Explct Extenson Operators on Herarchcal Grds G. Haase y S.V. Nepomnyaschh z Abstract Extenson operators extend functons dened on the boundary of a dom

Transcription

1 JOHANNES KEPLER UNIVERSIT AT LINZ Explct Extenson Operators on Herarchcal Grds Gundolf Haase Department of Mathematcs Johannes Kepler Unversty, A{44 Lnz, Austra Sergej V. Nepomnyaschh omputng enter Sberan Branch of Russan Academy of Scences Novosbrs, 639, Russa Insttutsbercht Nr. 524 June 997 INSTITUT F UR MATHEMATIK A{44 LINZ, ALTENBERGERSTRASSE 69, AUSTRIA

2 Explct Extenson Operators on Herarchcal Grds G. Haase y S.V. Nepomnyaschh z Abstract Extenson operators extend functons dened on the boundary of a doman nto ts nteror. Ths paper presents explct extenson operators by means of multlevel decompostons on herarchcal grds. It s shown that the norm-preservng property of these operators holds for the 2D as well for the 3D case wth constants ndependent on dscretzaton and doman sze. These constants can be further mproved by an addtonal teraton scheme appled to the extenson operator. Some mplementaton of these technques s presented for a doman decomposton precondtoner and numercal experments are gven. Keywords : Boundary value problems, trace theory, multlevel methods, doman decomposton, precondtonng, nte element method. Introducton The purpose of ths paper s to dscuss the constructon of norm-preservng explct extenson operators of functons at a boundary nto the nteror of the doman. The theorems on traces of functons from Sobolev spaces play an mportant role n studyng boundary value problems of mathematcal physcs [2, 24, 3]. These theorems are commonly used for dervng a pror estmates of the stablty wth respect to boundary condtons. For the case The second author was apponted guest professor at the Faculty for Techncal and Natural Scences of the Unversty of Lnz for the perod from March untl June 997. Ths wor was partally supported by the Russan Basc Research Foundaton (RBRF) under the grant y Johannes Kepler Unversty Lnz, Inst. of Math., Altenberger Str. 69, A{44 Lnz, Austra, ghaase@numa.un-lnz.ac.at z omputng enter, Sberan Branch of Russan Academy of Scences, Novosbrs, 639, Russa, svnep@oapmg.sscc.ru

3 of grd functons the rst constructve analyss of ths problem seems to be carred out n [], where the case of rectangular grds was consdered. For a numercal soluton of non-homogeneous Drchlet problems t s mportant to have a good extenson of the Drchlet condtons nsde the doman. If the trval extenson,.e. extenson by zero at nteror nodes of the grd, s used then nstead of the numercal soluton of a boundary value problem wth a smooth soluton, we have to compute a soluton wth a huge gradent n the vcnty of the boundary. The use of norm-preservng explct extenson operators gves a "good" ntal guess for teraton processes and reduces the boundary value problems wth non-homogeneous Drchlet condtons to the boundary value problems wth homogeneous Drchlet condtons. The soluton of the last s unformly bounded wth respect to the grd sze. An other mportant applcaton of the explct extenson operators s connected wth doman decomposton methods [7, 8, 2]. Usng the explct extenson operators n doman decomposton methods, nstead of exact solvers n the subdomans, local precondtonng operators can be utlzed. Usng ths operators, optmal estmates of the convergence rate of the doman decomposton methods and optmal arthmetc cost are obtaned. The rst constructon of the norm-preservng explct extenson operators for unstructured grds seems to be suggested n [7] and used for the constructon of precondtonng operators n [7, 8, 9, 2]. For the case of herarchcal grds, near norm-preservng explct extenson operators has been suggested n [3] whch can be easy realzed. Its teratve mprovng was dscussed n [9, ]. In ths paper we follow [2]. To construct the explct extenson operators, functons at the boundary are decomposed nto seres of functons at herarchcal grds. Then, each component of ths decomposton wll be extended n a very smple way. To prove the trace theorem for herarchcal grds, a multlevel decomposton of the boundary has been used n [23]. In the suggested paper, we desgn the multlevel decomposton of functons on the boundary wth very cheap arthmetc costs and the arthmetc costs of the resultng explct extenson operators (as well as adjont operators) s proportonal to the number of grd nodes. The paper s organzed as follows. In Secton 2, the constructon of explct extenson operators s suggested. Realzaton aspects of these extensons are dscussed n Secton 3. The mprovement of the extensons by an teratve scheme s proposed n Secton 4. The applcaton to doman decomposton methods s consdered n Secton 5 and some numercal experments are presented n Secton 6. 2

4 2 onstructon of Extenson Operators Let be a bounded, polygonal doman and? be ts boundary. consder a coarse grd trangulaton of h = [ M () = ; dam( () ) = O() Let us whch wll be rened several tmes. Ths results n a sequence of nested trangulatons h ; h; : : : ; h J such that h = M [ = () ; = ; ; : : : ; J ; where the trangles (+) are generated by subdvdng trangles () nto four congruent subtrangles by connectng the mdponts of the edges. Introduce the spaces W and V of FE (nte element) functons. The space W conssts of real-valued functons whch are contnuous on and lnear on the trangles n h. The space V s the space of traces on? of functons from W : V = f' h j' h = u h j? ; wth u h 2 W g We consder W and V as the subspaces of the Sobolev spaces H () and H 2 (?), respectvely, wth correspondng norms [3]. The man goal s the constructon of some norm{preservng explct extenson operator t from V J to W J : t : V J! W J : Ths constructon s based on the dea from [3], but nstead of Yserentant's herarchcal decomposton [26, 27] of the space V J we use some analogous of the so{called BPX{decomposton of V J [6, 25]. Denote by ' () ; = ; 2; : : : ; N the nodal bass of V and by () the one{dmensonal subspace spanned by ths functon ' () wth the support (). Dene Q () : L 2 (?)! () the L 2 {le projecton from L 2 (?) onto () : h = d () ( h )' () Q () ; 3

5 where d () = d () = ( h ; ' () (' () ) L2 (?) ; ) L2 (?) ; or (a) meas () ( h ; ) L2 ( () )) : (b) Denote by Q = N X = Q () ; = ; ; : : : ; J? : For = J, Q s dened as the L 2 {orthoprojecton from L 2 () onto V J. LEMMA. There exst postve constants c, c 2, ndependent of h, such that for any ' h 2 V J c ' h 2 H 2 (?) Q ' h 2 L 2 (?) + c 2 ' h 2 H 2 (?): = 2 (Q? Q (?) )' h 2 L 2 (?) PROOF The result above s a smple consequence of well-nown propertes of Q and a technque from [4, 8, 22, 23, 27]. However, for completeness we gve the proof. It s easy to see that Q s the lnear projecton onto V and there exsts a constant c 3, ndependent of h, such that Q ' L2 ( () c ) 3 ' L2 ( () ) 8 ' 2 L 2 (?) (2) holds, where () s the unon of all () j pont wth (). Also the followng approxmaton property of Q holds : '? Q ' 2 L 2 (?) = N X = N X = 2( + c 3 ) '? Q ' 2 L 2 ( () ) = N whch posses at least one common X = '? + Q (? ') 2 L 2 ( () ) N X = '? 2 L 2 ( () ) 4 '? +? Q ' 2 L 2 ( () ) :

6 Here s an arbtrary constant functon. Then the estmate '? Q ' 2 L 2 (?) c 4 2? j'j H (?) 8' 2 H (?) s vald wth some h-ndependent constant c 4. Accordng to [23] there exsts also an h-ndependent constant c 5 such that c 5 ' h 2 H =2 (?) nf h 2V JP h = 'h = = Q ' h 2 L 2 (?) + Q ' h 2 L 2 (?) h 2 L 2 (?) = = 2 (Q? Q? ) ' h 2 L 2 (?) 2 ' h? Q ' h 2 L 2 (?) : Let us estmate ' h? Q ' h L2 (?). For any 2 H (?) we have ' h? Q ' h L2 (?) = ' h? +? Q + Q? Q ' h L2 (?) ' h? L2 (?) +? Q L2 (?) + Q (? ' h ) L2 (?) ( + c 3)' h? L2 (?) + c 4 2? j j H (?) ; where the constant c 3 arses from (2) by summng of (). Tang the nmum over all 2 H (?), we obtan the K-functonal ' h? Q ' h L2 (?) c 6 nf 2H (?) ' h? + 2? j j H (?) = c 6 K (2? ; ' h ) ; where the constant c 6 s ndependent of h. Usng the equvalence of norms (see, e.g., [8, 23]) ' h 2 H =2 (?) 'h 2 L 2 (?) + one obtans the statement of Lemma. = 2 K (2? ; ' h ) ; Denote by x (), =,2, : : :,L the nodes of the trangulaton h (we assume that nodes x () are enumerated rst on? and then nsde ) and dene the extenson operator t n the followng way. For any ' h 2 V J set h = Q ' h ; (3a) h = (Q? Q? ) ' h ; = ; 2; : : : ; J : (3b) 5

7 Then ' h = h + h + : : : + h J : Dene the extenson u h 2 W of the functon h u h (x () ) = u h (x () ) = ( ( h (x () ) ; x () 2? ; x () 62? ; h (x () ) ; x () 2? ; ; x () 62? ; accordng to [3, 2]: (4a) (4b) For we choose ether the mean value of the boundary functon h or the soluton of the proper PDE on the coarsest grd wth Drchlet boundary condtons h. Now, we dene the extenson t ' h := u h u h + u h + : : : + u h J : (5) By settng t ' h := v J the denton above can be wrtten n a recursve way : v := u h (6a) v := v? + u h = ; : : : ; J : (6b) Note, that v 2 W s the extenson of Q ' h on level = ; : : : ; J. REMARK. We can use the L 2 {orthoprojecton from L 2 () onto V nstead of Q, = ; ; : : : ; J?. But n ths case the cost of the decomposton (3) s expensve (especally for three dmensonal problems). LEMMA 2. There exsts a postve constant c 4, ndependent of h, such that u h H () c 4 2 h L2 (?); = ; ; : : : ; J: PROOF The proof of ths lemma s obvous and was done n [3]. THEOREM. There exsts a postve constant c, ndependent of h, such that t' h H () c ' h H 2 (?) 8' h 2 V J : Here the operator t s dened n (5). 6

8 PROOF We have (see,e.g., [23]) t' h 2 H () = u h 2 H () c 2 nf w h 2W c 2 = JP 4 u h 2 L 2 () : w h = uh = = 4 w h 2 L 2 () Here c 2 s ndependent of h and the u h, = ; : : : ; J are dened n (4). Then t follows from Lemma, 2 and from the specal structure of the functons u h that = 4 u h 2 L 2 () c 3 = holds, where c 3, c 4 are ndependent of h. 2 h 2 L 2 (?) c 4 ' h 2 H =2 (?) REMARK 2. The constructon of the extenson operator t for three dmensonal problems can be done n the same way. The Theorem s vald too. If the orgnal doman s splt nto many subdomans n doman decomposton methods [8], then the dameters of the subdomans depend on some small parameter " and we need the extenson operator t such that the constant c from the Theorem s ndependent of ". To do ths, let us assume that by mang the change of varables x = " s; x 2 (7) the doman s transformed nto the doman wth the boundary? and that the propertes of are ndependent of ". From [8, 9] we have the followng. LEMMA 3. There exsts a postve constant c 2, ndependent of h and ", such that c 2 ' h H 2 " (?) u h H () for any functon u h 2 W J, where ' h 2 V J s the trace of u h at the boundary?. And there exsts a postve constant c 3, ndependent of h and ", such that for any ' h 2 V J there exsts u h 2 W J : u h (x) = ' h (x); x 2?; u h H () c 3 ' h H 2 " (?) 7 :

9 Here ' h 2 = "' h 2 L H 2 2 (?) + j' h j 2 " (?) H 2 (?) ' h 2 L 2 (?) = j' h j 2 H 2 (?) = Z? Z? (' h (x)) 2 dx ; Z? (' h (x)? ' h )y)) 2 jx? yj 2 dx dy : LEMMA 4. There exsts a postve constant c 4, ndependent of h and ", such that for any ' h 2 V J Here ' h 2 + H 2 (?) " 'h 2 L 2 (?) + j' h j 2 c H 2 4 ' h 2 (?) H The followng lemma s vald. ' h = Q ' h ; ' h = ' h? ' h : ; 2 " (?) : (8) LEMMA 5. There exsts a postve constant c 5, ndependent of h and ", such that ' h 2 H 2 " (?) + " Q ' h 2 L 2 (?) + Here ' h; 'h, are from (8). = PROOF Usng (7) and Lemma, we have 2 (Q? Q? )' h 2 L 2 (?)! c 5 ' h 2 H 2 (?) " 'h 2 L 2 (?) + j' h j 2 H 2 (?) = ' h 2 L 2 (? ) + j' h j 2 H 2 (? ) c (Q 'h 2 L 2 (? ) + = "c (Q ' h 2 L 2 (?) + = = 2 (Q? Q?)' h 2 L 2 (? )) 2 (Q? Q? )' h 2 L 2 (?) : Here Q s the projecton whch corresponds to Q wth the change of varables. 8

10 THEOREM 2. There exsts a postve constant c 6, ndependent of h and ", such that t' h H () c 6 ' h Here the operator t s dened n (5). PROOF For ' h ; 'h For the functon ' h Q ' h 2 + H 2 " (?) from (8) we have = Q ' h 2 + H 2 " (?) + = H 2 " (?) 8' h 2 V J : 2 (Q? Q? )' h 2 L 2 (?) = 2 (Q? Q? )' h 2 L 2 (?) 2 (Q? Q? )' h 2 L 2 (?) : let us consder the followng decomposton: ' h = ' h ; + ' h ; ; R ' h ; = const = ' h meas(?) (x)dx? ' h ; = ' h? ' h : ; It s easy to see that (Q? Q? )' h ; = ; = ; 2; ; J : Then we can use the evdent trc from [3] wth the Poncare nequalty n H 2 (? ): = 2 (Q? Q? )' h 2 L 2 (?) = = " = = 2 (Q? Q? )' h ; 2 L 2 (?) 2 (Q? Q?)' h ; 2 L 2 (? ) c 2 "' h ; 2 H 2 (? ) c 7 "j' h ;j 2 H 2 (? ) = c 7 "j' h ;j 2 H 2 (?) = c 7 "j' h j 2 H 2 (?) Here c 7 s from the Poncare nequalty. It s easy to see that there exsts a postve constant c 8, ndependent of h and ", such that u h H () c 8 h H 2 " (?) where h = ' h = Q ' h, and u h 2 W s from (4). The rest of the estmates for Theorem 2 and Theorem s the same. ; : 9

11 3 Realzaton of the Extenson onsder the symmetrc, nd u 2 H () : Z (x) r T u(x) rv(x) dx = H {ellptc and H {bounded varatonal problem Z f(x) v(x) dx 8v 2 H () ; (9) arsng from the wea formulaton of a scalar second{order, symmetrc and unformly bounded ellptc boundary value problem gven n a plane bounded doman R 2 wth a pecewse smooth boundary? The materal coecents (x) > 8x 2 have to be restrcted for certan extenson technques. Dene the usual nte elements (FE) nodal bass = [ ; I ] = # J ; ; #J ; # J ; ; # J ; () N (J) N (J) + N (J) +N(J) I where the rst N (J) = N J bass functons on the nest level J correspond to nodes on?, the remanng bass functons are n the nteror. The proper nodes wll be denoted by \" and \I". Smlarly, N () and N () I represent the number of bass functons on the boundary and n the nteror on level. Then the FE somorphsm leads to the symmetrc and postve dente system of equatons on each level! K; K K u := I; u; f = ; =: f K I; u I; f ; () I; K I; where K I; s symmetrc, postve dente. In the followng, the matrces I ;, I I; denote the proper denttes on level and the matrces P +, ; P +, I; P + represent I; the usual lnear FE nterpolaton matrces (Fg. ) on the proper subsets of nodes. The multlevel extenson of a functon P h N = (J) ' (J) 2 V J represented by the vector 2 R N (J) P nto a functon u h N = (J) (J) +N I u # (J) 2 W J represented by the vector u 2 R N (J) I conssts of three steps : A 2 2 AAU A + v A A v 3 A A AK A 2 A 2 A A - v A coarse grd nodes new ne grd nodes 2 Fgure : 2 Lnear FE nterpolaton

12 . Determne the rectangular N () (J) N projecton matrx Q and dene the coecents of the projecton Q h n the FE nodal bass of level := Q = ; : : : ; J : (2) 2. Accordng to (3) splt the vectors nto the coecents of the multlevel nodal bass presentaton of h mne the coecent vectors = P J = P N () () ' () and deter- := (3a) :=??P ;? I ;? are deter- P N 3. The coecents v of the extensons v = () mned by v; I; v = := v I; B I; I; P ;? v = v; v I; := P I;? P I;? = ; : : : ; J : () +N I = v # () v ;? A : v I;? (3b) (4a) (4b) Denote by E the matrx representaton of the extenson t (5), then we set E' := v J. The matrx B I; can be chosen as N ()? N () I, mappng the N () mean value of the boundary data nto the nteror. Another approach s the dscrete harmonc extenson on the coarsest grd wth respect to the PDE,.e., B I; =?K? I; K I;. 4 Improvng the Extenson by an Iteraton Scheme Denote by z I; some extenson of boundary data ;, then the functonal J (z I; ) = K ; ; represents the square of the energy norm of z I; ; z I; that extenson. The extensons gven n (6) are just an approxmaton of the followng mnmzaton problem v I; = arg mn z I; J (z I; ) ; (5)

13 whch s equvalent to the system of equatons v ; = ; K I; v I; =?K I; ; = ; : : : ; J : (6) To mprove the qualty of those extensons gven n (6),.e., decrease the constants n Theorems and 2, we apply some teraton scheme on (6) v j I; By denng the teraton matrx M I; := Q? := vj?? I; j B I; K I; v j? + I; K I; ; : (7)? II;? j B I;K I; the teratons can be rewrtten to j= v I;? := M I; v I;? I I;? M I; K? I; K I; ; (8) wth some ntal guess v. To dene the precondtoner I; B I;, for nstance, we can splt K I; nto the strctly upper, dagonal and lower part,.e., K I; = L I; + D I; + U I;, then the Gau-Sedel teraton matrx s dened va M I; := (D I; + L I; )? U I;. There are two opportuntes to choose the ntal guess v I; and the boundary data ;. Frst, we start the teraton wth the zero ntal guess and the dscrete representaton of h,.e.,, on the boundary. Ths changes (4b) v ; A I ; P ;?? v I; )K?(I I;? M I; I; K I; P I;? P I;? AB@ v ;? v I;? A : (9) The second opportunty conssts n usng the actual extenson on level as ntal guess and the proper boundary data v ;. The relaton v ; = + P v ;? ;? leads drectly to the second reformulaton of v ; A I ; v I;?(I I;? M? )K I; I; K I; M I; I ; P ;? P I;? P I;? AB@ v ;? v I;? (2) In the followng, we omt the level ndex. Recallng the functonal J (z I ), the proper error functonal s E(z I ) = z I? w I 2 wth w K I I =?K? I K I as the exact dscrete extenson. A : 2

14 LEMMA 6. Denote by w j I the j-th terate n teraton scheme (7), whch converges n the K I -energy norm,.e., there exsts a postve constant q < such that holds for all j >. Then the extenson wth some ntal guess w I. w j I? w I KI q w j? I? w I KI (2) w j I fullls J (w j I ) = J (w I) + E(w j I ) J (w I) + q 2j E(w I) PROOF Wth the dentons for J and E we wrte J (w j I ) = K ;? w j I w j I?? = K I w j I I ; wj + KI ; wi j + KI w j I ; {z }? =(w j I ;K I ) = K I w j I ; wj I + K I K? j I K I {z ; wi + } =?w I + (K I w I ; w I )? (K I w I ; w I ) + (K ; ) + (K ; ) I {z } =?w I w j I ; K I K? K I =? K I (w j I? w I); (w j I? w I)? (K I w I ; w I ) + (K ; ) = w j I? w I 2 K I {z } E(w j I ) 2? w I K I + 2 K {z } J (w I ) (2) q 2 w j? I? w I 2 K I +J (w I ) = q 2 E(w j? I ) + J (w I ) (2) q 2j w I? w I 2 K I +J (w I ) = q 2j E(w I) + J (w I ) : 5 Applcaton to a DD Precondtoner The doman wll be decomposed S nto p non-overlappng subdomans p s (s = ; : : : ; p) such that = s= s. The grd trangulatons h ; wll be dstrbuted analogously for all level = ; : : : ; J. Now, the submatrces n system () are changed nto blocmatrces, especally K I = dag (K I; ) =;::: ;p. Ths new system wll be solved by some 3

15 parallelzed precondtoned teratve method, e.g., G-method. As a precondtoner we use the ASM{DD precondtoner = I?B?T I O I O O I I O I?B I I I : (22) Ths precondtoner contans the three components I = dag ( I; ) =;::: ;p, and the bloc matrx B I, whch can be chosen freely n order to adapt the precondtoner to the partculars of the problem under consderaton. For the choce B I; =?B I; K I; see [2]. As precondtoner for the Schur complement S = K? K I K? I K I the BPS [5] s used. The precondtonng step w =? r can be rewrtten n the form Algorthm : The ASM-DD Precondtoner [2] w =? pp = A T ;? r; + B T I; r I; w I; =? I; r I; + B I; w ; ; = ; 2; : : : ; p A I;? A where A = ; A I; A I; denotes the subdoman connectvty matrx whch s used for a convenent notaton only. The subdoman FE assembly process whch s connected wth nearest neghbour communcaton stands behnd ths notaton. For further nvestgatons on DD precondtoners see [6, 2,, 5, 2, 9]. Assume postve, h-ndependent spectral equvalence constants ; ; I ; I fulllng the spectral equvalence nequaltes S and I I K I I I : If we have addtonally a constant c E so that v B I v K c E v S 8v 2 R Nc (23) holds then the upper and lower bounds of the condton number (? K) [2, 7] can be estmated as O(c 2 E) (? K) O(c 4 E) : (24) Estmate (23) represents the result from Theorem n a dscrete sense, so that B I can be chosen as the dscrete extenson operator dened n (4, 9, 2). Addtonally, Algorthm requres B I = B T I so that the transposed of 4

16 that extenson have to be appled. Whereas transposng (4) s qute smple one have to tae care when smoothng s ncluded. If we denote by M I; the adjont operator to M I; wth respect to the K I; nner product, then the transposed operaton to (9) can be wrtten v ;? v I;? A = M I; K? v ;? K I; I I;?? P T ;?? v; + PI;? T? vi; P I;? T vi; I; v I; A ; (25).e., we have to perform sweeps of the teraton procedure dened by M I; wth the rght hand sde v I; and a zero ntal guess. If M I; represents a lexcographcally forward Gau-Sedel teraton, then the adjont teraton s the lexcographcally bacward one. In the second case (2) we acheve v ;? v I;? A = I ; T P ;? P I;? P I;? T T B v ;?K I; M I; K? I I;? M I; T v I; The rst lne n the second matrx s smlar to (25). Notng that?? M T I; v I; = v I;? K I; K? I; v I; + MI; T v I; = v I;? K I; I I;? M I; K? I; v I; ; I; v I; A : the second lne s just a defect calculaton usng the result of the teraton n the rst lne. For algorthmc mprovements n combnaton wth the nner precondtoner I, see [, 9]. (26) 6 Numercal Experments In the numercal experments we used 2 smple and one challengng examples. Example :?4 u = n = [; ] [; :5] u = Example 2:?4 u = n = [; ] 2 u = For Example, the doman was subdvded nto 2 squares, the doman n 5

17 Example 2 was parttoned nto 6 squares. Example 3 (Electrcal machne): As a more challengng example we calculated the magnetc potental n an electrcal machne wth a rather complex geometry and large jumps n the coecents (see also [4]), for the decomposton of the doman nto 6 subdomans see Fg. 2. Fgure 2: Materal adapted decomposton and ntal mesh of Example 3 All calculatons were done on a 6 processor Parsytec Power-Xplorer wth 32 MByte memory per node. All examples were solved wth the precondtoned parallelzed G usng Algorthm as precondtong step untl an accuracy, measured n the K? K-energy norm, of?6 was acheved. As Schur complement precondtoner the BPS [5] was used. When not explaned further, the nner problem was solved exactly,.e., I = K I. For comparson we used n example 3 also a multgrd V-cycle wth one preand one post-smoothng sweep (V) for denng I. The teraton procedure (8) mplemented va (2) was appled, at most, one tme ( 2 f; g). In tables - 3 the projecton (a) was tested, the tables 4-6 present the results usng projecton (b). For measurng the qualty of the extensons we calculate aver = E' K '?K? I K I ' K, the rato between an approxmate extenson E' and the exact one n the energy norm. 6

18 proj. (a) J = J = J = 2 J = 3 J = 4 J = 5 J = 6 Iteratons aver Iteratons aver Table : # G-teratons for Example usng 2 processors proj. (a) J = J = J = 2 J = 3 J = 4 J = 5 J = 6 Iteratons aver Iteratons aver Table 2: # G-teratons for Example 2 ( d.o.f.) usng 6 processors proj. (a) J = J = J = 2 J = 3 J = 4 J = 5 # unnowns I exact: Iteratons Qualty aver I exact: Iteratons Qualty aver I - V: Iteratons solver n sec I - V: Iteratons solver n sec Table 3: # G-teratons for Example 3 usng 6 processors For all three examples, the behavor of the teraton counts reects the logarthmc grow of the condton number (? K) for the BPS Schur complement precondtoner,.e., the constant c E (23) seems to be h-ndependent. Also the qualty rato aver seems to be bounded wth growng level number J, especally n example 3. Both observatons conrm the result of Theorem. The addtonal teraton ( = ) for denng the extenson decreases the teraton count and really mproves the qualty of the extenson. The solver tmes n Table 3 ndcate that the addtonal teraton does not accelerate the soluton process for the example presented. 7

19 proj. (b) J = J = J = 2 J = 3 J = 4 J = 5 J = 6 Iteratons aver Iteratons aver Table 4: # G-teratons for Example usng 2 processors proj. (b) J = J = J = 2 J = 3 J = 4 J = 5 J = 6 Iteratons aver Iteratons aver Table 5: # G-teratons for Example 2 ( d.o.f.) usng 6 processors proj. (b) J = J = J = 2 J = 3 J = 4 J = 5 # unnowns Iteratons Qualty aver Iteratons Qualty aver I - V: Iteratons solver n sec I - V: Iteratons solver n sec Table 6: # G-teratons for Example 3 usng 6 processors Although the teraton counts n tables 4-6 are slghtly hgher, we can draw the same conclusons for projecton (b) as for the projecton accordng to (a). 7 onclusons The extenson technque (5) presented n Secton 2 s a fast and qualtatvely good approxmaton of the homogeneous extenson of Laplacan-le derental operators when usng the projectons (). The ndependence of the 8

20 constants n Theorems and 2 from the dscretzaton parameter h and the dameter " of the doman s stll vald n the 3D-case but has to be tested n future. In combnaton wth a proper teraton procedure (8), the extenson wors also wth respect to more complcated second order derental operators (see []). For the examples gven, the addtonal teraton step per level dd not result n an accelerated overall soluton tme. But ths property may change by the examples nvestgated. When usng the extensons n a DD precondtoner, the transposed of those extensons s needed. In combnaton wth a proper precondtoner I a sophstcated mplementaton reduces the tme per teraton (for more detals see [, 9]) sgncantly. Here, agan the 3D-case has to be nvestgated. Usng an ecent h-ndependent precondtoner I, e.g. multgrd, the asymptotc behavor of the condton number (? K) depends only on the asymptotc behavor of the Schur complement precondtoner. Therefore, the numercal eort of the whole parallel algorthm s nearly optmal (or logarthmcally optmal). References [] V. B. Andreev. The stablty of nte derence schemes for ellptc equatons wth respect to the Drchlet boundary condtons. Zh. Vychsl. Mat. Mat. Fz., 2:598{6, 972. [2] N. Aronszajn. Boundary value of functons wth nte Drchlet ntegral. In onference on partal derental equatons, Studes n egenvalue problems, 955. [3] O. V. Besov, V. P. Il'n, and S. M. Nol's. Integral presentatons of functons and embeddng theorems. Naua, Moscow, 975. n Russan. [4] F. A. Bornemann and H. Yserentant. A basc norm equvalence for the theory of multlevel methods. Numer. Math., 64:455{476, 993. [5] J. H. Bramble, J. E. Pasca, and A. H. Schatz. The constructon of precondtoners for ellptc problems by substructurng I { IV. Mathematcs of omputaton, 986, 987, 988, , 3{34, 49, {6, 5, 45{43, 53, {24. [6] J. H. Bramble, J. E. Pasca, and J. Xu. Parallel multlevel precondtoners. Mathematcs of omputaton, 55(9): { 22, 99. 9

21 [7] H. heng. Iteratve Soluton of Ellptc Fnte Element Problems on Partally Rened Meshes and the Eect of Usng Inexact Solvers. PhD thess, ourant Insttute of Mathematcal Scence, New Yor Unversty, 993. [8] W. Dahmen and A. Kunoth. Multlevel precondtonng. Numer. Math., 63:35{344, 992. [9] G. Haase. Multlevel extenson technques n doman decomposton precondtoners. In DD9 Proceedngs. John Wley and Sons Ltd., 997. [] G. Haase. Herarchcal extenson operators plus smoothng n doman decomposton precondtoners. Appled Numercal Mathematcs, 23(3):327{346, May 997. [] G. Haase and U. Langer. The non-overlappng doman decomposton multplcatve Schwarz method. Internatonal Journal of omputer Mathemathcs, 44:223{242, 992. [2] G. Haase, U. Langer, and A. Meyer. The approxmate drchlet doman decomposton method. Part I: An algebrac approach. Part II: Applcatons to 2nd-order ellptc boundary value problems. omputng, 47:37{5 (Part I), 53{67 (Part II), 99. [3] G. Haase, U. Langer, A. Meyer, and S. V. Nepomnyaschh. Herarchcal extenson operators and local multgrd methods n doman decomposton precondtoners. East-West Journal of Numercal Mathematcs, 2:73{93, 994. [4] B. Hese and M. Jung. Parallel solvers for nonlnear ellptc problems based on doman decomposton deas Submtted for publcaton. Avalable as Report 494, Insttute of Mathematcs, Unversty Lnz. [5] A. M. Matson and S. V. Nepomnyaschh. Norms n the space of traces of mesh functons. Sov. J. Numer. Anal. Math. Modellng, 3:99{26, 988. [6] A. Meyer. A parallel precondtoned conjugate gradent method usng doman decomposton and nexact solvers on each subdoman. omputng, 45:27{234, 99. [7] S. V. Nepomnyaschh. Doman Decomposton and Schwarz method n subspace for approxmate soluton of ellptc boundary value problems. PhD thess, Academy of Scence, Novosbrs, 986. n Russan. 2

22 [8] S. V. Nepomnyaschh. Doman Decomposton method for ellptc problems wth dscontnous coecents. In R. G. et al., edtor, Doman Decomposton methods for PDEs, pages 242{252. SIAM, 99. [9] S. V. Nepomnyaschh. Mesh theorems on traces, normalzaton of functon traces and and ther nverson. Sov. J. Numer. Anal. Math. Modellng, 6(3):223{242, 99. [2] S. V. Nepomnyaschh. Method of splttng nto subspaces for solvng ellptc boundary value problems n complex-form domans. Sov. J. Numer. Anal. Math. Modellng, 6(2), 99. [2] S. V. Nepomnyaschh. Optmal multlevel extenson operators. Report 95-3, TU hemntz, 995. [22] A. A. Oganesyan and L. A. Ruhovets. Varatonal derence methods for the soluton of ellptc equatons. Izdat. Aad. Nau Armanso SSR, Erevan, 979. (n Russan). [23] P. Oswald. Multlevel Fnte Element Approxmaton: Theory and Applcaton. Teubner, Stuttgart, 994. [24] L. N. S. V. M. Babch. On the boundness of the Drchlet ntegral. Dol. Aad. Nau SSSR, 6(4):64{67, 956. [25] J. Xu. Iteratve methods by space decomposton and subspace correcton: A unfyng approach. SIAM Revew, 34:58{63, 99. [26] H. Yserentant. On the mult-level splttng of nte element spaces. Numer. Math., 49(4):379{42, 986. [27] H. Yserentant. Two precondtoners based on the mult-level splttng of nte element spaces. Numer. Math., 58:63{84, 99. 2