Konrad-Zuse-Zentrum für Informationstechnik Berin Heibronner Str. 10, D-10711 Berin - Wimersdorf Raf Kornhuber Monotone Mutigrid Methods for Eiptic Variationa Inequaities II Preprint SC 93{19 (Marz 1994)
Raf Kornhuber Monotone Mutigrid Methods for Eiptic Variationa Inequaities II Abstract. We derive fast sovers for discrete eiptic variationa inequaities of the second kind as resuting from the approximation by piecewise inear nite eements. Foowing the rst part of this paper, monotone mutigrid methods are considered as extended underreaxations. Again, the coarse grid corrections are ocaized by suitabe constraints, which in this case are xed by ne grid smoothing. We consider the standard monotone mutigrid method induced by the mutieve noda basis and a truncated version. Goba convergence resuts and asymptotic estimates for the convergence rates are given. The numerica resuts indicate a signicant improvement in eciency compared with previous mutigrid approaches. Key words: convex optimization, adaptive nite eement methods, mutigrid methods AMS (MOS) subect cassications: 65N30, 65N55, 35J85
Chapter 1 Introduction Let be a poygona domain in the Eucidean space R 2. We consider the optimization probem u 2 H 1 0() : J (u) + (u) J (v) + (v); v 2 H 1 0(); (1.1) where the quadratic functiona J, J (v) = 1 a(v; v) `(v); (1.2) 2 is induced by a continuous, symmetric and H 1 0(){eiptic biinear form a(; ) and a inear functiona ` 2 H 1 (). The convex functiona of the form (v) = Z (v(x)) dx; (1.3) is generated by a scaar convex function. Denoting z = min fz; 0g and z + = max fz; 0g for z 2 R, then is taken to be the piecewise quadratic convex function (z) = 1 2 a 1(z 0 ) 2 s 1(z 0 ) + 1 2 a 2(z 0 ) 2 + +s 2 (z 0 ) + ; z 2 R; (1.4) with xed 0 2 R and non{negative constants a 1 ; a 2 ; s 1 ; s 2 2 R. More genera boundary conditions can be treated in the usua way. It is we{known (c.f. Gowinski [8]) that (1.1) can be equivaenty rewritten as the eiptic variationa inequaity of the second kind u 2 H 1 () : 0 a(u; v u) + (v) (u) `(v u) ; v 2 H 1 0 (); (1.5) and admits a unique soution u 2 H 1 0 (). Note that (1.1) becomes a ower (or upper) obstace probem, if s 1 (or s 2 ) tends to innity. Non{smooth optimization probems of the form (1.1) arise in a arge scae of appications, ranging from friction probems or non{inear materias in easticity to the spatia probems resuting from the impicit time{discretization of two{phase Stefan probems. Roughy speaking, the underying physica situation is smooth in the dierent phases u < 0 and u > 0, respectivey, but changes in a discontinuous way as u passes the threshod 0. We refer to Duvaut and Lions [4], Gowinski [8] and Eiot and Ockendon [7] for numerous exampes and further information. Let T be a given partition of in trianges t 2 T with minima diameter of order 2. The set of interior nodes is caed N. Discretizing (1.1) by 1
continuous, piecewise inear nite eements S H 1 0(), we obtain the nite dimensiona probem u 2 S : J (u ) + (u ) J (v) + (v); v 2 S : (1.6) Observe that the functiona is approximated by S {interpoation of the integrand (v), giving (v) = Z X p2n (v(p)) () p (x) dx; (1.7) where = f () p ; p 2 N g stands for the noda basis in S. Of course, (1.6) is uniquey sovabe and can be reformuated as the variationa inequaity u 2 S : a(u ; v u ) + (v) (u ) `(v u ); v 2 S : (1.8) For convergence resuts we refer to Eiot [6]. In this paper we wi derive fast sovers for the discrete probem (1.6). Cassica reaxation methods based on the successive optimization of the energy J + in the direction of the noda basis are discussed to some extend by Gowinski [8]. To overcome the we{known drawbacks of such singe{grid reaxations, Hoppe and Kornhuber [15] have derived a mutigrid agorithm, which was appied successfuy to various practica probems [13, 16]. As a basic construction principe, the dierent phases must not be couped by the coarse grid correction. Using advanced reaxation strategies of Hackbusch and Reusken [11, 12], Hoppe [14] recenty derived a gobay dampened version dispaying a considerabe improvement in asymptotic eciency rates. The construction of the previous mutigrid methods was based on the fu approximation scheme so that the possibe impementation as a mutigrid V{ cyce was cear from the very beginning. However, suitabe conditions for convergence were ess obvious. Foowing the rst part of this paper [18], we wi derive monotone mutigrid methods by extending the set of (high{frequent) search directions by additiona (intentionay ow{frequent) search directions. As a consequence, our construction starts with a gobay convergent method, which then is modied in such a way that the ecient impementation as a mutigrid V{cyce becomes possibe whie the goba convergence is retained. It is the main advantage of our approach that such modications can be studied in an eementary way. The corresponding theoretica framework wi be derived in the next section. We formay introduce extended reaxation methods and describe so{caed quasioptima approximations, preserving the goba convergence and asymptoticay optima convergence rates. The actua construction of quasioptima approximations takes pace in Section 3. The reasoning is guided by the basic observation that the standard 2
V{cyce for inear probems reies on simpe representations of inear operators and inear functionas on the coarse grid spaces. For noninear probems such (approximate) representations can be expected ony ocay. Consequenty, the coarse-grid corrections of our monotone mutigrid methods are obtained from certain obstace probems, which are xed by the preceding ne grid smoothing. In this way, the couping of dierent phases is not excuded. Foowing the rst part of this paper [18], we consider a standard monotone mutigrid method and a truncated variant, reying on the mutieve noda basis and its adaptation to the actua guess of the free boundary, respectivey. Both methods can be regarded as permanent extensions of the cassica mutigrid method and of the corresponding agorithms presented in [18]. By construction, we obtain goba convergence and the asymptotic convergence rates are bounded by 1 O( 3 ). In our numerica experiments reported in the na section, we basicay found the same behavior as for obstace probems (c.f. [18]). In particuar, for good initia iterates as obtained by nested iteration, the overa convergence is dominated by the optima asymptotic convergence rates, which are inherited from the reated inear case. Compared to previous mutigrid methods, this eads to a signicant improvement in asymptotic eciency. Of course, our approach is not restricted to the specia probem (1.6). We chose the very simpe functiona (and the reated functionas ) in order to keep the exposition as cear as possibe. However, the basic convergence resuts to be presented extend without change to any functiona of the form (1.7) with repaced by arbitrary scaar, convex functionas p, p 2 N. For exampe, the restriction of the optimization (1.1) to a convex subset K H 1 0 () of obstace{type woud cause no changes of the theoretica resuts and ony minor modications of the mutigrid agorithms. If not expicity otherwise stated, a our agorithmic considerations and convergence resuts are independent of the space dimension. 3
Chapter 2 Extended Reaxation Methods Let (M ) 0 be a given sequence of nite subsets M S, 0, with the property = f () p p 2 N g M ; 0: (2.1) Reca that () p, p 2 N, denote the noda basis functions of the given nite eement space S. Each set M = f ; : : : ; 1 m g is ordered in a suitabe way and we assume that a functions are non{negative, i.e. that 0 (p); p 2 N ; (2.2) hods for a 2 M, 0. The eements of M c = M n are intended to pay the roe of coarse grid functions with arge support, in contrast to the ne grid functions contained in. The extended reaxation method induced by (M ) 0 is resuting from the successive minimization of the energy J + in the search directions 2 M. More precisey, we introduce the spitting S = Xm =1 V ; V = spanf g; 0; (2.3) of S in the one{dimensiona subspaces V S. Then, for a given iterate u 2 S, we compute a sequence of intermediate iterates w, = 0; : : : ; m, from the m oca subprobems v 2 V : J (w 1 + v ) + (w 1 + v ) J (w 1 + v) + (w 1 + v); v 2 V ; (2.4) setting w = 0 u and w = w + 1 v, = 1; : : : ; m. The next iterate is given by u +1 = wm. Of course (2.4) is ust the noninear mutipicative Schwarz method induced by the spitting (2.3). Observe that M may change in each iteration step, so that the corresponding spitting can be iterativey adapted to the actua discrete free boundary. By construction, the extended reaxation (2.4) is monotone in the sense that J (w ) + (w ) J (w ) + 1 (w 1 ): (2.5) For notationa convenience, the index wi be frequenty suppressed in the seque. 4
Before investigating the convergence of extended reaxation methods, we wi consider the (approximate) soution of the oca subprobems (2.4). It is easiy seen that (2.4) admits a unique soution and can be equivaenty rewritten as the foowing variationa inequaity v 2 V : a(v ; v v ) + (w 1 + v) (w 1 + v ) `(v v ) a(w 1; v v ); v 2 V : (2.6) This formuation avoids the derivative of the convex functiona, which does not exist in the cassica sense. However, using subdierentia cacuus (c.f. Ekeand and Temam [5] or Carke [2]), we can reformuate (2.4) as the dierentia incusion v 2 V : 0 2 a(v ; v) + a(w 1; v) `(v) + @ (w 1 + v )(v); v 2 V : (2.7) Here, the subset @ (w) S 0 denotes the set of subgradients of at w 2 S. Denoting v = z, the incusion (2.7) can be rewritten as the scaar dierentia incusion z 2 R : 0 2 a z r + @ (z ); (2.8) where we have used the denitions a = a( ; ); r = `( ) a(w 1; ) and @ (z) R denotes the subdierentia of the scaar convex function (z) = (w 1 + z ); z 2 R: Reca that = is depending on. Using the abbreviation p = R () (x) dx and the representation (1.7), we obtain p (z) = (w 1(p) + z (p))p; p2n z 2 R: (2.9) Expoiting (2.2), the subdierentia @ is a scaar, maxima monotone mutifunction consisting of a weighted sum of transated subdierentias of the given scaar, convex function, @ (z) = X p2n (p) @(w 1(p) + z (p))p; z 2 R: (2.10) Note that the subdierentia @ is the maxima monotone mutifunction @(z) = 8 >< >: a 1 (z 0 ) s 1 if z < 0 [ s 1 ; s 2 ] if z = 0 : (2.11) a 2 (z 0 ) + s 2 if z > 0 5
For ne grid functions = () p 2, the sum in (2.10) is reducing to @ (z) = @(w 1(p ) + z () p )p ; () p 2 : Hence, the subdierentias @ corresponding to coarse grid functions 2 M c are the sum of their ne grid counterparts. In mutigrid terminoogy this means that the evauation of the subdierentias on coarse grids can be performed by canonica weighted restriction. For ne grid functions = () p 2, the oca probems (2.8) can be easiy soved by means of z = 0 w 1(p ) + denoting 8 >< >: (r p + s 1 )=(a p + a 1 ); r p < s 1 0; r p 2 [ s 1 ; s 2 ] ; (2.12) (r p s 2 )=(a p + a 2 ); r p > s 2 a p = a =p ; r p = (r a ( 0 w 1(p )))=p : The situation is more dicut if 2 M c. The main reason is that the number of critica vaues of @, where @ is set{vaued, is growing with the number of nodes p 2 N \ int supp. Reca that supp is assumed to be arge for 2 M c. This motivates the approximation of @ by scaar mutifunctions @ for 2 M c. In abuse of our preceding notation, the mutifunctions @ do not need to be subdierentias. Assume that @ is maxima monotone on D R, D 6=. Then D must be a (possiby degenerated) interva. If D is bounded from above, say sup D = z 0, then sup @ (z) tends to 1 as z tends to z 0. Hence, we formay set @ (z) = 1 for a z =2 D, z z 0. In the same way, we extend @ by 1, if D is bounded from beow. A maxima monotone mutifunction @ is caed a monotone approximation of @, if sup @ (z) sup @ (z); z 0; (2.13) inf @ (z) inf @ (z); z 0: In particuar, we have @ (0) @ (0). This motivates the trivia choice @ = @ 1, with @ 1(0) = ( 1; 1) dened on D 1 = f0g. As a further exampe, consider the nite dierences @ (z) = ( ((q + 1)z) (z))=(qz), with some xed q 6= 0, providing a monotone approximation for z 6= 0. Other variants of practica interest wi be described in the next section. The approximations @, 2 M c, give rise to the approximate subprobems z 2 R : 0 2 a z r + @ (z ); 2 M c : (2.14) The resuting approximate coarse grid corrections are given by v = z. We wi need the foowing ocation principe, which can be shown by standard arguments from convex anaysis. 6
Lemma 2.1 Assume that F is a scaar, strongy maxima monotone mutifunction on D F R, which is extended to R n D F as described above. Let [z 0 ; z 1 ] R and inf F (z 0 ) 0 sup F (z 1 ): Then there is a unique 2 [z 0 ; z 1 ], such that 0 2 F (). If @ is a monotone approximation, then Lemma 2.1 appied to F (z) = a z r + @ (z); z 2 R; (2.15) shows that the approximate subprobem (2.14) admits a unique soution z. We now generaize a reated resut from the rst part of this paper [18]. Lemma 2.2 Assume that @ is a monotone approximation of @. Then the corrections v and v, computed from (2.8) and (2.14), respectivey, are reated by v =! v ;! 2 [0; 1]: (2.16) Proof. We wi make use of the strongy maxima mutifunction F (z) de- ned in (2.15). Assume that the soution z of (2.8) is non{negative. Utiizing (2.13), we easiy get inf F (0) 0 sup F (z ) and Lemma 2.1 yieds 0 z z. In the remaining case, the assertion foows in a symmetrica way. An approximate scheme based on exact ne grid corrections v, 2, and dampened coarse grid corrections v =! v,! 2 [0; 1], 2 M c, respectivey, is caed extended underreaxation. Lemma 2.2 states that an extended underreaxation is induced by a sequence of monotone approximations (@ ) 0. Note that the cassica singe grid reaxation is recovered by the trivia choice @ = @ 1 for 2 M c. It foows from the convexity of J + that extended underreaxations preserve the monotonicity (2.5). The foowing Theorem is an immediate consequence of this property and the convergence of the ne grid reaxation. Theorem 2.1 An extended underreaxation is gobay convergent. We omit the proof, which can be amost iteray taken from [18]. As a by{ product, we obtain the convergence of the whoe sequence of intermediate iterates w, w! u ;! 1: (2.17) 7
We have described a genera approach to construct convergent iterative schemes by seecting suitabe search directions (M ) 0 and monotone approximations ( ) 0. Note that ony the representation (2.12) of the exact soution of the ne grid probems makes use of the actua choice of the scaar function. As a consequence, Theorem 2.1 remains vaid for a functionas of the form (1.7), which are represented by a famiy of arbitrary scaar, convex functions p, p 2 N. In the remainder of this section, we wi investigate the asymptotic behavior of extended underreaxations. Denote N (v) = fp 2 N v(p) = 0 g; v 2 S ; and N (v) = N n N (v). The critica points p 2 N (v) wi take the roe of the active points occurring in soution of obstace probems. The discrete probem (1.6) is caed non{degenerate, if p 2 N (u ) ) `( () ) a(u p ; () ) 2 int @ (u )( () ): (2.18) p This condition describes the stabiity of the critica nodes N (u ) with respect to sma perturbations of u. The discrete phases N (v) and N + (v) of a function v 2 S consist of a nodes p 2 N with v(p) < 0 and v(p) > 0, respectivey. We say that M is ordered from ne to coarse, if = () p and p 2 int supp impies < 0 0 for a 2 and 0 2 M. The sequence c (M ) 0 is caed positive and bounded, if there are positive constants c, C not depending on, such that p 0 < c (p) C; p 2 int supp \ N ; 2 M ; (2.19) hods uniformy for 0. A positive, bounded sequence (M ) 0 is caed reguar, if N (w ) = N (u ), 0, impies that the sets M aso remain invariant for 0. Lemma 2.3 Assume that the discrete probem (1.6) is non{degenerate. If (M ) 0 is positive, bounded and ordered from ne to coarse, then the phases of the intermediate iterates w, = 1; : : : ; m, resuting from an extended underreaxation induced by (M ) 0, converge to the phases of u that in the sense N (w ) = N (u ); N (w ) = N (u ); N + (w ) = N + (u ) (2.20) hods for 0, = 1; : : : ; m, and some 0 0. 8
It is easiy seen that the convergence (2.17) of the whoe sequence impies that there is a 1 0 with the property Proof. w N (u ) N (w ); N + (u ) N + (w ); 1 : (2.21) Then, the assertion easiy foows from the incusion N (u ) N (w ) for arge. This is what we are going to show now. As a rst step, we derive the extended non{degeneracy condition `( ) a(u ; ) 2 I int @ (u )( ); 0; (2.22) for a 2 M with the property int supp \ N (u ) 6=. The cosed intervas I R are dened by I = fz 2 R z (`( ) a(u ; )) "g and " is independent of or. Indeed, as a consequence of the non{ degeneracy condition (2.18), we can nd an " > 0 such that (2.22) hods for a = () p 2. Taking the constant c from (2.19), it is easiy checked that (2.22) is vaid for a 2 M, if " satises 0 < " c ". Because (M ) 0 is bounded, the functionas a(; ) 2 S 0 are uniformy bounded in,. Hence, utiizing (2.22) and the convergence of w, we can nd a threshod 2 1 such that `( () p ) a(w ; () p ) 2 int @ (u )( () p ); 2 ; (2.23) hods for a p 2 N (u ). Consider some xed p 2 N (u ) and reca that w is resuting from the ne grid correction associated with () p. This property can be rewritten as `( () p ) a(w ; () p ) 2 @ (w )(() p ): (2.24) Using the representation @ (w)( () ) = @(w(p))p, w 2 S p, and the monotonicity of @, it foows from (2.23) and (2.24) that w (p ) = u (p ) = 0. Hence, the ne grid correction makes sure that for arge each critica point of u is a critica point of the corresponding intermediate iterate. We sti have to show that these critica points are not aected by the coarse grid correction, i.e. that int supp \ N (u ) 6= ) v = v = 0; 3 ; (2.25) hods for 2 M and a suitabe c 3 2. Let 2 M and int supp \ c N (u ) 6=. As (M ) 0 is ordered from ne to coarse, we can assume inductivey that the vaues of w 1 in p 2 int supp \ N (u ) were xed to 0 by the preceding ne grid corrections and were not changed by possibe 9
preceding coarse grid corrections. In this case, we can use (2.22) and the continuity of the derivative @(z) in z 6= 0 to nd a 3 2 such that `( ) a(w 1 ; ) 2 @ (w 1 )( ); 3: (2.26) Using our `scaar' notation (2.8), (2.26) can be rewritten as r 2 @ (0), giving z = 0. This competes the proof. Once the correct phases N = N (u ) [ N (u ) [ N + (u ) (2.27) are known, we can dene the biinear form b u (v; w), b u (v; w) = X p2n (u ) and the functiona f u (v), f u (v) = X p2n (u ) a 1 v(p)w(p)p + + X p2n + (u ) (s 1 + a 1 0 )v(p)p X p2n + (u ) a 2 v(p)w(p)p; v; w 2 S ; (s 2 a 2 0 )v(p)p; v 2 S : (2.28) (2.29) Denoting a u (v; w) = a(v; w) + b u (v; w); `u (v) = `(v) + f u (v); (2.30) it is easiy checked that the desired soution u satises the variationa equaity a u (u ; v) = `u (v); v 2 S ; (2.31) where the reduced subspace S S is dened by S = fv 2 S v(p) = 0; p 2 N (u )g: If M is reguar and we asymptoticay have M = M, then the reduced set M = f 2 M (p) = 0; p 2 N (u )g M ; is inducing an extended reaxation method for the iterative soution of (2.31). The corresponding corrections v 2 V in the direction of 2 M are computed from the inear oca subprobems v 2 V : a u (v ; v) = `u (v) a(w 1; v); v 2 V : (2.32) 10
Assuming that the origina discrete probem (1.6) is non{degenerate, it is easiy seen that an extended reaxation induced by a reguar sequence (M ) 0 is asymptoticay reducing to the inear scheme (2.32). In order to obtain a reated resut for extended underreaxations, we have to impose further restrictions on the oca approximations. A sequence of monotone approximations (@ ) 0 is caed quasioptima, if the convergence of the intermediate iterates w and of their critica vaues (w ) impies that there is a 0 0 and an open interva I R, which contains 0 and is not depending on,, such that N @ (z) = @ (z); z 2 I; 0; (2.33) hods for a with (p) = 0, p 2 N (u ). Now we are ready to state the main resut of this section. Theorem 2.2 Assume that the discrete probem (1.6) is non{degenerate. Then the extended underreaxation induced by reguar search directions (M ) 0 and quasioptima oca approximations (@ ) 0 is reducing to the extended reaxation (2.32) for 0 and some 0 0. Proof. It foows from Lemma 2.3 that N ) = N (u (w ) hods for 1 and some suitabe 1 0. The exact oca corrections v ; = z ; tend to zero. Hence, we can nd a 0 1 so that z ; 2 I, 0. Then it foows from (2.33) that z = z ;, 0. This competes the proof. Theorem 2.2 states that for non{degenerate probems a extended underreaxations, which are induced by a xed sequence (M ) 0 and various quasioptima approximations, asymptoticay coincide. This incudes the origina extended reaxation itsef. In the case of good initia iterates (\good" with respect to the stabiity of the actua critica set N (u )), this optima asymptotic behavior dominates the whoe iteration process. We refer to the numerica experiments reported beow. 11
Chapter 3 Monotone Mutigrid Methods Assume that T is resuting from renements of an intentionay coarse trianguation T 0. In this way, we obtain a sequence of trianguations T 0 ; : : : ; T and corresponding nested nite eement spaces S 0 : : : S. Though the agorithms and convergence resuts to be presented can be easiy generaized to the non{uniform case, we assume for notationa convenience that the trianguations are uniformy rened. More precisey, each triange t 2 T k is subdivided in four congruent subtrianges in order to produce the next trianguation T k+1. Coecting the noda basis functions from a renement eves, we dene the mutieve noda basis, = f () ; p1 () : : : ; p2 () ; : : : ; (0) pn ; : : : ; p1 (0) pn 0 g; (3.1) with m = n + : : : + n 0 eements. As indicated in (3.1), is ordered from ne to coarse. An extended underreaxation induced by a reguar sequence (M ) 0 and quasioptima oca approximations ( ) 0 is caed monotone mutigrid method, if the reduced mutieve noda basis = f 2 (p) = 0, p 2 N (u )g is contained in the corresponding reduced set M. We rst consider the constant search directions M =, 0, with coarse grid functions given by c = n. In this way, we wi generaize the standard monotone mutigrid method proposed in the rst part of this paper [18]. It is cear that is reguar. Due to the ordering of the search directions, each iteration step starts with a ne grid smoothing of the given iterate u, invoving the search directions 2. Reca that the corresponding oca ne grid corrections can be easiy computed from (2.12). Then, we basicay want to improve the resuting intermediate iterate wn by successive minimization of the energy J + in the coarse grid directions 2 c. To take advantage of the simpe representation of inear operators and inear functionas on the coarse spaces S k S, 0 k <, which is crucia for the optima compexity of cassica mutigrid methods, we want to restrict the scaar corrections z to such intervas, on which the subdierentias @ (z) = @ (w 1 + z )( ) are inear. In this case, we can evauate the coarse grid corrections v = z without visiting the ne grid. Foowing this basic idea, we dene the cosed, convex subset K S, K = fv 2 S ' (p) v(p) ' (p); p 2 N g; 12
where the obstaces ' ; ' 2 S are given by ( 1; w ' (p) = n (p) < 0 0 ; wn (p) 0 ; ' (p) = ( 0 ; w n (p) 0 1; w n (p) > 0 (3.2) for a p 2 N. As usua, the index wi be frequenty skipped in the seque. By construction of the obstaces ' and ', the functiona on K can be rewritten in the form (v) = 1 2 b wn (v; v) f wn (v); v 2 K : (3.3) The biinear form b wn (; ) and the functiona f wn on S are dened by (2.28) and (2.29), respectivey, repacing u by w n. Observe that the underying approximate spitting N = N (w n ) [ N (w n ) [ N + (w n ) (3.4) is xed by the ne grid smoothing. We wi impose the condition w 2 K on the remaining intermediate iterates w, = n + 1; ; m. Equivaenty, the coarse grid corrections must not cause a change of phase. In particuar, the vaues w n (p) = 0 at the critica points p 2 N (w n) remain invariant. We emphasize, that the couping of the phases by the coarse grid correction is not excuded. The restricted successive minimization of the energy functiona J + on K in the directions 2 c eads to the same type of oca obstace probems as we have aready considered in the rst part of this paper [18]. Hence, we can directy appy a the arguments and agorithms presented therein. In particuar, the exact soution of the resuting oca obstace probems is sti not avaiabe at reasonabe cost. For an approximation we use quasioptima oca obstaces, 2 V = spanf g generated by monotone recursive restriction of the defect obstaces ' w 1, ' w 1 2 S. Introducing the biinear form a wn (; ) and the functiona `wn on S according to (2.30) and the oca constraints D V, D = fv 2 V (p) v(p) (p); p 2 N g; the (approximate) coarse grid corrections v are nay computed from v 2 D : a wn (v ; v v ) `wn (v v ) a wn (w 1 ; v v ); v 2 D ; (3.5) for a = n +1; ; m. Note that the resuting standard monotone mutigrid method can be impemented as a cassica V{cyce. We refer to [18] for detais. 13
To appy the convergence theory deveoped in the preceding section, we reformuate (3.5) as a scaar incusion of the form (2.14). For this reason, we dene the scaar, convex functions, (z) = (w 1 + z ) + (z); z 2 R; 2 c ; (3.6) with denoting the characteristic function of I = fz 2 R z 2 D g R. Then, it is easiy checked that (3.5) can be reformuated as z 2 I : 0 2 a z r + @ (z ) (3.7) and v = z. Reca the notation a = a( ; ) and r = `( ) a(w 1; ). Lemma 3.1 The subdierentias of the scaar functions ( (3.6) are quasioptima approximations (@ ) 0. ) 0 dened in Proof. Consider some arbitrary, xed 0 and a xed, 2 c. Being monotone restrictions of the defect obstaces ' w 1 and ' w 1, the oca defect obstaces and satisfy ' w 1 0 ' w 1: (3.8) Hence, 0 2 I. Now the monotonicity (2.5) foows from and simpe arguments from convex anaysis. (z) = (z) + (z); z 2 R; (3.9) Assume that the intermediate iterates w and their critica points N ) (w converge to u and N (u ), respectivey. Choose 0 0 such that N ) = (w N (u ) for 0, = 1; : : : ; m. Then the obstaces ', ' and the corresponding constraints K remain invariant, say = K K for 0. It is easiy checked that u is the soution of the doube obstace probem u 2 K : a u (u ; v u ) `u (v u ); v 2 K : Note that the corresponding active set of u coincides with the critica set N (u ). By the denition of the quasioptimaity of, and (c.f. [18, 19]), there is a positive number 2 R and a threshod 1 0, such that (p) < 0 < (p); p 2 N \ int supp ; 1 ; (3.10) hods if is vanishing on N (u ). Setting I = ( ; ), it is obvious that 0 2 I I so that @ (z) = (z); z 2 I; @ 1; (3.11) 14
is vaid for a with int supp \ N (u ) =. This competes the proof. Expoiting recent estimates of the convergence rates for the inear reduced probem (c.f. [18, 19]), the foowing theorem is an immediate consequence of Lemma 3.1 and Theorem 2.2. Theorem 3.1 The standard monotone mutigrid method induced by the oca coarse grid probems (3.5) is gobay convergent. If additionay the discrete probem (1.6) is non{degenerate, then the phases aso converge and the a posteriori error estimate ku u +1 k (1 c( + 1) 3 )ku u k (3.12) hods for 0 with suitabe 0 0. Here k k 2 = a(; ) denotes the energy norm and the positive constant c < 1 depends ony on the eipticity of a(; ) and on the initia trianguation T 0. Note that the error estimate (3.12) requires no additiona reguarity assumptions. On the other hand, this resut is restricted to two space dimensions. We refer to [18, 19] for a detaied discussion. Obviousy, there are no contributions from coarse grid functions 2 c n, once the correct phases are xed. However, the reduced spitting induced by may be rather poor, eading to unsatisfying asymptotic convergence rates (c.f. [18, 19]). Foowing [18], we wi extend the set by suitabe truncations of the coarse grid functions 2 c n. In each iteration step, we adapt c to the critica set N (wn ) of the smoothened iterate wn. More precisey, the actua coarse grid search directions ~ c are given by ~ c = f ~ ~ = T ;k (k) p ; (k) p 2 c ; p 2 N n N (w n )g: (3.13) The truncation operators T ;k, T ;k = I S : : : I S ; k = 0; : : : ; 1; (3.14) k+1 are resuting from recursive Sk {interpoation denoted by I S k reduced spaces Sk S k, : S! S k. The S k = fv 2 S k v(p) = 0; p 2 N k g; k = 0; : : : ; ; (3.15) consist of the functions v 2 S k vanishing on the restricted critica sets Nk = N k \ N (w n ), k = 0; : : : ;. The ordering of ~ = f~ c n +1; : : : ; ~ m g is inherited from c. It is easiy checked that ~ = [ ~ c, 0, is reguar. 15
In particuar, we have ~ = ~, 0, with some xed ~, if the phases remain invariant for 0. Note that ~ hods by construction. As before, we use quasioptima restrictions ~ and ~ of the defect obstaces ' w 1 and ' w 1 to dene the oca constraints ~ D ~ V = spanf ~ g, ~D = fv 2 ~ V ~ (p) v(p) ~ (p); p 2 N g; ~ 2 ~ c : For a ~ 2 ~ c, the coarse grid corrections ~v are computed from ~v 2 ~ D : a wn (~v ; v ~v ) `wn (v ~v ) a wn (w 1 ; v ~v ); v 2 ~ D ; (3.16) In this way, we have derived a truncated monotone mutigrid method. The next theorem foows amost iteray in the same way as Theorem 3.1. Theorem 3.2 The truncated monotone mutigrid method induced by the oca coarse grid probems (3.16) is gobay convergent. If additionay the discrete probem (1.6) is non{degenerate, then the phases aso converge and the a posteriori error estimate hods for 0 ku u +1 k (1 c( + 1) 3 )ku u k (3.17) with suitabe 0 0. The positive constant c < 1 depends ony on the eipticity of a(; ) and on the initia trianguation T 0. Both the standard and the truncated version can be impemented as a V{ cyce with non{inear Gauss{Seide smoothing (2.12) on the ne grid and proected Gauss{Seide smoothing on the coarse eves. This carries over to the adaptive case. Other variants incuding W{cyces or symmetric Gauss{ Seide smoothing can be obtained in a simiar way. 16
Chapter 4 Numerica Experiments The non{inear evoution equation @ @t H(U) U = F; in (0; T ); (4.1) with suitabe initia and boundary conditions describes the heat conduction in undergoing a change of phase. H is a generaized enthapy or heat content, U is a generaized temperature and F is a body heating term. The enthapy H is a scaar maxima monotone mutifunction, H(z) = 8 >< >: c 1 (z 0 )= 1 if z < 0 [0; L] if z = 0 ; z 2 R; (4.2) c 2 (z 0 )= 2 + L if z > 0 which is set{vaued at the phase change temperature 0. The positive constants c i ; i, i = 1; 2, describe the therma properties in the two dierent phases and L > 0 stands for the atent heat. Discretizing (4.1) in time by the backward Euer scheme with respect to a uniform step size > 0, the spatia probems at the dierent time eves t k = k can be identied with probems of the form (1.1). The soution u = U (; t k ) is the approximation at the actua time step, the biinear form a(v; w) = (rv; rw) is generated by the Lapacian and the functiona ` is given by `(v) = ( F k + H k 1; v) with F k = F (; t k ) and a suitabe function H k 1 2 H(U (; t k 1)). The brackets (; ) denote the canonica scaar product in L 2 (). Finay, we choose a i = c i = i, i = 1; 2, and s 1 = 0, s 2 = L so that the piecewise quadratic function dened in (1.3) satises @ = H. This semi{discretization has been used by Jerome [17] to estabish existence and uniqueness of the continuous soution U and aso provides a genera framework for a variety of numerica methods. We refer to Hoppe [14] and the iterature cited therein. To iustrate the numerica properties of our monotone mutigrid methods, we wi concentrate on a simpe mode probem, which has been aready considered by Hoppe and Kornhuber [15] and Hoppe [14]. The space{time domain (0; T ) is specied by = (0; 1) 2 and T = 0:5, whie the physica data are c 1 = 2, 1 = 1, c 2 = 6, 2 = 2 and 0 = 0, L = 1. Using the (physica) temperature, (x 1 ; x 2 ; t) = (x 1 0:5) 2 + (x 2 0:5) 2 exp( 4t)=4; (x 1 ; x 2 ) 2 ; t > 0; 17
the source term F is given by ( c F (x 1 ; x 2 ; t) = 1 exp( 4t) 4 1 if < 0 c 2 exp( 4t) 4 2 if > 0 ; (x 1; x 2 ) 2 ; t > 0: Then the generaized temperature U, U = 1 if 0; U = 2 if 0; is the soution of (4.1). Initia and boundary conditions taken from the exact soution U. As in [14, 15], we choose the time step = 0:0125. To obtain an initia trianguation T 0, a partition of in two trianges is reguary rened. Starting with T 0, we appy successive uniform renement to obtain a sequence of trianguations T 0 ; : : : ; T 7. The resuting discrete probems (1.6) are soved iterativey by the standard monotone mutigrid method STDKH (c.f. Theorem 3.1) and the truncated version TRCKH (c.f. Theorem 3.2). The impementation was carried out in the framework of the nite eement code KASKADE (c.f. Erdmann, Lang and Roitzsch [1]) and we used a SPARC IPX Workstation for the computations. Figure 4.1: Iteration History Let us consider the convergence behavior for the spatia probem resuting from the initia time step. In our rst experiment the renement eve is xed to = 6 and we appy both mutigrid methods to the initia iterate u 0 = 0. The resuting iterative errors with respect to the energy norm are depicted in Figure 4.1. Obviousy, the iteration history can be separated in three dierent parts. First, we observe a rapid decrease due to the fast eimination 18
of the high frequent terms. In the foowing transient phase the agorithm determines the correct free boundary unti nay the asymptotic behavior of the reduced inear iteration is reached. Obviousy, TRCKH heaviy benets from the adaptive truncation of the standard search directions, providing a tremendous improvement of the asymptotic convergence rates. Figure 4.2: Asymptotic Convergence Rates We now concentrate on the variation of the convergence behavior with increasing renement eve. For the xed initia iterate u 0 = 0 the transient convergence rates seem to be uniformy bounded but the number of transient steps grows consideraby with increasing. However, using reasonabe initia iterates as resuting from nested iteration, we found that the transient steps were vanishing competey or (for arge ) were reduced to a very sma number. Starting with the interpoated soution from the previous eve, we consider the asymptotic convergence rates given by q = 0 " 0 ="0 ; = 0; : : : ; 7; (4.3) where " denotes the iterative error after iteration steps. To be compatibe with [14, 15], the error is measured in the 2 {norm and we choose 0 such that " 0 < 10: 8. The resuting asymptotic convergence rates of STDKH and TRCKH over the eves = 1; : : : ; 7 are shown in Figure 4.2. Obviousy, the convergence rates ony sighty deteriorate with increasing. To compare TRCKH with previous mutigrid methods, we consider the agorithms MGSTEF2 (c.f. [15]) and the dampened version DMGSTEF (c.f.[14]). As a basic construction principe of both methods, the coarse grid correction is restricted to the interior of the (approximate) phases, which have been 19
xed by ne grid smoothing. In addition, DMGSTEF uses advanced reaxation strategies in the spirit of Hackbusch and Reusken [11, 12], eading to goba convergence resuts and signicanty improved asymptotic eciency rates. The asymptotic eciency rates q are obtained by mutipying the number 0 of iterations appearing in (4.3) by a certain work unit. A work unit corresponds to one symmetric Gauss{Seide step on the nest eve. Tabe 1 beow dispays the resuting asymptotic eciency rates q5 for TR- CKH, MGSTEF2 and DMGSTEF at the time eves t = 10k, k = 1; : : : ; 5. The vaues for MGSTEF2 and DMGSTEF are taken from [14]. Simiar resuts are obtained for the remaining time steps. t=0.10 t=0.20 t=0.30 t=0.40 t=0.50 TRCKH 0.20 0.23 0.21 0.19 0.19 DMGSTEF 0.34 0.31 0.33 0.30 0.29 MGSTEF2 0.50 0.45 0.50 0.44 0.43 Tabe 4.1: Asymptotic Eciency Rates Though we did not (yet) appy a suitabe ordering of the unknowns or additiona reaxation techniques, TRCKH performs best for a time eves. Unike the other two methods, TRCKH aows the couping of the phases by the (truncated) search directions. This eads to a arger coarse grid space, which is the reason for the improved convergence. The author wants to thank R. Roitzsch for compu- Acknowedgements. tationa assistance. 20
Bibiography [1] B. Erdmann, J. Lang, R. Roitzsch: KASKADE Manua, Version 2.0 Technica Report TR 93{5, Konrad{Zuse{Zentrum Berin (1993) [2] F.H. Carke: Optimization and Nonsmooth Anaysis. John Wiey and Sons, New York (1983) [3] P. Deuhard, P. Leinen, H. Yserentant: Concepts of an Adaptive Hierarchica Finite Eement Code. IMPACT Comput. Sci. Engrg. 1, p. 3{35 (1989) [4] G. Duvaut, J.L. Lions: Les inequations en mecanique et en physique. Dunaud, Paris (1972) [5] I. Ekeand, R. Temam: Convex Anaysis and Variationa Probems. North Hoand, Amsterdam (1976) [6] C.M. Eiot: On the Finite Eement Approximation of an Eiptic Variationa Inequaity Arising from an Impicit Time Discretization of the Stefan Probem. IMA J. Numer. Ana. 1, p. 115{125 (1981) [7] C.M. Eiot, J.R. Ockendon: Weak and Variationa Methods for Moving Boundary Probems. Research Notes in Mathematics 53, Pitman, London (1982) [8] R. Gowinski: Numerica Methods for Noninear Variationa Probems Springer, New York (1984) [9] R. Gowinski, J.L. Lions, R. Tremoieres: Numerica Anaysis of Variationa Inequaities. North{Hoand, Amsterdam (1981) [10] W. Hackbusch: Muti{Grid Methods and Appications. Springer, Berin (1985) [11] W. Hackbusch, A. Reusken: Anaysis of a Dampened Noninear Mutieve Method. Numer. Math. 55, p. 225{246 (1989) [12] W. Hackbusch, A. Reusken: On Goba Mutigrid Convergence for Noninear Probems. In: Robust Mutigrid Methods, W. Hackbusch (ed.), Vieweg, Braunschweig, p. 105{113 (1988) [13] R.H.W. Hoppe: Numerica Soution of Muticomponent Aoy Soidication by Mutigrid Techniques. IMPACT Comput. Sci. Engrg. 2, p. 125-156 (1990) 21
[14] R.H.W. Hoppe: A Gobay Convergent Muti{Grid Agorithm for Moving Boundary Probems of Two{Phase Stefan Type. IMA J. Numer. Ana. 13, p. 235{253 (1993) [15] R.H.W. Hoppe, R. Kornhuber: Mutigrid Methods for the Two{Phase Stefan Probem. In: Mutigrid Methods: Theory, Appications and Supercomputing, St. McCormick (ed.), Marce Dekker, New York, p. 267{ 297 (1988) [16] R.H.W. Hoppe, R. Kornhuber: Muti{Grid Soution of Two Couped Stefan Equations Arising in Induction Heating of Large Stee Sabs. Int. J. Numer. Methods Eng., Vo. 30, p. 779{801 (1990) [17] J.W. Jerome: Approximation of Noninear Evoution Equations. Academic Press, New York (1983) [18] R. Kornhuber: Monotone Mutigrid Methods for Eiptic Variationa Inequaities I. To appear in Numer. Math. (1994) [19] R. Kornhuber, H. Yserentant: Mutieve Methods for Eiptic Probems on Domains not Resoved by the Coarse Grid. Preprint SC 93{21, Konrad{Zuse{Zentrum Berin (1993) [20] J. Xu: Iterative Methods by Space Decomposition and Subspace Correction. SIAM Reviews 34, p. 581{613 (1992) [21] H. Yserentant: Od and New Convergence Proofs of Mutigrid Methods. Acta Numerica, p. 285{326 (1993) 22