Efficient numerical solution of Neumann problems on complicated domains. Received: July 2005 / Revised version: April 2006 Springer-Verlag 2006

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1 CACOO 43, (2006) DOI: /s CACOO Springer-Verag 2006 Efficient numerica soution of Neumann probems on compicated domains Serge Nicaise 1, Stefan A. Sauter 2 1 Macs, Université de Vaenciennes, Vaenciennes, France e-mai: snicaise@univ-vaenciennes.fr 2 Institut für Mathematik, Universität Zürich, Zürich, Switzerand e-mai: stas@math.unizh.ch Received: Juy 2005 / Revised version: Apri 2006 Springer-Verag 2006 Abstract. We consider eiptic partia differentia equations with Neumann boundary conditions on compicated domains. The discretization is performed by composite finite eements. The a priori error anaysis typicay is based on precise knowedge of the reguarity of the soution. However, the constants in the reguarity estimates possiby depend criticay on the geometric detais of the domain and the anaysis of their quantitative infuence is rather invoved. Here, we consider a poyhedra ipschitz domain with a possiby huge number of geometric detais ranging from size O(ε) to O(1). We assume that is a perturbation of a simper ipschitz domain. We prove error estimates where ony the reguarity of the partia differentia equation on is needed aong with bounds on the norm of extension operators which are expicit in appropriate geometric parameters. Since composite finite eements aow a mutiscae discretization of probems on compicated domains, the inear system which arises can be soved by a simpe muti-grid method. We show that this method converges at an optima rate independent of the geometric structure of the probem. 1 Gaerkin discretization of Neumann probems on compicated domains by composite finite eements 1.1 The mode probem et R d be a bounded domain with a ipschitz poyhedra boundary Ɣ. We are interested in appications where has a rough boundary, i.e., the

2 96 S. Nicaise, S.A. Sauter 9 9 Fig. 1. Physica domain and simper domain 0. number of straight segments in the poyhedra boundary is possiby huge. We assume that can be regarded as a perturbation of a simper poyhedra ipschitz domain in the sense that area ( diff) =: ε, where diff := ( \ ) ( \ ), (1) is sma (cf. Fig. 1). Athough the method does not require the existence of such a domain we state the condition area ( diff) is sma here, because the convergence anaysis is strongy based on this fact. For any bounded domain D R d, we define the Soboev space H s (D), s 0, in the usua way (see, e.g., [12] or [17]). Its norm and semi-norm are denoted by H s (D) and H s (D), respectivey. For s R, weset { H s H s (D) s 0, (D) := ( H s (D) ) s<0. If D is equa to, for brevity we write H s instead of H s ( ). The 2 - scaar product is denoted by (, ) 2 (D) and is identified with its continuous extension to the dua pairing, H s (D) H s (D). Suppose that the right-hand side f 2 ( ) is given. Consider the foowing probem. Find u H 1 such that a (u, v) := u, v + uv = fv v H 1. (2) 1.2 Composite finite eement discretization et G := {τ i : 1 i N} be a shape-reguar trianguation in the sense of Ciaret of an overapping domain of. The maxima mesh width is denoted by h G. In the case that ε = O (h G ), the geometric structure can be

3 Efficient numerica soution of Neumann probems on compicated domains 97 resoved by a finite eement mesh in a standard way. In this ight, we assume that the measure for the size of the non-resoved geometric detais satisfies ε C res h 1+θ G (3) for some θ>0. et SG denote the standard finite eement space on, S G := { u C 0 ( ) τ G : u τ P 1 }. The composite finite eement space on the domain is defined as the restriction of SG to, namey, S G := SG := { u : u SG}. The Gaerkin discretization of (2) via composite finite eements is given as foows. Find u G S G such that a (u G,v) = fv v S G. (4) As a basis for the space S G we choose the restrictions ( ϕ z of the )z G standard noda basis eements ϕ z for the space SG. Here and in the seque, G denotes the set of mesh points in G. The basis representation of (4) eads to the inear system Au = f, (5) where the system matrix A R G G and the right-hand side f R G are given by A x,y = a ( ) ϕ y,ϕ x and f x := fϕ x. The soution of (5) is inked to the soution of (4) via u G = z G u z ϕ z, where u = (u z ) z G. The efficient assembing of A and f is expained in [13] and we do not discuss this aspect here.

4 98 S. Nicaise, S.A. Sauter 1.3 Muti-grid methods for Neumann probems on compicated domains The basis representation of the Gaerkin method eads to a system of inear equation of the form Au = f. (6) Typicay, the dimension of A is huge and iterative sovers are empoyed for its soution. Muti-grid methods are among the fastest iterative sovers and we formuate and anayze a muti-grid method for the Neumann probem on compicated domains. For a detaied description of muti-grid methods we refer to [11]. The efficiency of muti-grid methods is based on a mutiscae discretization of the boundary vaue probem. It is a combination of an iterative sover (caed a smoother) on each discretization eve and a recursive coarse grid correction. Formay, we introduce a parameter N with 0 describing the discretization eve. We start with the given fine grid equations (6) and rename them as A u = f, where the number of eves is not known a priori. Anaogousy, we rename the finite eement space S G as S and its basis as ϕ,x, x, where is the set of a mesh points in G. 1.4 Muti-grid agorithm et (G ) =0 be a sequence of finite eement meshes which arise by appying recursivey standard refinement strategies to an initia mesh G 0. Notation: The domain covered by a finite eement mesh G is denoted by. The set of mesh points in G is denoted by. The precise requirements on the mesh sequence (G ) =0 are as foows. 1. Overapping property. a) b) For a 0 and τ G, area (τ ) > Nestedness. For a 0 1 and τ G, there exists a set of sons sons (τ) G +1 such that t τ. t sons(τ)

5 Efficient numerica soution of Neumann probems on compicated domains 99 et ϕ,x, x, denote the standard continuous, piecewise inear agrange basis on G. For any grid function u = (u z ) z R, we associate a finite eement function on the overapping domain by P [u](x) := z u z ϕ,z (x). (7) From the incusion +1 we concude that the function P [u] can be evauated at the grid points +1 of the finer mesh. In this ight, the inter-grid proongation p +1, : R R +1 is defined by and the matrix representation is p +1, [u](x) := P [u](x), x +1, p +1, R +1 : p +1, (x, y) = P [ϕ,y ](x) for a x +1 and y. The restriction is the transpose matrix of p +1,, i.e., r,+1 R +1 : r,+1 (x, y) = p +1, (y, x). Coarse grid operators A are recursivey defined, for <, via the Gaerkin product A := r,+1 A +1 p +1,. (8) In order to define the muti-grid agorithm we have to specify a (cassica) iterative sover on each singe grid. We restrict ourseves here to inear sovers of the form u (i+1) := u (i) N ( A u (i) f ). (9) The appication of ν iterations of the form (9) defines a mapping ( ) S (ν) u (i), f := u (i+ν). The muti-grid agorithm is a recursive procedure which requires as input parameters ν 1, ν 2 N specifying the number of pre- and post-smoothing steps and a parameter γ {1, 2} controing whether a V- or a W-cyce is empoyed (for detais we refer to [11]). The muti-grid agorithm is caed by and defined by u := 0; mg (u, f,) ;

6 100 S. Nicaise, S.A. Sauter procedure mg(u, f,); begin if ( = 0) then u := A 1 f ese begin u := S (ν 1) (u, f ) ; d := A u f ; d 1 := r 1, d ; v 1 := 0; for j := 1 to γ do mg(v 1, d 1, 1) ; u := u p, 1 v 1 ; u := S (ν 2) (u, f ) ; end; end; 2 Convergence anaysis for the Gaerkin discretization In this section, we derive convergence estimates for the Gaerkin soutions. Emphasis is paced on the expicit tracking of constants on the parameters describing the geometry of the domain. For the space dimension d, we assume in the seque that d {1, 2, 3}. 2.1 Anaysis of perturbations in the domain Since the quantitative reguarity of the soution u H 1 of (2) might be very compicated, we compare this soution with the soution of a reated probem on the simper domain (cf. (1)). In this ight, we extend the data f to by zero and denote the resuting function again by f. et û H ( ) 1 denote the unique soution of a 0 (û, v) := û, v + ûv = fv v H ( ) 1. (10) First, we investigate the error e := u û in the H 1 -norm. As prerequisite we discuss the dependence of the norm of extension operators for Soboev spaces on the geometric parameters describing the domain Extension operators. In this section (cf. Def. 7), we define extension operators E k : H k ( ) H k ( ) for any k N so that the supremum sup E k v H k ( )/ v H k ( ) =: Cext k < v H k ( )\{0}

7 Efficient numerica soution of Neumann probems on compicated domains 101 Fig. 2. Compicated domain and ocay cuboid neighborhoods. After two iterations the resuting extended domain is a rectange.? A A Fig. 3. Scaing of oca cuboid neighborhoods. is moderatey bounded for a arge cass of domains, which may contain a huge number of geometric detais. We empoy the extension operator which was deveoped in [26,20] as a refinement of extension operators in [24,18, 16]. The proofs of the theorems in this section can be found in [26,20]. The construction consists of severa steps. 1. The rough boundary (detais of size ε) of the domain is simpified by extending to ocay cuboid neighborhoods Q i of the origina domain. We assume that, after a few iterations, the extended domain contains ony detais of size O(1) (cf. Fig. 2). 2. The extension operator is defined ocay from the intersections Q i to Q i. The diameters of these cubes shoud be of the same order as the size of the underying detais of the origina domain (cf. Fig. 3) 3. These ε-cubes Q i aong with their intersections Q i are scaed to reference cubes Q i and scaed intersections Q i of diameter O(1). We prove that the norm of the extension operator mainy depends on the norm of ( the minima ) extension operator for the reference cubes, i.e., : H k Q i H ( Q ) k i. Note that the minima extension operator E min k E min k : H k (ω) H k ( ) for two domains ω is characterized, for any

8 102 S. Nicaise, S.A. Sauter, /, M B = C N = > Fig. 4. A bounded open set D R 2 with property X. The intersection D ω is the X- boundary Ɣ D. u H k (ω), by the conditions E min k u H k ( ), E min k u ω = u ω, E min k u = min H k ( ) v H k ( ) { v H k ( ) : v ω = u ω }. Definition 1 For ε>0, N N and M>0, the domain R d is of cass (ε, M, N) if there exists a famiy U = (U i ) i N of subsets in R d such that: 1. for a x there exists i N such that B ε (x) U i ; 2. for any x, card {U U : x U} N; 3. for any i, the intersection U i is ocay the graph of a ipschitz curve with ipschitz constant ess than or equa to M. Definition 2 A bounded open set D R d has property X (cf. Fig. 4) if there d is a cuboid Q = (a i,b i ) R d (with edges parae to the axes) such i=1 that the set ω := Q\D contains one fu side of the cube Q, i.e., there is 1 i d and r { a i,b i } such that i 1 (a i,b i ) {r} i=1 d i=i +1 (a i,b i ) ω. The boundary Ɣ D := D ω is the X-boundary of D. A pair (ω,u) of bounded domains of R d with ω U is admissibe for extension if 1. ω is a ipschitz domain, 2. ω c := U\ω has property X, 3. Ɣ ω c ( ω) ( ω c ), where Ɣ ω c is the X-boundary of ω c.

9 Efficient numerica soution of Neumann probems on compicated domains 103 Theorem 3 et k N and et ω R d be of cass (ε, M, N). Suppose that (ω,u) is admissibe for extension. Then there exists an extension operator E k : H k (ω) H k (U) which satisfies E k f H k (U) C 2 (1 + C 4 ) f H k (ω) f H k (ω), E k f H k (U) C 3 (1 + C 4 ) f H k (ω) f H k (ω). The constants C 2 and C 3 ony depend on ε, M, N and diam ω c, where ω c := U\ω. Further, C 4 is the constant appearing in the so-caed Poincaré inequaity, i.e., f H ( k ω c) : f 2 H k (ω c ) C 4 f 2 H k (ω c ) + C 2 4 D α f. α <k The essentia observation for extension on compicated domains is that the constants in Theorem 3 remain unchanged if ω is scaed. In this ight, et ω R d be a subset with positive diameter diam ω>0. We define the scaing operator χ ω : R d R d by χ ω (x) := x diam ω and ˆω := {χ ω (x) : x ω}. Theorem 4 et k N and et ω U R d. Set Û = {χ ω (x) : x( U} ). Assume that the normaized domain ˆω is of cass (ε, M, N) and that ˆω, Û is admissibe for extension. et Ê k : H k ( ω) H ( k Û ) be the extension operator as in Theorem 3. Then the operator E k := H k (ω) H k (U) defined by E k f = ( ( )) Ê k f χ 1 ω χω (11) is an extension operator with E k H k (U) H k (ω) Ĉ 2 ( 1 + Ĉ4 )( 1 + diam k ω ), where Ĉ 2, Ĉ 4 are the constants as in Theorem 3 for the operator Ê k on the normaized domains ˆω, Û. Now we compose the goba extension operator from to an overapping (simper) domain. Definition 5 et be a bounded domain. A finite famiy of disjoint cubes Q = {Q i : 1 i q} with edges parae to the axes is admissibe for extension for the domain if, for a 1 i q, the pairs (Q i,q i ) are admissibe for extension. The oca extension operator E oc k,i : H k (Q i ) H k (Q i ) is as in (11) where ω is repaced by Q i and U by Q i. et

10 104 S. Nicaise, S.A. Sauter Q := q i=1 Q i. The singe stage goba extension operator E ss H k ( ) H k ( Q ) is ( E ss k f ) { ( ) E oc (x) := k,i f Qi (x) if x Qi, f (x) otherwise. Theorem 6 et be a bounded domain and suppose that k : Q = {Q i : 1 i q} (12) is admissibe for extension for. The extension operator E ss k : H k ( ) H ( ) k Q as in Definition 5 is bounded as foows: E ss H k ( 1 + max E oc 2 Q) H k ( ) H k (Q i ) H k ( Q =: C i ) Q. k 1 i q Finay, we may aow finite iterations of the extension to famiies of cubes. Definition 7 et be a bounded domain and et Q = (Q i : 1 i p) be a finite sequence of famiies of axes parae cubes. Recursivey, we put 0 := and, for 1 i p, i = i 1 Q. Q Q i If, for a 1 i p, the famiy Q i is admissibe for extension for the domain i 1, we say that Q is admissibe for extension from to p. et E ss k,i : H k ( i 1 ) H k ( i ) be constructed as in Definition 5. Then the goba extension operator E k : H k ( 0 ) H ( ) k p is the composition E k := E ss k,p Ess k,p 1... Ess k,1. Theorem 8 et be a bounded domain and suppose that Q = {Qi : 1 i p} (13) is admissibe for extension from to p. The extension operator E k : H k ( ) H k ( p ) as in Definition 5 is bounded as foows: E k H k ( Q) H k ( ) k p C Qi. In summary, we have shown that the extension operator E k from a domain to a domain p which is the iterated cuboid extension of, is independent of 1. the number q of geometric detais (cf. (12)), 2. the size diam Q i of the geometric detais (cf. (12)), i=1

11 Efficient numerica soution of Neumann probems on compicated domains 105 but depends on ( ) 1. the norm of the oca extension operator Ê k : H k Q i H ( Q ) k i on the normaized domains Q i and Q i, 2. the number p of iterations in the extension process (cf. (13)) Bounds on the perturbation error. et be the given (ipschitz) domain and the simpified domain as expained in Sect. 1. To reduce technicaities, we assume that. et u (resp. û) denote the soution of (2) (resp. (10)) and e := u û. et E min 1 : H 1 ( ) H ( ) 1 be the minima extension operator. We consider Eq. (10) and empoy test functions of the form E min 1 v H ( ) 1 for any v H 1 ( ). Subtracting this equation from (2) yieds a (e, v) \ for a v H 1 ( ). Choosing v = e,weget ( û, E min 1 v + ûe min 1 v ) = 0 e 2 H 1 ( ) = ( û, E min 1 e ) H 1 ( \ ). The Cauchy-Schwarz inequaity and the boundedness of E min 1 ead to e 2 H 1 ( ) C1 ext û H 1 ( \ ) e H 1 ( ). (14) Hence, the perturbation error e H 1 ( ) can be estimated by the norm of the soution û in the sma strip \. This norm is estimated in emma 10 under the weak assumption that \ S ε, where S ε is a strip of width O(ε) aong the boundary of, i.e., S ε := {x : dist(x, )<c w ε} (15) for some c w > 0. Furthermore, we assume a minima reguarity for the homogenous Neumann probem. Assumption 9 (H λ -reguarity) There exists λ ] 3 2, 2] and C 1 > 0 such that, for any µ [1,λ] and f H µ 2 ( ), the soution û to (10) satisfies û H µ ( ) C 1 f H µ 2 ( ).

12 106 S. Nicaise, S.A. Sauter emma 10 Assume that the Neumann probem on is H ( ) λ -reguar for some λ ] 3 2, 2] (cf. Assumption 9). Then, for any 3 <δ µ λ and 2 f H ( ) µ 2, ( ε ) û H 1 ( \ ) C f H δ 2 ( ) + εµ 1 f H µ 2 ( ). (16) The constant C depends continuousy on C 1, δ, µ, λ, and c w and, possiby, tends to infinity as δ, µ, λ 3/2. Proof Part I: By using a finite system of oca charts and changes of variabe, we can ocaize and rescae the estimate, so that it is sufficient to consider the case of a hypercube = (0, 1) d = B (0, 1), where B = (0, 1) d 1 and the boundary is reduced to γ := B {0}. In this case, the above-mentioned transformation of the strip S ε is contained in S ε = B (0,C I ε) for some C I > 0 which ony depends on and c w (cf. (15)). The transformed function û is denoted by ũ. First, we show that there exists C>0such that, for a 1/2 <κ s 1, ( ε ) v 2 ( S ε) C v H κ ( ) + εs v H s ( ) v H ( ) s. (17) We prove (17) by using finite eement approximation theory on an auxiiary mesh. In order to avoid technicaities we assume that (C I ε) 1 N. This aows us to define a conforming, uniform, simpicia mesh G aux, where a trianges are transations and rotations of the simpex { } d x (R >0 ) d : x i <C I ε. Further, we may assume that there is a subset G ε G aux which defines a partitioning of S ε. The auxiiary finite eement space S aux is given by S aux := { u C ( ) } 0 τ G aux : u τ P 1. Now et 1/2 <s 1 and v H ( ) s. Then, it is we-known (see [5, 21 and 22, Section 4.8]) that there is a projection operator P : H ( ) s S aux such that v Pv H r ( S ε) v Pv H r ( ) C IIε t r v H t ( ) (18) i=1

13 Efficient numerica soution of Neumann probems on compicated domains 107 for a 0 r t s. The constant C II depends ony on s (since G aux is a uniform grid, no mesh parameters enter in the approximation error estimates (18)). We concude that v 2 ( S ε) v Pv 2 ( S ε) + Pv 2 ( S ε) C II ε s v H s ( ) + Pv 2 ( S ε). (19) We prove in Part III that, for any 1/2 <κ s, ( ε ) Pv 2 ( S ε) C III Pv Hκ ( ) + εs Pv H s ( ), (20) where C III ony depends (continuousy) on s, κ, and C II and possiby deteriorates if s 1/2 orκ 1/2. From (18) we concude for k {κ, s} that Pv H k ( ) Pv v H k ( ) + v H k ( ) (C II + 1) v H k ( ). (21) The combination of (19) (21) yieds ( ε ) v 2 ( S ε) C IV v H κ ( ) + εs v H s ( ), (22) where C IV ony depends on C II and C III. Part II: Next, we derive (16) from (22). Appying estimate (22) to i ũ H ( ) µ 1 for a i = 1,,d,we obtain ( ε ) ũ H 1 ( S ε ) C V ũ H δ ( ) + εµ 1 ũ H µ ( ) for a 3/2 <δ µ; here C V ony depends on C IV. The concusion foows by the estimates ũ H δ ( ) C 1 f H δ 2 ( ) and ũ H µ ( ) C 1 f H µ 2 ( ). Part III: In this part, we estabish (20). et w S aux and note that w H 1 ( ). We empoy the representation w ( ) ( y,y d = w y, 0 ) yd + 0 w ( y,t ) t The Cauchy-Schwarz inequaity eads to dt w ( ) 2 y,y ( d 2w 2 y, 0 ) CI ε + 2C I ε 0 y = ( y,y d ) B (0,CI ε). ( ( w y,t ) ) 2 dt. t

14 108 S. Nicaise, S.A. Sauter Integrating over S ε shows that w 2 2 ( S ε) 2ε w 2 2 (γ ) + 2 (C Iε) 2 w 2 2 ( S ε). (23) Since the trace operator γ 0 : H ( ) κ 2 (γ ) is continuous for any κ>1/2, we obtain { ε } w 2 ( S ε) C w H κ ( ) + ε w 2 ( S ε), (24) where C ony depends on C I and κ and may deteriorate as κ 1/2. Finay, we empoy an inverse inequaity for the (uniform) mesh G aux (cf. [5]) to obtain w 2 ( S ε) Cεs 1 w H s ( S ε) (25) for any s [0, 1]. The combination of (24) and (25) yieds the proof of Part III. Theorem 11 Suppose that the assumptions in emma 10 hod. Then, for any 3 2 <δ µ λ and f ( ) Hµ 2, ( ε ) e H 1 ( ) C 0 f H δ 2 ( ) + εµ 1 f H µ 2 ( ) (26) with C 0 := C 1 ext C and C is as in emma 10. Proof Combine (14) with (16). 2.2 Error anaysis of the Gaerkin soution The convergence anaysis for composite finite eement discretization is based on the quasi-optimaity of the Gaerkin discretization and transformed as usua to an estimate of the interpoation error. However, since the intersections τ are neither shape-reguar nor affine equivaent to a reference eement, we cannot appy the standard interpoation estimates straightforwardy but have to empoy an extension operator first. First, we introduce the constants which appear in the error estimates. Reguarity on. We aways assume that the Neumann probem on is H ( ) λ -reguar for some λ ] 3 2, 2] (cf. Assumption 9). Bounds for the minima extension operator. Reca that is the domain which is covered by the overapping finite eement mesh. et E min 2 : H ( ) 2 H 2 ( ) denote the minima extension operator with norm C I ext := sup { E min 2 v H 2 ( ) : v H 2 ( ) v H 2 ( ) = 1 } <.

15 Efficient numerica soution of Neumann probems on compicated domains 109 Note that Soboev s embedding theorem impies that, for d = 1, 2, 3, E min 2 : H ( ) 2 C ( 0 ) and Cext II := E min 2 C 0 ( ) H 2 ( ) <. We put C ext := max { Cext ext} I,CII. Bounds for the interpoation error. et be the overapping domain which is covered by the finite eement mesh G. Since the embedding H 2 ( ) C 0 ( ) is continuous, the noda interpoation I G : H 2 ( ) SG is we defined. It is we-known that, for m = 0, 1, the constants C apx,m ony depend on the minima anges in the mesh G, C apx,m := h m 2 G We put C apx := max { C apx,0, C apx,1 }. sup v I G v H m( ). v H 2 ( ) v H 2 ( ) =1 Theorem 12 Suppose that Assumption 9 hods. et f H ( ) λ 2. Then, for any 3/2 <δ λ, the Gaerkin soution u G (cf. (4)) satisfies the error estimate u u G H 1 ( ) C 0 ε f H δ 2 ( ) + ( C 0 ε λ 1 + C 5 h λ 1 ) G f H λ 2 ( ). (27) The constant C 5 ony depends on C 1, C ext and C apx. Proof The continuity and eipticity constants for the Neumann probem on are 1, i.e., a (u, v) u H 1 ( ) v H 1 ( ) u, v H 1 ( ), a (u, u) = u 2 H 1 ( ). et û denote the soution of the extended probem (10). Hence, the quasioptimaity of the Gaerkin discretization and Theorem 11 yied u u G H 1 ( ) = inf u v H v S 1 ( ) u û H 1 ( )+ inf û v H G v S 1 ( ) G ( ε ) C 0 f H δ 2 ( ) + ελ 1 f H λ 2 ( ) + inf û v H v S 1 ( ). G It remains to estimate the infimum in the estimate above. First, we assume that the Neumann probem (10) on is H 2 -reguar. The infimum can be estimated by introducing v int = ( I G E min 2 û ).

16 110 S. Nicaise, S.A. Sauter This eads to the estimate inf û v H v S 1 ( ) û v int H 1 ( ) E min 2 û I G E min 2 û H 1 ( ) G C apx h G E min 2 û H 2 ( ) C extc apx h G û H 2 ( ) C ext C apx C 1 h G f 2 ( ). The resut for intermediate Soboev spaces H λ ( ), λ ] 3 2, 2], foows by interpoation appied to the operator u := u P(u), where P(u) is the H 1 -orthogona projection of u onto S G. Coroary 13 Suppose that the assumptions of Theorem 12 hod and that (3) hods. Then, for any 3/2 <δ λ, the Gaerkin soution u G (cf. (4)) satisfies the error estimate u u G H 1 ( ) C 0 ε f H δ 2 ( ) + C 6 h λ 1 G f H λ 2 ( ). The constant C 6 ony depends on C 5 and C res (cf. (3)). The Aubin-Nitsche duaity argument makes it possibe to obtain error estimates with respect to weaker norms. In this ight, for any given ϕ H 1, we define the function v ϕ H 1 as the unique soution of a ( v ϕ,w ) = (ϕ,w) 2 ( ) w H 1. Coroary 14 Suppose that the assumptions of Theorem 12 hods and that (3) hods. et 3/2 <δ µ λ and 3 <δ s λ. Then the Gaerkin 2 soution u G (cf. (4)) satisfies the error estimate ( u u G H 2 µ ( ) ) C 0 ε + C6 h µ 1 G ( C0 ε f H δ 2 + C 6 h s 1 G f H s 2). Proof et e = u u G. By duaity, for 3/2 <µ λ, wehave e H 2 µ ( ) = (e, ϕ) sup 2 ( ) = sup ϕ H 2+µ \{0} ϕ H 2+µ ( ) e, vϕ ṽ ϕ = sup a inf ϕ H 2+µ \{0} ṽ ϕ S G e H 1 ( ) ϕ H 2+µ \{0} a ( e, v ϕ ) ϕ H 2+µ ϕ H 2+µ v ϕ ṽ H ϕ sup inf 1 ( ). ϕ H 2+µ ϕ H 2+µ \{0} ṽ ϕ S G

17 Efficient numerica soution of Neumann probems on compicated domains 111 By choosing ṽ ϕ as the Gaerkin approximation of v ϕ, we may appy Coroary 13 twice to obtain ( ) e H 2 µ ( ) C 0 ε + C6 h µ 1 G ( C0 ε f H δ 2 + C 6 h s 1 G f H s 2). Coroary 15 Suppose that Assumption 9 hods. et f H ( ) 1. Assume that (3) hods. et 1 µ λ. Then the Gaerkin soution u G (cf. (4)) satisfies the error estimate ( u u G H 2 µ ( ) 2 ) C 0 ε + C6 h µ 1 f H 1 ( ). The constant C ony depends on C 0, C 1, C ext and C apx. Proof et e = u u G. As in the proof of Coroary 14 one shows that ( ) e H 2 µ ( ) C 0 ε + C6 h µ 1 G e H 1 ( ). The eipticity of the biinear form a (, ) impies that e H 1 ( ) u H 1 ( ) + u G H 1 ( ) 2 f H 1 ( ). G 3 Muti-grid convergence In this section, we investigate the convergence of the muti-grid method foowing the genera muti-grid convergence theory from [11]. However, the proofs require an approximation property for finite eement spaces which might depend on the geometric detais in a compicated way. Here, we prove the approximation property by combining the perturbation estimates with reguarity estimates on the simpified domain. Since our focus in this paper is more on the approximation property of composite finite eements and ess on the choice of an optima smoother we restrict our attention here to a damped Jacobi-type method as the smoothing iteration. For the system of inear equations A u = f, this is given by where u (i+1) = u (i) N 1 N 1 and M denotes the mass matrix (M ) x,y := ( ϕ x,,ϕ y, ) ( A u (i) f ), (28) := ωh 2 M 1 (29) 2 ( ) x,y. The parameter ω>0 is a suitabe damping parameter. We prove convergence in the framework of geometric muti-grid methods (cf. [11]).

18 112 S. Nicaise, S.A. Sauter Remark 16 We have chosen N as in (29) in order to simpify the anaysis of the smoothing property as much as possibe and to focus on the approximation property. For the practica reaization one has to sove in each iteration step a inear system of the form M x = y. (30) Aspects of this are discussed beow. 1. For the numerica experiments, we have aways repaced N by the diagona part of A, i.e., N := ω diag [ ] (A ) x,x : x and obtained convergence rates which are independent of the geometric detais in the domain (see Sect. 4). 2. In standard cases, the condition number of the mass matrix is of order 1 and the soution of (30) requires ony a sma number of iteration steps which, in particuar, is independent of dim M. This can be proved as ong as the areas of the intersections ( ) supp ϕ x, are of order h d. 3. The case that ( ) supp ϕ x, is degenerate (i.e., much smaer than h d ) for some x, is anayzed in [23] for a mode probem and it was shown that the muti-grid convergence is not affected by such scaing effects. The numerica soution of boundary vaue probems on compicated domains is a topic of active research. Our approach differs from such techniques as agebraic muti-grid methods ([19, 15, 4, 25]), aggomeration methods ([1, 3, 6, 2, 9, 7]), or subspace correction methods ([14, 27, 28]) since our construction is based on the coarse scae discretization of the boundary vaue probem where the asymptotic convergence order is preserved on coarser grids. Hence, it can be used not ony for constructing a spectra equivaent preconditioner for the fine scae equations but aso for a ow dimensiona discretization of the partia differentia equation for a given prescribed (moderate) accuracy. 3.1 Smoothing and approximation property We start by considering the two-grid method. The iteration can be written as an affine map in the form u (i+1) = K TGM u (i) + RTGM f with the two-grid iteration matrix K TGM := K ν ( 2 A 1 p, 1A 1 1 r ) 1, A K ν 1 and the iteration matrix K := I N 1 A of the inear sover (9). For the Jacobi-type smoother (cf. (28)), K = I ωh 2 M 1 A.

19 Efficient numerica soution of Neumann probems on compicated domains 113 The iteration converges if and ony if the spectra radius ρ ( ) K TGM is smaer than one. The convergence proof is based on a mutipicative spitting of K TGM and an estimate of the factors in appropriate norms. In this ight, we introduce, for α [ 1, 1], a scae of norms α, : R R. et : S S be defined by ( u, v) 2 ( ) = a (u, v) u, v S. Remark 17 The operator can be expressed by means of the system matrix A, the mass matrix M and the proongation P as = P M 1 A P 1. It is easy to see that is sef-adjoint with respect to the 2 ( )-scaar product and satisfies ( u, u) 2 ( ) = u 2 H 1 ( ) u S. Hence, powers of are we-defined for any rea α R. The operator aows us to define a scae of norms on S.Forα [ 1, 1], weset (u, v) α, := ( α u, v) 2 ( ) and u α, := (u, u) 1/2 α,. Remark 18 Note that, for α = 0, 1 and u S,wehave u 0, = u 2 ( ) and u 1, = u H 1 ( ). The discrete counterparts of the scaar product (, ) α, and of the norm α, are given by u, v α, := (P u,p v) α, and u α, := u, u 1/2 α,. Throughout this section we assume that the Neumann probem on is H λ -reguar for some λ ]3/2, 2]. We estabish the muti-grid convergence with respect to the s 2, -norm, where { s := min λ, θ } with θ as in (3). (31) 2

20 114 S. Nicaise, S.A. Sauter The proof of the foowing emma requires an inverse assumption. Since we assume that the difference diff (cf. (1)) has sma measure ε<c res h 1+θ (cf. (3)), the constant C inv, > 0in C inv, := h sup u H 1 ( ) u S \{0} u 2 ( ) (32) shoud have a moderate size (cf. [8, Coroary 1]). emma 19 Suppose that the assumptions of Coroary 15 hod and that (3) hods for some θ>0and et C inv, be bounded independent of the refinement eve. Then there exists C P > 0, independent of the refinement eves, such that P f sup H s 2 C P. (33) f R \{0} f s 2, Proof et R : S R be the adjoint to P so that R u, v 0, = (u, P v) 2 ( ) u S v R. (34) Then, R P = I is the identity matrix and Pˆ := P (R P ) 1 equas P. Hence, the proof foows by using emmas 25 and 26 beow and appying [11, emma (ii)]. Coroary 20 Suppose that the assumptions of emma 19 hod. Then P 1 v 2 s, sup C P. (35) v S \{0} v H 2 s Proof First note that P 1 u = P u u S. Consider the Banach spaces S µ := ( ) S, µ, and H µ := ( ) R, µ,. Then, the eft-hand side in (35) is the operator norm of the adjoint operator P 1 H 2 s S 2 s = P H 2 s S 2 s = P S s 2 H s 2. Since the norm on the right-hand side equas the eft-hand side in (33), the assertion foows. The convergence proof for the two-grid method is spit into the smoothing property M 1 A K ν s 2, 2 s, C S h 2 2s η (ν) (36) with η (ν) 0asν, and the approximation property ( A 1 p, 1A 1 1 r ) 1, M 2 s, s 2, C A h 2s 2.

21 Efficient numerica soution of Neumann probems on compicated domains 115 Theorem 21 If the parameter ω is sma enough so that ωh 2 ρ( )<1, then the smoothing property hods: M 1 A K ν s 2, 2 s, C S h 2 2s η (ν). Proof The definition of the norms impies that M 1 A K ν s 2, 2 s, = = (s 2)/2 s/2 P M 1 A K ν P 1 (s 2)/2. 0, 0, P K ν P 1 Powers of the iteration matrix can be written in the form P K ν P 1 and, hence, M 1 A K ν s 2, 2 s, = s/2 = ( I ωh 2 ) ν ( I ωh 2 ) ν (s 2)/2 (s 2)/2 0, 0, 0, 0, where η 0 (ν) = = ( ) ωh 2 1 s ( ) ωh 2 s 1 ( ) I ωh 2 ν 0, 0, ( ( )) ν s 1 ω 1 s h 2 2s η 0, s 1 ν ν (ν + 1) ν+1 = 1 e ν + O ( ν 2). Hence, the choice of s as in (31) impies that η (ν) := ω ( ( 1 s η ν 0 tends to zero as ν. s 1 )) s 1 For the approximation property, we empoy the theory in [11, Chapter ]. We assume that there is a constant C s such that h 1 C s h 1. (37) Theorem 22 Suppose that the assumptions of Coroary 14, emma 19, and (37) hod. Then, for any 3/2 <δ s, ( A 1 p, 1A 1 1 r 1,) M f 2 s, C ( ε + h s 1 )( ε f δ 2, + h s 1 f s 2,), where C ony depends on C 0, C 6 and C s (as in (37)).

22 116 S. Nicaise, S.A. Sauter Proof et f R and define f S by f := x α,x f x with α := M 1 f = (α,x) x. et u (resp. u 1 ) denote the composite finite eement soution to probem (4) with S G and f being repaced by S and f (resp. by S 1 and f ). The exact soution in H 1 ( ) for the right-hand side f is denoted by u. Then ( A 1 Coroary 20 impies that p, 1A 1 1 r 1, sup v S \{0} P 1 ) f = P 1 (u u 1 ). v 2 s, v H 2 s ( ) This estimate and the triange inequaity yied ( A 1 p, 1A 1 1 r 1,) f 2 s, C P. C P ( u u H 2 s ( ) + u 1 u H 2 s ( )). From the convergence estimates for the Gaerkin soution and the reguarity properties (cf. Coroary 14), we deduce that u u H 2 s ( ) ( C 0 ε + C6 h s 1 )( C0 ε f H δ 2 + C 6 h s 1 f ) H s 2 = ( C 0 ε + C6 h s 1 )( ( ) C0 ε P M 1 f H δ 2 + C 6 h s 1 ( ) P M 1 f H s 2) C ( ε + h s 1 ) ( ε M 1 f δ 2, + h s 1 M 1 f ) s 2,, where C depends on C 0, C 6 and C P. The estimate of the difference u 1 u is competey anaogous with h being repaced by h 1. However, the compatibiity of consecutive step widths (cf (37)) eads to ( A 1 p, 1A 1 1 r 1,) f 2 s, C ( ε + h s 1 ) ( ε M 1 f δ 2, + h s 1 M 1 f ) s 2,. Substituting f by M f yieds the assertion. Coroary 23 Suppose that the assumptions of Theorem 22 hod and that (3) hods. Then the approximation property hods: ( A 1 p, 1A 1 1 r ) 1, M f 2 s, Ch 2s 2 f s 2,.

23 Efficient numerica soution of Neumann probems on compicated domains 117 Theorem 24 Suppose that the assumptions of Theorems 21 and 22 hod. Then the norm of the two-grid operator can be estimated by s 2, s 2, Cη (ν), K TGM where the function η (ν) 0 is independent of h and tends to zero as ν. Since estimate [11, (7.1.2)] hods with C p = C p = 1 and [11, (7.1.1)] foows from Theorem 21, the convergence of the W-cyce is impied by [11, Theorem 7.1.2]. In summary, we have proved that the muti-grid method on compicated domains converges robusty with respect to the area measure 0 <ε Ch 1+θ under very weak geometric assumptions on the domains. 4 Numerica experiments We performed numerica experiments to study the convergence behavior of the muti-grid method based on composite finite eements for a Neumann probem on the compicated domain of the Batic Sea (cf. Fig. 5). We empoyed thev-cyce muti-grid agorithm with two symmetric Gauss- Seide smoothing steps. The stopping criterion is A u (i) f 10 8.In Tabe 1 we dispay the number of iterations of our muti-grid agorithm as a function of the eves. Since these numbers are independent of the eve, the convergence rates are sma and independent of the refinement eve. Fig. 5. is the two-dimensiona surface of the Batic Sea.

24 118 S. Nicaise, S.A. Sauter asty, we dispay the mesh sequence which shows that the coarsest mesh has ony 9 degrees of freedom. The overaps of trianges with the domain are of rather genera shape and neither quasi-uniform or shape-reguar. Note that the underying mesh which resoves is ony used for numerica integration and is not reated to the degrees of freedom. A Proof of norm equivaences In this section, we prove (33). We empoy [11, emma (ii)]. In this ight, we prove that [11, emma ] aso hods in our setting. Reca the definitions of the finite eement proongation P : R S H 1 as in (7) and its adjoint R : H 1 R (cf. (34)). et := A and define the operator M 1 X := I Q with Q := P 1 R. (38) emma 25 et Q be defined as in (38). Then Q H 1 H 1 1. Tabe 1. Number of iterations for the muti-grid agorithm eve mg-iteration

25 Efficient numerica soution of Neumann probems on compicated domains 119 Proof The continuity constant of the eiptic boundary vaue probem equas 1 and, hence, sup f R \{0} H 1 H 1 = 1. The definition of the norm 1, impies that P u sup H 1 R f 1, = sup = 1. u R \{0} u 1, f H 1 \{0} f H 1 Finay, the eipticity constant for the biinear form a (, ) equas 1 and this eads to (see [12, emma 6.5.3]) 1 f 1, 1. f 1, emma 26 Suppose that the assumptions of Coroary 15 hod and that (3) hods for some θ>0. Then, for any 1 µ min { } λ, 3+θ 2 and any u H 1, (I Q ) u H 2 µ H 1 2 (C 0 + C 6 ) h µ 1 u H 1. (39) Proof Note that I Q = ( 1 P 1 R ). For u H 1, by using Coroary 14 and the reguarity of the boundary vaue probem, we obtain ( 1 P 1 R ) u H 2 µ 2 ( C 0 ε + C6 h µ 1) u H 1 2 ( C 0 ε + C6 h µ 1) u H 1. Condition (3) impies the assertion. Acknowedgements. Thanks are due to Profs. Monique Dauge and Wofgang Hackbusch for fruitfu discussions. References [1] Bank, R., Smith, R.: An agebraic mutieve mutigraph agorithm. SIAM J. Sci. Comput. 23, (2002) [2] Bank, R., Xu, J.: A hierarchica basis mutigrid method for unstructured grids. In: Hackbusch, W., Wittum, G. (eds.): Fast sovers for fow probems. Braunschweig: Vieweg 1995, pp [3] Bank, R., Xu, J.: An agorithm for coarsening unstructured meshes. Numer. Math. 73, 1 36 (1996) [4] Braess, D.: Towards agebraic mutigrid for eiptic probems of second order. Computing 55, (1995)

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