cyce mutigrid methods. The numerica resuts pretty we conform the robustness and optimaity of the mutigrid methods proposed. Mathematics Subject Cassification (1991): 65N55 1. Introduction We consider the inear easticity probem to find u [H0 1(Ω)]2 such that (1) 2µ e(u) :e(v) dx + λ div u div vdx= f T v dx, Ω Ω with the positive constants λ and µ of Lamé, the strain operator e(u) := 0.5( u +( u) T ) and the voume force f [L 2 (Ω)] 2. We are interested in the neary incompressibe case, i.e. the Poisson ration ν is cose to 0.5. Then the bad parameter ε := 2µ/λ becomes sma. This work is supported by the Austrian Science Fund - Fonds zur Förderung der wissenschaftichen Forschung - under project P TEC Ω

2 98 J. Schöber For conforming ow order finite eement methods the parameter ε enters disadvantageousy into the discretization error estimate. This effect is aso verified numericay and is we known as ocking effect, [2]. Various nonconforming discretization methods ead to discretization errors robust for ν 0.5, see [12,10]. We use the mixed formuation for u and div u to obtain a stabe sadde-point system. By the choice of non-continuous finite eements for the dua variabe, it can be eiminated at eement eve, and we return to a symmetric positive definite finite eement method. Aso the convergence rate of standard mutigrid methods appied to sove the positive definite inear system deteriorates as ν 0.5. To overcome this difficuty, robust mutigrid methods have been designed for the equivaent mixed finite eement scheme with penaty term in [21,15,3,9,4]. Reated mutigrid methods for the Stokes probem are anayzed in [8,20]. The papers mainy differ in the kind of smoothing iteration used for the indefinite system. In [18], a new mutigrid method for parameter dependent probems in prima variabes has been suggested and the anaysis for the two-eve method was given. The key components are an overapping bock-smoother capturing the divergence free basis functions, and a grid transfer operator proongating coarse grid divergence free functions to fine grid divergence free functions. In this paper we estabish the approximation and the smoothing property necessary for the mutigrid anaysis [14,6]. During the anaysis we switch between both equivaent agorithms, the prima one and the mixed one. The outine of the paper is as foows. In Sect. 2 some avaiabe resuts are coected. The agorithmic aspects of the mutigrid method are formuated in Sect. 3, the anaysis is started in Sect. 4. Approximation property and smoothing property are proven in Sect. 5 and in Sect. 6, respectivey. Numerica resuts are given in Sect Stabiity and discretization We introduce the dua variabe p := ε 1 div u and obtain the equivaent mixed probem to find (u, p) X := V Q := [H 1 0 (Ω)]2 L 2 /R such that (2) B ((u, p), (v, q))=(f,v) 0 (v, q) X, with f =(2µ) 1 f and the biinear-form (3) B ((u, p), (v, q)) =(e(u),e(v)) 0 +(div u, q) 0 +(div v, p) 0 ε (p, q) 0,

3 Mutigrid methods for a parameter dependent probem in prima variabes 99 where (.,.) 0 denotes the inner product in L 2 of scaar, vector vaued or tensor vaued functions. Ceary, B is continuous on X X with the product norm (u, p) X = ( u p 2 0 )1/2. The proper stabiity criterion on some subspace X := V Q X is the condition B((u, p), (v, q)) (4) sup c (u, p) X (u, p) X. (v,q) X (v, q) X Here and throughout the paper c wi be a generic constant which is independent of the parameter ε and the mesh-size defined beow and which may be different in different equations. It foows from the second inequaity of Korn, the LBB condition of the Stokes probem and further estimates due to the penaty term that B is stabe on X = X, see [11,1]. We assume that Ω is a convex poygona domain and get from [10] the reguarity theorem (5) u 2 + p 1 c f 0. For finite eement discretization we choose the subspace X L = V L Q L X, where V L consists of continuous, piecewise quadratic functions, and Q L of piecewise constant functions on a trianguar mesh with mesh-size parameter h L. The integer L defines the number of mutigrid eves. We get the finite dimensiona probem find (u L,p L ) X L such that (6) B((u L,p L ), (v L,q L ))=(f,v L ) 0 (v L,q L ) X L. The LBB condition is fufied for the pair of spaces V L and Q L, see [11, p. 211], which impies the stabiity condition (4) on X = X L. The essentia fact is the non-continuity of the functions in Q L eading to an easiy invertibe matrix for the L 2 inner product. The dua variabe p L can be eiminated eement-wise and the probem can be reduced to the non-conforming symmetric and positive definite probem find u L V L such that (7) A L (u L,v L )=(f,v L ) 0 v L V L, with the biinear-form (8) A L (u, v) =(ε(u),ε(v)) 0 + ε 1 (I Q L div u, div v) 0. The operator I Q L denotes the L 2-orthogona projection onto Q L. We mention that this projection can be impemented in the eement matrix assembing subroutine. Due to the equivaence of the prima and the mixed finite eement method we get bounds for the discretization error that are independent of ε aso for the prima version. The author is aware of the sub-optima convergence rate O(h) for the P 2 P 0 eement pairing. There exist severa eements with non-continuous pressure and optima convergence rate, see [13,11]. The eement is chosen for reasons of simper notation and impementation, but the foowing anaysis is not imited to the specia eement.

4 100 J. Schöber Fig. 1. Subspace V,T used in proongation 3. The mutigrid agorithm For the appication of mutigrid sovers a sequence of uniformy refined trianguations T of mesh-size h and the corresponding nested P 2 P 0 finite eement spaces (9) X 1 = V 1 Q 1 X 2 = V 2 Q 2... X L = V L Q L are used. By means of the computabe Q-orthogona projection operators I Q : Q Q we define the biinear-forms (10) A (u, v) =(e(u),e(v)) 0 + ε 1 (I Q div u, div v) 0 u, v V, for =1,...,L. We mention that the forms are defined on the infinite dimensiona space V. We define norms u A := A (u, u) 1/2 and the L 2 sef-adjoint operators A : V V as (A u,v ) 0 = A (u,v ), u,v V, =1,...,L. It is cear that A (.,.) estimates A +1 (.,.) from beow, i.e. A +1 (u, u) A (u, u) u V, but the converse estimate does not hod with some constant independenty bounded in ε. This fact requires specia grid transfer operators, which are constructed as foows. On each eve =2,...,Lwe define the subspace of functions which vanish on the boundaries of the coarse grid eements (11) V,T := [H0 1 (T )] 2 V. T T 1 It spits orthogonay into T 1 subspaces. Each of them is generated by the basis functions beonging to the nodes inside a triange of the coarser grid, see Fig. 1. We define the projection operator P A,T : V V,T such that (12) A (P A,T u, v,t )=A (u, v,t ) u V, v,t V,T.

5 Mutigrid methods for a parameter dependent probem in prima variabes 1 Fig. 2. Basis functions for kerne of I Q div The co-projection I P A,T is the discrete harmonic extension on each coarse grid triange. It can be computed fast and wi be used as proongation operator. We use the natura embedding V 1 V without denoting it by any symbo. (13) Proongation u 1 u : u =(I P A,T )u 1 The idea of this proongation is to ift coarse grid div-free functions to fine grid div-free functions. As we wi see ater, this proongation is continuous in the sense of (I P A,T ) u 1 A c u 1 A 1 u 1 V 1. We define the operator E 1 : V V as E 1 proongation can be rewritten as = P A,T A 1 such that the I P A,T = I E 1 A. The operator E is sef-adjoint with respect to (.,.) 0. In matrix form E is the restriction of A to the degrees of freedom spanning the space V,T. Using the L 2 -orthogona projection P L 2 1 : V V 1, the (.,.) 0 - adjoint restriction operator is P L 2 1 (I A E 1 ). We mention that the projection P L 2 1 is required for notation ony, it does not enter into the computation. Aso the smoother must be propery designed. A damped Richardson smoother (I τa ) woud need a damping parameter τ proportiona to ε. Thus the components of the error in the kerne of A woud be smoothed out very sow, as ε becomes sma. The suggested smoother is a bock Jacobi smoother, which takes care of the kerne of I Q div. On a simpy connected domain with ony one part of natura boundary conditions, the kerne of I Q div is spanned by basis functions drawn in Fig. 2, simiar to [11, pp 268].

6 102 J. Schöber V,i Fig. 3. Subspaces containing div-free basis functions These kerne basis functions are captured by subspaces V,i generated by noda basis functions beonging to the nodes drawn in Fig. 3. For a =2,...,L, this eads to the definition of the n subspaces (14) V,i =[H 1 0 (Ω i )] 2 V assigned to the corner nodes N,i,i =1,...,n, of the trianguation T. Here, Ω i is the cosure of the union over eements in T adjacent to the node N,i. We define the projections P A,i : V V,i as A (P A (15),i u, v,i) =A (u, v,i ) u V v,i V,i. By means of these subspaces we define the bock Jacobi smoother as (16) Smoother S A : V V : S A = I τ n i=1 P A,i. The necessary damping parameter depends ony on the number of overapping spaces, which is bounded for shape reguar eements. Especiay, it does not depend on ε and. We assume that τ is sma enough to ensure ony positive eigenvaues of S A. In the numerica exampes, we wi aso use the according bock Gauss-Seide smoother, which does not need any damping at a. We aso define the operator D 1 : V V as (17) D 1 = n i=1 P A,i A 1, which corresponds in matrix form to the sum over oca inverses. By means of D the smoother can be written as (18) S A = I τd 1 A. Now we can state the mutigrid agorithm in recursive form. We appy m smoothing steps on the eve and perform either the V -cyce (q =1)orthe W -cyce (q =2).

7 Mutigrid methods for a parameter dependent probem in prima variabes 103 Agorithm 1 Procedure MG(u, f, ) if =1 MG(u, f, ) =A 1 1 f ese u 1,0 = u do j =1,...,m u 1,j = u 1,j 1 + τd 1 (f A u 1,j 1 ) d 1 = f A u 1,m d 2 = P L 2 1 (I A E 1 )d 1 u 2,0 =0 do j =1,...,q u 2,j = MG(u 2,j 1,d 2, 1) u 3 =(I E 1 A )u 2,q u 4,0 = u 1,m + u 3 do j =1,...,m u 4,j = u 4,j 1 + τd 1 (f A u 4,j 1 ) MG(u, f, ) =u 4,m 4. The mutigrid anaysis In this section we start the anaysis of the mutigrid method. First, we specify a mutigrid method for the mixed form and prove equivaence to the agorithm stated in the ast chapter. We define an L 2 -ike norm, for which we wi prove the approximation property and the smoothing property in the foowing two chapters. For =1,...,Lwe define the subspaces } (19) X,0 := {(u,p ) X : I Q div u = εp. We wi use the reation (20) B((u,p ), (0,q ))=0 (u,p ) X,0, q Q ater. From (10) and (19) it foows that (21) B((u,p ), (v, 0)) = A (u,v ) (u,p ) X,0, v V is vaid. We extend definition (14) to oca mixed spaces (22) X,i := V,i Q,i with Q,i =(L 2 (Ω,i )/R) Q for i = 1,...,n, = 1,...,L. We aso extend the spaces V,T of the proongation defined in (11) to the mixed spaces (23) X,T := V,T Q,T with Q,T := (L 2 (T )/R) Q, T T 1

8 104 J. Schöber =2,...,L. The spaces are designed such that the orthogona decomposition (24) Q = Q 1 Q,T and (25) B((u T,p T ), (0,q 1 ))=0 (u T,p T ) X,T, q 1 Q 1 hod. In addition to the norm. X we define the energy-norm (u, p) B := B((u, p), (u, p)) 1/2 =( e(v) ε p 2 0) 1/2, which by Korn s inequaity is equivaent to the norm ( u ε p 2 0 )1/2. For any subspace X X the projection P B : X X is defined by (26) B(P B (u, p), (v, q)) = B((u, p), (v, q)) (v, q) X. We wi use projections to X 1, X,T and X,i. The next emma coects some of their properties. Lemma A subspaces X, X,i and X,T fufi the stabiity condition (4) with one common constant c. 2. The projections P 1 B, P,i B, P,T B are we defined and uniformy bounded on X with respect to the. X -norm. 3. P 1 B maps X,0 into X 1,0. 4. P,i B and P,T B map X,0 into itsef, and they are bounded by 1 with respect to the. B -norm on X,0. 5. The co-projection I P,T B maps X 1,0 into X,0. Proof. 1. Stabiity condition for the spaces are standard. The spaces V,i and the factor-spaces of V,T can be derived from a finite number of spaces by scaing and transation, and these transformations do not change the stabiity constant. Thus the common constant is the maximum of a finite number of stabiity constants. 2. The continuity of P B with respect to. X -norm foows from the stabiity condition (4) P B B(P B (u, p), (v, q)) (u, p) X c sup (v,q) X (v, q) X 3. For (u, p) X,0 we get = c sup (v,q) X B((u, p), (v, q)) (v, q) X c (u, p) X. B(P B 1 (u, p), (0,q 1)) = B((u, p), (0,q 1 ))=0 q 1 Q 1.

9 Mutigrid methods for a parameter dependent probem in prima variabes Next set (û, ˆp) = P,i B(u, p). We now decompose a function q Q orthogonay into q 1 Q,i and q 2 =(q, 1) 0,Ω,i / Ω,i in Ω,i and q 2 = q in Ω \ Ω,i. B((û, ˆp), (0,q 1 )) vanishes by definition of the projection, B((û, ˆp), (0,q 2 ))=(div û, q 2 ) 0,Ω,i ε (ˆp, q 2 ) 0,Ω,i =0is achieved by Green s theorem and definition of Q,i. By (24), (25) and the same arguments P,T B X,0 is proven. Now, et (u, p) X,0 and X such that (û, ˆp) =P B (u, p) X,0. Then, (û, ˆp) 2 B = B((û, ˆp), (û, ˆp)) = B((u, p), (û, ˆp)) = B((u, p), (û, ˆp)) + B((u, p), (0, ˆp)) B((0,p), (û, ˆp)) =(e(u),e(û)) 0 + ε (p, ˆp) 0 ( e(u) ε p 2 0) 1/2 ( e(û) ε ˆp 2 0) 1/2 = (u, p) B (û, ˆp) B gives the upper bound Let (u, p) X 1,0. Then B((I P B,T )(u, p), (0,q 1))=0 q 1 Q 1 hods because of the assumption and (25). The same form tested with q Q,T vanishes because of the definition of the projection, and Q can be decomposed by (24). We aso define projections P A,A k : V k V V by (27) A (P A,A k u, v) =A k (u, v) v V and set P A = P A,A. This definition is consistent with (12) and (15). Lemma 2. Let X = V Q X and (û, ˆp) =P B (u, p). 1. If (u, p) X k,0 and (û, ˆp) X,0, then û = P A,A k u. 2. If (u, p) (û, ˆp) X,0, then û = P A u. Proof. To verify statement 1 we use (21) and obtain A (û, v) =B((û, ˆp), (v, 0)) = B((u, p), (v, 0)) = A k (u, v) v V. Statement 2 is checked by A (û u, v) =B((û u, ˆp p), (v, 0))=0 v V. Agorithm 1 eads to the mutigrid operator ML A, which for =2,...,L fufis the recursion M A 1 =0, M A (28) =(S A )m (S A )m. ( I (I P A,T )(I (M 1 A )q )P A 1,A 1 (I P A,T ) )

10 106 J. Schöber We define the corresponding mutigrid operator for the mixed system as M B 1 =0, M B =(S B ) m ( I (I P,T B )(I (M 1 B )q )P 1 B (I P,T B )) (S B ) m, with the smoothing operators S B n := I τ i=1 P B,i for 2 L. The iteration for the mixed system is we defined for the imit case ε =0, too. Theorem 1 (Equivaence of agorithms). Both mutigrid procedures are equivaent, namey for (u,p ) X,0 there hods (29) (û, ˆp ):=M B (u,p ) fufis û = M A (u ). Proof. By induction on, Lemma 1 and Lemma 2. In the foowing chapters we wi need the approximation properties of the finite eement spaces. Let I V be the Lagrange interpoator into V. Reca ( the (oca) ) L 2 projector I Q to Q and define the product operator I X = I V,I Q : X X. Then the approximation inequaities (30) hod. u I V u 1 ch u 2 and p I Q p 0 ch p 1 We define the norm (31) (u, p) 2,0 := h 2 u ε p I Q 1 p 2 0. On the space X,0, it is identica to the norm u 2,0 (32) := h 2 u ε 1 I Q div u ε 2 I Q 1 div u 2 0 on V. These norms wi be used in the mutigrid proof for measuring smoothness. The main theorem of this paper is Theorem 2 (Two-grid convergence). The two-grid operator ˆM A can be estimated by (33) ˆM A A cm 1/4, with a constant c independent of and ε. The two-grid operator ˆM B maps X,0 into itsef and is bounded on X,0 by (34) ˆM B B cm 1/4 with a constant c independent of and ε.

11 Mutigrid methods for a parameter dependent probem in prima variabes 107 Proof. Define for (u 0,p 0 ) X,0 (35) (u 1,p 1 )= ( I (I P B,T )P B 1 (I P B,T )) (u 0,p 0 ) and (36) (u 2,p 2 )=(S B ) m (u 1,p 1 ). By the previous emmata we get (u 1,p 1 ) X,0 and ( ) u 1 = I (I P A,T )P A 1,A 1 (I P A,T ) u 0. In Sect. 5 we wi prove the approximation property (see Theorem 4) (37) (u 1,p 1 ),0 = u 1,0 c u 0 A = c (u 0,p 0 ) B using the mixed form. We aso get (u 2,p 2 ) X,0 and u 2 =(S A )m u 1. In Sect. 6 we wi prove the smoothing property (see Theorem 5) (38) (u 2,p 2 ) B = u 2 A cm 1/4 u 1,0 = cm 1/4 (u 1,p 1 ),0 using the prima form. Combining both properties proves the theorem. The foowing theorem foows by standard techniques [14,6,5]. Theorem 3 (Mutigrid convergence). The norm of the W-cyce operator is bounded independenty of L and ε if the number of smoothing steps m is sufficienty arge. The variabe V-cyce operator with m =2 L eads to a preconditioner C 1 L := (I ML A)A 1 L with condition number κ(c 1 L A L) bounded independenty of L and ε. 5. Approximation property The coarse grid operator (u 1,p 1 ) X,0 (u 5,p 5 ) X,0 is spit into (39) (u 2,p 2 )=(I P,T B )(u 1,p 1 ), (u 3,p 3 )=P 1 B (u 2,p 2 ), (u 4,p 4 )=(I P,T B )(u 3,p 3 ), (u 5,p 5 )=(u 1,p 1 ) (u 4,p 4 ). Theorem 4 (Approximation property). Let (u 1,p 1 ) X,0 and compute (u 5,p 5 ) by (39). Then the approximation property (40) is vaid. (u 5,p 5 ),0 c (u 1,p 1 ) B

12 108 J. Schöber Proof. We use the triange inequaity and the three emmata proven beow to obtain the resut (u 5,p 5 ),0 = (u 1,p 1 ) (u 4,p 4 ) 0 (u 1,p 1 ) (u 2,p 2 ),0 + (u 2,p 2 ) (u 3,p 3 ),0 + (u 3,p 3 ) (u 4,p 4 ),0 c (u 1,p 1 ) B. Lemma 3. With the notation of (39) there hods (41) (u 2,p 2 ) B + (u 2,p 2 ) (u 1,p 1 ),0 + p 2 I Q 1 p 2 0 c (u 1,p 1 ) B. Proof. Lemma 1 gives P,T B (u 1,p 1 ) B (u 1,p 1 ) B, which bounds the first term. The second term is bounded due to the norm equivaence. B.,0 on X,T. From p 2 I Q 1 p 2 Q,T, stabiity (4) of X,T, orthogonaity (25) and the definition of P,T B we obtain p 2 I Q 1 p B((0,p 2 I c sup p 2), (v, q)) (v,q) Q,T (v, q) X B((0,p 2 ), (v, q)) = c sup (v,q) Q,T (v, q) X = c sup (v,q) Q,T B(( u 2, 0), (v, q)) (v, q) X c u 2 1. Lemma 4. With the notation of (39) there hods Q (42) (u 3,p 3 ) B + (u 3,p 3 ) (u 2,p 2 ),0 c (u 1,p 1 ) B. Proof. By stabiity of X 1, the definition of (u 3,p 3 ), continuity of B(.,.) and Lemma 3 we get u p 3 I Q 1 p B((u 3,p 3 I c sup p 2), (v, q)) (v,q) Q 1 (v, q) X and in combination with Lemma 3 Q B((u 2,p 2 I 1 = c sup p 2), (v, q)) (v,q) Q 1 (v, q) X c ( u p 2 I Q 1 p 2 0 ) c (u 1,p 1 ) B, p 3 p 2 0 p 3 I Q 1 p p 2 I Q 1 p 2 0 c (u 1,p 1 ) B. Q

13 Mutigrid methods for a parameter dependent probem in prima variabes 109 This gives aso ε p ε p 2 p ε p c (u 1,p 1 ) 2 B. We state the dua probem on X B((ϕ, ψ), (v, q))=(u 2 u 3,v) 0 (v, q) X and get the L 2 estimate by Gaerkin orthogonaity, approximation (30) and reguarity (5) u 2 u = B((ϕ, ψ), (u 2 u 3,p 2 p 3 )) = B((ϕ, ψ) (I 1 V ϕ, IQ 1 ψ), (u 2 u 3,p 2 p 3 )) c ( ϕ I 1 V ϕ 1 + ψ I Q 1 ψ 0)( u 2 u p 2 p 3 0 ) ch( ϕ 2 + ψ 1 )( u u p 2 p 3 0 ) ch u 2 u 3 0 (u 1,p 1 ) B. Dividing by u 2 u 3 0 we obtain the resut. Lemma 5. With the notation of (39) there hods (43) (u 4,p 4 ) (u 3,p 3 ),0 c (u 1,p 1 ) B. Proof. Friedrichs inequaity on V,T, stabiity (4), Gaerkin - and orthogonaity (25) give (u 4,p 4 ) (u 3,p 3 ),0 c (u 4,p 4 ) (u 3,p 3 ) X B((u 3 u 4,p 3 p 4 ), (v, q)) c sup (v,q) X,T (v, q) X B((u 3,p 3 ), (v, q)) = c sup (v,q) X,T (v, q) X = c sup (v,q) X,T B((u 3, 0), (v, q)) (v, q) X u 3 V, and the proof if compete. 6. Smoothing property In this chapter we prove the smoothing property (44) (I τd 1 A ) m u A cm 1/4 u,0.

14 110 J. Schöber Reca that we have chosen τ such that τd 1 A A 1. The estimate (I τd 1 A ) m u 2 A =(D 1 A (I τd 1 A ) 2m u, u) D cm 1 u 2 D (45) is we estabished in mutigrid theory [14]. By additive Schwarz techniques [22,17] the induced norm u D = (D u, u) 1/2 0 can be expressed by u 2 D = inf u= u,i 2 A u.,i u,i V,i If the estimate u D c u,0 woud be true, the smoothing property woud be proven. Unfortunatey, it is not. The essentia part of this section is the proof of the estimate (46) u [D,A ] 1/2 c u,0, where. [D,A ] 1/2 is the interpoation norm between. D and. A with parameter 1/2. We use the rea method of interpoation of Lions and Peetre [16], see aso [7]. Beside inequaity (45), the energy estimate (I τd 1 A ) m u A u A hods true by the choice of τ. Interpoation gives (47) (I τd 1 A ) m u A cm 1/4 u [D,A ] 1/2, and therefore inequaity (46) impies the smoothing property (44). We define the biinear-form for the imit case ε =0as (48) B 0 ((u, p), (v, q))=(e(u),e(v)) 0 +(div u, q) 0 +(div v, p) 0. To estabish (46) we spit u = u 1 + u 2 + u 3 by soving for (u i,p i ) X such that B 0 ((u 1,p 1 ), (v, q)) = B 0 ((u, 0), (v, 0)), (49) B 0 ((u 2,p 2 ), (v, q)) = B 0 ((u, 0), (0,q I Q 1 q)), B 0 ((u 3,p 3 ), (v, q)) = B 0 ((u, 0), (0,I Q 1 q)) (v, q) X. The spitting is constructed such that u 1 is discrete divergence free, u 2 has non-smooth divergence and u 3 has smooth divergence. Theorem 5 (Smoothing property). The estimate (46) and therefore the smoothing property (44) are vaid.

15 Mutigrid methods for a parameter dependent probem in prima variabes 111 Proof. We spit u using (49), appy the triange inequaity, Lemma 7-9, and Lemma 6 beow to obtain (46) by u [D,A ] 1/2 u 1 [D,A ] 1/2 + u 2 [D,A ] 1/2 + u 3 [D,A ] 1/2 c ( u 1,0 + u 2,0 + u 3,0 ) c u,0. The smoothing property (44) foows by the estimates (47). Lemma 6. The decomposition (49) is stabe in.,0 norm, namey (50) u 1,0 + u 2,0 + u 3,0 c u,0. Proof. By stabiity (4) we get the bounds u p 1 0 c u 1 and u p 2 0 c I Q div u 0. First, we bound u 1 2,0 = h 2 u The soution of the dua probem find (ϕ, ψ) X such that B 0 ((ϕ, ψ), (v, q))=(u 1,v) 0 (v, q) X, is bounded by ϕ 2 + ψ 1 c u 1 0. By Gaerkin orthogonaity, approximation, reguarity, and the inverse inequaity h u 1 c u 0 we obtain u = B 0 ((ϕ, ψ), (u 1,p 1 )) = B 0 ((ϕ, ψ) I X (ϕ, ψ), (u 1,p 1 )) + B 0 (I X (ϕ, ψ) (ϕ, ψ), (u, 0)) + B 0 ((ϕ, ψ), (u, 0)) c (h ( ϕ 2 + ψ 1 )( u p 1 0 )+h( ϕ 2 + ψ 1 ) u 1 +( ϕ 2 + ψ 1 ) u 0 ) c (h u 1 0 u 1 + u 1 0 u 0 ) c u 1 0 u 0. Next, we estimate (51) h 2 u c I Q div u 2 0 cε u 2,0. Therefore et B 0 ((ϕ, ψ), (v, q))=(u 2,v) 0 (v, q) X, then we get by B((u 2,p 2 ), (v,q 1 ))=0 v V,q 1 Q 1 u = B 0 ((ϕ, ψ) (I V ϕ, I Q 1 ψ), (u 2,p 2 )) ch( ϕ 2 + ψ 1 )( u p 2 0 ) c u 2 0 h I Q div u 0. The ast term u 3 is bounded by the triange inequaity. The discrete divergence free part u 1 is estimated by ifting to the potentia space and Soboev-Space interpoation in the next emma.

16 112 J. Schöber Lemma 7. Let u 1 be defined in (49). Then the estimate (52) u 1 [D,A ] 1/2 c u 1,0 is vaid. Proof. First, we define a ifting procedure E : V V and a eft-inverse interpoation operator Π : V V between discrete divergence free and continuous divergence free functions. Therefore et X + := V + Q + := T (H1 0 (T ) L 2(T )/R) and set Eu 1 = u 1 w, with (w, p) X + such that (53) B 0 ((w, p), (v, q)) = B 0 ((u 1, 0), (0,q)) (v, q) X +. Hence, div Eu 1 =0, and by stabiity and Friedrichs inequaity Eu h 1 Eu 1 0 ch 1 u 1 0. Because Ω is assumed to be convex, there exists a potentia ϕ H0 2(Ω) such that Eu 1 = rot ϕ ϕ 2 + h 1 ϕ 1 ch 1 u 1 0. The interpoation is a modification of the Scott-Zhang interpoation [19]. First, shrink a Ω i to Ω i = N i +0.9(Ω i N i ). For each node N i seect an edge e i and a set σ i such that N i σ i e i, σ i c e i and σ i Ω j = corner nodes N i N j, see the figure beow. We set σ = σ i. Ni σi ~ Ωj Foowing [19], we construct a L 2 (σ)-biorthogona basis { i L 2 (σ i )} to the noda basis {p i }. The projection operator Π 1 : V V Π 1 v := (v, i ) L2 (σ)p i is we defined on H 1 and the approximation is of optima order (54) v Π 1 v m ch 1 m v 1, m =0, 1. If v V is such that supp v Ω i, then Π 1 v V,i. The operator Π 2 : V V defined by Π 2 v(n i )=0 corner nodes N i Π 2 vds= vds edges e i e i e i

17 Mutigrid methods for a parameter dependent probem in prima variabes 113 is standard for Stokes probems [11, p. 211], and fufis Π 2 v 1 c ( v 1 + h 1 v 0 ). Then the projection operator Π := Π 2 (I Π 1 )+Π 1 fufis Π 1 c and I Q div v = I Q div Π v, and thus (55) Π A c. Because Eu 1 u 1 V + and Π vanishes on V +, it is a eft-inverse to the ifting defined above. Because Π 2 preserves support in Ω i, aso Π maps H0 1( Ω i ) into V,i. Let {Ψ i } be a partition of unity fufiing Ψi =1, supp Ψ i Ω i, h 2 Ψ i 2, + h Ψ i 1, + Ψ i 0, c. The product rue and integration by parts gives (56) Ψ i ϕ 2 c (h 2 ϕ 0,Ωi + ϕ 2,Ωi ). Using Π rot (Ψ i ϕ) V,i, (55), (56), we get Π rot ϕ 2 D Π rot (Ψ i ϕ) 2 A c rot (Ψ i ϕ) 2 A c Ψ i ϕ 2 2 c ( h 4 ϕ 2 0,Ω i + ϕ 2 ) 2,Ω i c ( h 4 ϕ ϕ 2 ) 2 By oca L 2 -projection onto a C 1 continuous FE-space of the same meshsize h we can spit ϕ = ϕ + ϕ such that and the inverse inequaity and the approximation inequaity ϕ j c ϕ j j =0, 1, 2, ϕ 2 ch 2 ϕ 0 ϕ 0 ch 2 ϕ 2

18 114 J. Schöber are fufied. This gives Π rot ϕ D ch 2 ϕ 0 and Π rot ϕ D c ϕ 2. Using operator interpoation and norm equivaence H 1 (Ω) [L 2 (Ω),H 2 (Ω)] 1/2 we get u 1 [D,A ] 1/2 = Π rot ϕ [D,A ] 1/2 Π rot ϕ [D,A ] 1/2 + Π rot ϕ [D,A ] 1/2 ( ) c ϕ [h 2 L 2,H 2 ] 1/2 + ϕ H 2 c ( ϕ h 1 H 1 + ϕ H 2) c ( h 1 ϕ 1 + ϕ 2 ) ch 1 u 1 0 c u 1,0. The component u 2 is orthogona to divergence free functions and has non-smooth divergence. Lemma 8. Let u 2 be defined in (49). Then the estimate (57) u 2 [D,A ] 1/2 c u 2,0 is vaid. Proof. We use. 2 A c. 2 D,. 2 A ch 2 ε and the intermediate resut (51) to obtain u 2 2 [D,A ] 1/2 c u 2 2 D = c inf u 2 = ui 2 A u i c inf u 2 = h 2 ε 1 u i 2 0 ch 2 ε 1 u c u 2,0. u i The part u 3 with smooth divergence wi now be estimated by better approximation of the coarse grid interpoant of the dua variabe. Lemma 9. Let u 3 be defined in (49). Then the estimate (58) u 3 [D,A ] 1/2 c u 3,0 is vaid. Proof. By definition of u 3 we have I Q div u 3 = I Q 1 div u, and together with stabiity of X 1 we get u 3 1 I Q 1 div u 0. This gives u 3 2 A c ( u ε 1 I Q div u 3 2 0) cε 1 I Q 1 div u 2 0 cε u 3 2,0. On the other hand, we have u 3 2 D c inf u 3 = h 2 ε 1 u i 2 0 cε 1 h 2 u cε 1 u 3 2,0. u i By operator interpoation we finish the proof.

19 Mutigrid methods for a parameter dependent probem in prima variabes 115 Fig. 4. Soution of probem A 7. Numerica resuts Severa versions of the mutigrid method in prima variabes deveoped and anayzed above have been tested numericay. The foowing probems have been investigated within the finite eement code FEPP on a SUN Utra 1 / 166 MHz workstation with 320 MB RAM. Probem A: driven cavity exampe We consider the unit square Ω =(0, 1) 2. The initia trianguation T 1 is given by two trianges, further meshes are obtained by successive refinement. We have used the finite eement space based on P 2 eements. The biinear-form A L (.,.) on the finest eve is defined in (8), where the projection I Q L maps into the piece-wise constant FE-space. The source term is set to f =0. Dirichet boundary conditions are specified as { (1, 0) u L = T at nodes [0, 1] {1}, (0, 0) T at nodes [0, 1] {1}, and incorporated by homogenization of the FE system. A pot of the soution at eve 5 is given in Fig. 4. Probem B: fow through a pipe The geometry and the soution at eve 4 are given in Fig. 5. The boundary is spit into the jacket Γ 1, inet boundary Γ 2 and outet boundary Γ 3. We specify homogeneous Dirichet boundary conditions at Γ 1 and natura boundary conditions esewhere. We sove the finite eement probem find u L V L such that à L (u L,v L )=(g, v L ) 0,Γ2 v L V L. The biinear-form Ã(u, v) is obtained from A(u, v) by repacing the term (e(u),e(v)) 0 by ( u, v) 0. This is done to obtain physicay correct boundary conditions. The boundary stress is defined as g =(0, 1) T. The probem

20 116 J. Schöber Fig. 5. Soution of probem B Fig. 6. Geometry and soution of probem C invoves curved boundary approximation, a non-convex domain and mixed boundary conditions. Probem C: neary incompressibe sub-domains We consider a probem of inear easticity with two incompressibe subdomains. The geometry and the soution at eve 4 are given in Fig. 6. Dirichet boundary conditions are introduced at the bottom, twisting voume forces are appied in Ω 2. The materia data are E = 100,ν =0.3in Ω 1 Ω 2 and E =1,ν =0.499 in Ω 3 Ω 4. At first, we investigate the behavior of the condition number κ(c 1 L A L) in dependence of the number of eves L and the parameter ε. The preconditioner C L is obtained by the appication of a symmetric mutigrid operator, either a W-2-2 cyce or a V-1-1 cyce. In addition to the additive smoother (16), we use the mutipicative counterpart (59) n S A = (I P A i=1,i ), for pre-smoothing and in reversed order for post-smoothing. It does not need damping at a. The numerica resuts for the condition number κ(c 1 L A L) for probem A obtained by the Lanzcos method are given in Tabe 1 for a V-1-1 cyce and in Tabe 2 for a W-2-2 cyce. For the W-2-2 the cacuated

21 Mutigrid methods for a parameter dependent probem in prima variabes 117 Tabe 1. Condition numbers for V-1-1 cyce Unknowns additive smoother mutipicative smoother ε = Tabe 2. Condition numbers for W-2-2 cyce Unknowns additive smoother mutipicative smoother ε = Tabe 3. Iteration numbers and CPU times for probem A, PCG with V-1-1 cyce Leve Unknowns Iterations Time[sec] condition numbers neither depend on the eve nor on the parameter, what is in correspondence with the anaysis provided. We do not have optima estimates for V-cyce convergence rate yet, but the numerica resuts seem to be very promising. Next, we used the V-1-1 mutigrid preconditioner in a preconditioned conjugate gradients sover for the soution of probem A, probem B, and probem C. The sma parameter is set to ε = 10 6 in probem A and probem B. The iteration is terminated after a reduction of the error in energy norm by a factor of The necessary iteration numbers and CPU times are shown in Tabe 3, Tabe 4, and Tabe 5, respectivey.

22 118 J. Schöber Tabe 4. Iteration numbers and CPU times for probem B, PCG with V-1-1 cyce Leve Unknowns Iterations Time[sec] Tabe 5. Iteration numbers and CPU times for probem C, PCG with V-1-1 cyce Leve Unknowns Iterations Time[sec] Finay, I want to express my thanks to D. Braess, U. Langer and Ch. Wieners for many discussions on this topic and for vauabe suggestions to improve the presentation. References 1. Braess, D. (1996): Stabiity of sadde point probems with penaty. M 2 AN 30, Braess, D. (1997): Finite Eements: Theory, Fast Sovers, and Appications in Soid Mechanics. Cambridge University Press 3. Braess, D., Bömer, C. (1990): A mutigrid method for a parameter dependent probem in soid mechanics. Numer. Math. 57, Braess, D., Sarazin, R. (1997): An efficient smoother for the Stokes probem. App. Num. Math. 23, Brambe, J.H. (1993): Mutigrid Methods. Longman Scientific & Technica, Longman House, Essex, Engand 6. Brambe, J.H., Pasciak, J.E., Xu, J. (1991): The anaysis of mutigrid agorithms with non-imbedded or non-inherited quadratic forms. Math. Comp. 55, Brenner, S., Scott, L.R. (1994): The Mathematica Theory of Finite Eement Methods. Springer, Berin, Heideberg, New York 8. Brenner, S.C. (1990): A nonconforming mutigrid method for the stationary Stokes equations. Mathematics of Computations 55(192), Brenner, S.C.(1996): Mutigrid methods for parameter dependent probems. Math. Mod. a. Num. Ana. 30, Brenner, S.C., L.-Sung, Y. (1992): Linear finite eement methods for panar inear easticity. Math. Comp. 59(200),

23 Mutigrid methods for a parameter dependent probem in prima variabes Brezzi, F., Fortin, M. (1991): Mixed and Hybrid Finite Eement Methods. Springer, Berin, Heideberg, New York 12. Fak, R.S. (1991): Nonconforming finite eement methods for the equations of inear easticity. Mathematics of Computation 57, Giraut, V., Raviart, P. (1986): Finite Eement Methods for Navier-Stokes Equations. Springer, Berin, Heideberg, New York 14. Hackbusch, W. (1985): Muti-Grid Methods and Appications. Springer, Berin, Heideberg, New York 15. Huang, Z. (1990): A muti-grid agorithm for mixed probems with penaty. Numer. Math. 57, Lions, J., Peetre, J. (1964): Sur une casse d espaces d interpoation. Institut des Hautes Etudes Scientifique, Pub.Math. 19, Oswad, P. (1994): Mutieve Finite Eement Approximation. Teubner Stuttgart 18. Schöber, J. (1998): Mutigrid Methods for Parameter Dependent Probems I: The Stokes-type Case. In: Hackbusch, W., Wittum, G., editors, Proceedings of the Fifth European Mutigrid Conference Springer (to appear) 19. Scott, L.R., Zhang, S. (1990): Finite eement interpoation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), Turek, S. (1994): Mutigrid techniques for a divergence-free finite eement discretization. East-West J.Numer.Math. 2(3), Verfürth, R. (1984): A mutieve agorithm for mixed probems. SIAM J. Numer. Ana. 21, Xu, J. (1992): Iterative Methods by Space Decomposition and Subspace Correction. SIAM Review 34(4),