Figure 1. Figure. Anticipating te use of two-level domain decomposition preconditioners, we construct a triangulation of in te following way. Let be d

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1 Lower Bounds for Two-Level Additive Scwarz Preconditioners wit Small Overlap * Susanne C. Brenner Department of Matematics University of Sout Carolina Columbia, SC 908 Summary. Lower bounds for te condition numbers of te preconditioned systems are obtained for two-level additive Scwarz preconditioners for bot second order and fourt order problems. Tey sow tat te known upper bounds are sarp in te case of a small overlap. Matematics Subject Classication (1991): 65N55, 65N Introduction Let = (0; 1) (0; 1), V = 1 0 () for te second order model problem and 0 () for te fourt order model problem, and te variational form a(; ) be dened by eiter (1:1) a(v 1 ;v )= for te second order case, or (1:) a(v 1 ;v )= X i;j=1; rv 1 rv dx for te fourt order case. Consider te following variational problem: Find u V suc tat (1:3) a(u; v) = were f L (). 8 v 1 ;v 1 0() (v 1 ) xi x j (v ) xi x j dx 8 v 1 ;v 0 () fvdx 8v V; Te variational problem (1:3) can be discretized using te P 1 conforming nite element (cf. Figure 1) in te second order case and te sie-cloug-tocer macro element (cf. Figure and [11]) in te fourt order case. Te nodal variables of tese elements are depicted in Figure 1 and Figure according to te conventions in [10] and [6]. * Tis work was supported in part by te National Science Foundation under Grant No. DMS

2 Figure 1. Figure. Anticipating te use of two-level domain decomposition preconditioners, we construct a triangulation of in te following way. Let be divided into J = k nonoverlapping squares b 1 ;:::;b J (cf. Figure 3 were k=). By adding a diagonal to eac b j we obtain a triangulation T of (cf. Figure 4). Ten we perform a dyadic subdivision of T to obtain te triangulation T (cf. Figure 5). ere and are te lengts of te orizontal edges in T and T respectively. Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Figure 3. Figure 4. Figure 5. Let V V be te nite element space associated wit T. Te discretization of (1:3) is: Find u V suc tat (1:4) a(u ;v)= fvdx 8v V : Let A : V! V 0 be te linear operator from V to its dual space dened by (1:5) A v 1 ;v i=a(v 1 ;v ) 8v 1 ;v V ; were ; i is te canonical bilinear form between a vector space and its dual. Te operator A is symmetric positive denite (SPD) in te sense tat A v 1 ;v i = A v ;v 1 i for all v 1 ;v V and A v; vi > 0 for 0 6= v V. Note tat if f V 0 is dened by f ;vi= R fvdx for all v V, ten (1:4) can be written as A u = f. Te two-level additive Scwarz preconditioner (cf. [8], [1] and te references terein) for A is constructed as follows. Let b j be enlarged in all directions by te amount = ` (` N) and j be te intersection of tis enlarged square wit (cf. Figure 6).

3 Ω 1 Ω 8 Ω 10 Figure 6. We dene V V to be te nite element space associated wit T, and V j to be te subspace of V wose members vanis identically outside j, for 1 j J. Te SPD operators A : V! V 0 and A j : V j! Vj 0 are dened by (1:6) (1:7) A v 1 ;v i=a(v 1 ;v ) 8v 1 ;v V ; A j v 1 ;v i=a(v 1 ;v ) 8v 1 ;v V j : Te operators I : V! V and I j : V j! V are just natural injections, and we denote by I t : V 0! V 0 and It j : V 0! Vj 0 teir transposes wit respect to te canonical bilinear forms, i.e., (1:8) (1:9) I t ; vi = ; I vi 8 V 0 ;v V ; I t j ; vi = ; I j vi 8 V 0 ;v V j: Te two-level additive Scwarz preconditioner B : V 0! V is dened by (1:10) B = I A 1 It + j=1 I j A 1 j I t j : It is easy to ceck tat BA : V! V is SPD wit respect to te bilinear form A; i = a(; ). It is known (cf. [13], [3]) tat for second order problems (1:11) (BA ) C 1+ ; and for fourt order problems (cf. [4], [3]) (1:1) (BA ) C 1+ 3 ; were te (generic) constant C in (1:11) and (1:1) is independent of,, J and. 3

4 In tis paper we will sow tat for = (minimal overlap) te following estimate olds for te second order model problem (1:13) (BA ) c ; wile te estimate (1:14) (BA ) c olds for te fourt order model problem, were te (generic) positive constant c is independent of,and J. ence, te known upper bounds are sarp in te case of a small overlap for bot second and fourt order problems. We note tat te sarpness of (1:11) as already been remarked upon in [13]. Te rest of te paper is organized as follows. Section contains some lemmas tat are needed in te subsequent sections. We prove te lower bound (1:13) for te second order model problem in Section 3 and te lower bound (1:14) for te fourt order model problem in Section Some Lemmas First we state an abstract result for additive Scwarz preconditioners. Let V and W j, 0 j J, be nite dimensional vector spaces, and A : V! V 0 and B j : W j! W 0 be linear SPD operators. Let te vectors spaces be connected by te linear operators I j : W j! V. Ten te additive Scwarz preconditioner B : V 0! V is dened by B = j=0 I j B 1 j I t j ; were Ij t : V 0! W 0 is te transpose of I j wit respect to te canonical bilinear forms. We ave te following lemma (cf. [17], [19], [0], [1], [4], [14]) on te eigenvalues of BA. Lemma.1. Te operator BA is symmetric positive semi-denite wit respect to A ; i. Te minimum eigenvalue min (BA ) and te maximum eigenvalue max (BA ) of BA ave te following caracterizations: (i) min (BA )= min v V v 6= 0 (ii) max (BA )= max v V v 6= 0 min v=p J j=0 I j w j w j W j min v=p J j=0 I j w j w j W j Av; vi j=0 Av; vi j=0 B j w j ;w j i B j w j ;w j i ; : 4

5 Next we state tree lemmas concerning discrete norms and semi-norms for nite element spaces. Tey can all be easily proved by straigt-forward calculations and standard scaling arguments. Te Sobolev semi-norms in tese lemmas are dened by jvj `(G) = X G jj=` (@ x v) dx 1= ; were G is an open subset of R x 1 x n x n and jj = n. p 3 p 3 m m 1 c p 1 p p 1 m p 3 Figure 7. Figure 8. Lemma.. Let v(x 1 ;x ) be a linear polynomial on an isosceles rigt-angled triangle T wit vertices p 1, p and p 3 (cf. Figure 7). Ten tere exists a positive constant C independent of diam T and v suc tat X j=;3 v(p1 ) v(p j ) Cjvj 1 (T ) : Lemma.3. Let v(x 1 ;x ) be a C 1 function on an isosceles rigt-angled triangle T suc tat v is piecewise cubic wit respect to te triangulation formed by te vertices p i (1 i 3) and te centroid c of T (cf. Figure 8). Let m i,1i3, be te midpoints of te tree sides of T. Ten tere exists a positive constant C independent of diam T and v suc tat X X i=1; j=;3 vxi (p 1 ) v xi (p j ) + v(p1 ) v(p ) jp 1 p j v(p1 ) v(p 3 ) + jp 1 p (m (m ) (T) ; denotes te normal derivative of v in te direction of te outer normal. Lemma.4. Let I be an interval wit endpoints p 1 and p. Let P 1 (I), P 3 (I) be respectively te space of linear and cubic polynomials dened on I. Ten tere exist positive constants C 1 and C independent X ofjij suc tat (i) kvk L (I) C 1jIj v (p i ) 8 v P 1 (I) ; i=1; X (ii) kvk L (I) C 3jIj v (p i )+jij (v 0 ) (p i ) 8 v P 3 (I) : i=1; 5

6 3. Te Second Order Case In tis section we consider te preconditioner B (cf. (1:10)) for te second order model problem, were V = 1 0 (), a(; ) is dened by (1:1), and te P 1 conforming nite element is used. Te overlap is taken to be, i.e., we consider te case of minimal overlap. In order to avoid te proliferation of constants, we will encefort use te notation A < B (or B > A) to represent te statement tat A constant B, were te constant is independent of,,jand te variables in A and B. Te notation A B means tat A < B and A > B. First we apply Lemma.1 to obtain a lower bound for max (BA ). In tis context we ave V =V, W 0 =V, W j =V j for 1 j J, A = A, B 0 = A, B j = A j for 1 j J, I 0 = I, and I j = I j for 1 j J. Lemma 3.1. Te following estimate olds: (3:1) max (BA ) 1 : P J Proof. Let 0 6= v V 1.Weave a trivial decomposition of v : v = v + j=1 v j, were 0=v =v = = v J and v 1 = v. It follows from (1:5){(1:7) and (ii) of Lemma.1 tat max (BA ) a(v ;v )= min a(v ;v )+ a(v j ;v j ) i a(v ;v ) a(v ;v ) =1: v =v + P J j=1 v j v V ;v j V j j=1 By (1:5){(1:7) and (i) of Lemma.1, in order to sow tat min (BA ) < (=), it suces to nd one function v y V suc tat (3:) a(v y ;v y ) < min v y =v + P J j=1 v j v V ;v j V j a(v ;v )+ j=1 i a(v j ;v j ) : We will construct v y as one of te discrete armonic functions associated wit te nonoverlapping decomposition 1 b ;:::;b J (cf. Figure 3). S Let = J j b be te skeleton of te nonoverlapping decomposition. Te subspace V ( n ) of V is dened by V ( n ) = fv V : v vanises on g : Te subspace V ( ) of V is te a(; )-ortogonal complement ofv ( n ), i.e., (3:3) V ( ) = fv V : a(v; w) =0 8wV ( n )g : Te functions in V ( ) are known as discrete armonic functions and tey are completely determined by teir nodal values along. Te property of discrete armonic functions tat we will use is stated in te following lemma, te proof of wic can be found in [] and []. 6

7 Lemma 3.. Te following estimate olds: jvj 1 (b j ) jvj 1= (@b j ) for 1 j J and 8 v V ( ) : Te fractional order Sobolev semi-norm jj in Lemma 3. is dened by 1= (@b j ) jvj = jv(x) v(y)j ds(x) ds(y) ; 1= (@b j j jx yj were ds denotes te dierential of te arc lengt. Let P 1 P be te common boundary of two subdomains b j1 and b j wic is parallel to te x 1 -axis, and Q 1,Q be two points on P 1 P suc tat jp 1 Q 1 j = jp Q j = =4 (cf. Figure 9). Ω j 1 P 1 P Q 1 Q Ω j Figure 9. Te restriction to of te function v y V ( ) tat we are going to construct will vanis outside te line segment Q 1 Q. Lemma 3. and a simple calculation sows tat for suc functions te following lemma olds. Lemma 3.3. Suppose tat v V ( ) and v vanises outside Q 1 Q. Ten we ave jvj 1 () jvj 1= (P 1 P ) ; were jvj 1= (P 1 P ) = P 1 P P 1 P jv(x) v(y)j jx yj dx 1 dy 1 : In view of Lemma 3.3, we can focus our construction to te reference interval I =[0;1]. Let T beadyadic subdivision of I wit mes size and L (I) be te space of continuous piecewise linear functions on I associated wit T. Since te dimension of te subspace fw L 1=8 (I): w= 0 outside (1=4; 3=4)g of L 1=8 (I) is tree, tere exists a nontrivial function ^g wit te following properties: 7

8 (i) ^g L 1=8 (I), (ii) ^g = 0 outside (1=4; 3=4), (iii) 3=4 1=4 ^g(x) dx = 3=4 1=4 x^g(x) dx =0. We denote by te constant (j^gj 1= (I) =j^gk L (I)), wic is of course independent of, and J. Te next lemma follows from te construction on I above and a scaling argument. Lemma 3.4. Tere exists a continuous function g dened on te line segment P 1 P (cf. Figure 9) wic is piecewise linear wit respect to te dyadic subdivision induced by T, for any (=8), and wic as te following properties: (3:4) (3:5) g vanises outside te line segment Q 1 Q (cf. Figure 9) ; Q 1 Q g(x)v(x) dx 1 =0 for any v wic is a linear polynomial on P 1 P ; (3:6) jgj 1= (P 1 P ) kgk L (Q 1 Q ) = : For (=) (1=8), we can now dene v y V ( ) to be te discrete armonic function wic vanises everywere on except te segment P 1 P,were it is identical to te function g in Lemma 3.4. It follows from (1:1), Lemma 3.3, (3:4) and (3:6) tat (3:7) a(v y ;v y )< 1 (v y;v y ) L (Q 1 Q ) : Given any decomposition (3:8) v y = v + were v V j=1 and v j V j for 1 j J, weave, since te overlap is minimal, (3:9) (v y v ) Q1 Q = v j1 Q1 Q + v j Q1 Q : It follows from (3:5) and (3:9) tat (3:10) (v y ;v y ) L (Q 1 Q ) (v y v ;v y v ) L (Q 1 Q ) < kv j 1 k L (Q 1 Q ) + kv j k L (Q 1 Q ) : Let p`, 1`L, be te dyadic subdivision points on Q 1 Q induced by T.Part (i) of Lemma.4 implies tat v j (3:11) kv j1 k L (Q 1 Q ) + kv j k L (Q 1 Q ) < L X`=1 v j 1 (p`)+v j (p`) : Since te overlap is minimal, eac p` belongs to a triangle T` T were v j vanises at all te vertices except p`. Tese triangles appear as te saded triangles in Figure 10. 8

9 Ω j 1 P 1 P (3:1) ence, by Lemma., we ave Similarly weave (3:13) LX `=1 v j (p`) < L X`=1 Q 1 Q Ω j LX `=1 Figure 10. jv j j 1 (T`) jv j j 1 () = a(v j ;v j ): v j 1 (p`) < a(v j1 ;v j1 ): Combining (3:7) and (3:10){(3:13) we nd a(v y ;v y ) < a(vj1 ;v j1 )+a(v j ;v j ) for any decomposition of v y given by (3:8), wic implies (3:) and ence te next lemma. Lemma 3.5. For (=) (1=8) we ave (3:14) min (BA ) < : Finally we can establis te estimate (1:13). Teorem 3.6. Tere exists a positive constant c independent of,and J suc tat (BA ) c olds for te second order model problem in te case of minimal overlap. Proof. For (=) (1=8) te estimate follows (3:1) and (3:14). On te oter and, te estimate follows from te trivial estimate 1 (BA ) wen (=) (1=8). Remark 3.7. It is easy to see tat Teorem 3.6 can be applied to many oter elements and tat te estimate (3:15) (BA ) c is valid under te condition tat (=) is bounded. Note also tat (3:15) can be extended to te second order model problem on te unit cube (0; 1) 3. 9

10 Remark 3.8. Te estimate (3:15) can also be extended to nonconforming nite elements. Remark 3.9. Wen is xed and te overlap, weave(ba ) 1, wic is also te estimate for te condition number of te Scur complement in nonoverlapping domain decomposition algoritms for second order problems (cf. [1], [18], [5]). 4. Te Fourt Order Case We consider in tis section te fourt order model problem were V = 0 (), a(; ) is dened by (1:) and te sie-cloug-tocer macro element is used. Te overlap is again taken to be. Te rst lemma is establised by te same argument in te proof of Lemma 3.1. Lemma 4.1. Te following estimate olds: (4:1) max (BA ) 1 : By (i) of Lemma.1, in order to sow tat min (BA ) < (=) 3, it suces to nd one function v y V suc tat (4:) a(v y ;v y ) < 3 min v y =v + P J j=1 v j We will construct v y v V ;v j V j a(v ;v )+ j=1 i a(v j ;v j ) : as one of te discrete biarmonic functions associated wit te nonoverlapping decomposition b 1 ;:::;b J (cf. Figure 3). Let = SJ b j be te skeleton. Te subspace V ( n ) is dened by V ( n ) = fv V : v vanises to te rst order on g : Te subspace V ( ) is ten dened as in (3:3). Te functions in V ( ) are known as discrete biarmonic functions and tey are completely determined by teir nodal values (i.e., derivatives up to order one at te vertices and normal derivatives at te midpoints (cf. Figure )) along. Te proof of te following property of discrete biarmonic functions can be found in [15], [16] and [7]. Lemma 4.. Te following estimate olds: X jvj jv (b j ) xi j for 1 j J and 8 v V 1= (@b j ) ( ) : i=1; Using Lemma 4. and referring to Figure 9, we obtain te following lemma by a simple calculation. 10

11 Lemma 4.3. Suppose tat v V ( ) and v vanises to te rst order outside Q 1 Q. Ten we ave jvj () X i=1; jv xi j 1= (P 1 P ) : Note tat te restriction of v V to P 1 P is a C 1 function wic is piecewise cubic. In view of Lemma 4.3 we can again focus our construction to te reference interval I =[0;1]. Let T be a dyadic subdivision of I wit mes size and C (I) be te space of C 1 functions wic are piecewise cubic wit respect to T. Since te dimension of te subspace fw C 1=8 (I): w= 0 outside (1=4; 3=4)g of C 1=8 (I) is six, tere exists a nontrivial g C 1=8 (I) wit te following properties: (i) ^g L 1=8 (I), (ii) ^g = 0 outside (1=4; 3=4), (iii) 3=4 1=4 x k^g(x) dx = 0 for k =0;1;;3. We denote by te constant (j^g 0 j 1= (I) =k^gk L (I)), wic is independent of,and J. Te following lemma is obtained by a scaling argument. Lemma 4.4. Tere exists a C 1 function g dened on te line segment P 1 P (cf. Figure 9) wic is piecewise cubic wit respect to te dyadic subdivision induced by T, for any (=8), and wic as te following properties: (4:3) (4:4) (4:5) g vanises outside te line segment Q 1 Q (cf. Figure 9) ; g(x)v(x) dx 1 =0 for any v wic is a cubic polynomial on P 1 P ; Q 1 Q jg x1 j 1= (P 1 P ) 3 = kgk : L (Q 1 Q ) For (=) (1=8) we dene v y V ( ) to be te discrete biarmonic function wic vanises to te rst order everywere on except te segment P 1 P.OnP 1 P it satises te following conditions: (4:6) (4:7) v y P1 P = g; (v y ) x P1 P =0; were g is te function in Lemma 4.4. It follows from (1:), Lemma 4.3, (4.3) and (4:5){ (4:7) tat (4:8) a(v y ;v y ) < 1 3 (v y ;v y ) L (Q 1 Q ) : 11

12 + 3 L X`=1 Given any decomposition of v y dened by (3:8), we ave, by (4:4), (4:9) (v y ;v y ) L (Q 1 Q ) (v y v ;v y v ) L (Q 1 Q ) < kv j 1 k L (Q 1 Q ) + kv j k L (Q 1 Q ) : Let p`, 1`Lbe te dyadic subdivision points on Q 1 Q induced by T. It follows from (ii) of Lemma.4 tat (4:10) kv j1 k L (Q 1 Q ) + kv j k L (Q 1 Q ) < L X`=1 v j 1 (p`)+v j (p`) (vj1 ) x 1 (p`)+(v j ) x 1 (p`) : Since te overlap is minimal, eac p` belongs to a triangle T` T were all te nodal values of v j vanis on te side opposite to p` (cf. Figure 10). ence, by Lemma.3, we ave (4:11) (4:1) LX LX `=1 v j (p`) < L X`=1 `=1(v j ) x 1 (p`) < L X`=1 jv j j (T`) jv j j () = a(v j ;v j ); jv j j (T`) jv j j () = a(v j ;v j ); and similarly, (4:13) (4:14) LX `=1 LX `=1 v j 1 (p`) < a(v j1 ;v j1 ); (v j1 ) x 1 (p`) < a(v j1 ;v j1 ): Combining (4:8){(4:14) we nd a(v y ;v y )< 3 a(vj1 ;v j 1 )+a(v j ;v j ) ; wic implies (4:) and ence te following lemma. Lemma 4.5. For (=) (1=8) we ave (4:15) min (BA ) < 3 : 1

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