Numerische Mathematik

Transcription

1 Numer. Math. 72: (1996) Numerische Mathemati c Sringer-Verlag 1996 Multilevel Schwarz methods for ellitic roblems with discontinuous coefficients in three dimensions Masymilian Dryja 1,, Marcus V. Saris 2,, Olof B. Widlund 3, 1 Deartment of Mathematics, Warsaw University, Banacha 2, 2-97 Warsaw, Poland dryja@mimuw.edu.l 2 Deartment of Comuter Science, University of Colorado, Boulder, CO , USA msaris@tigger.cs.colorado.edu 3 Courant Institute of Mathematical Sciences, 251 Mercer St, New Yor, NY 112, USA widlund@cs.nyu.edu Received Aril 6, 1994 / Revised version received December 7, 1994 Summary. Multilevel Schwarz methods are develoed for a conforming finite element aroximation of second order ellitic roblems. We focus on roblems in three dimensions with ossibly large jums in the coefficients across the interface searating the subregions. We establish a condition number estimate for the iterative oerator, which is indeendent of the coefficients, and grows at most as the suare of the number of levels. We also characterize a class of distributions of the coefficients, called uasi-monotone, for which the weighted L 2 -rojection is stable and for which we can use the standard iecewise linear functions as a coarse sace. In this case, we obtain otimal methods, i.e. bounds which are indeendent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse saces given by simle exlicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods. Mathematics Subject Classification (1991): 65F1, 65N3, 65N55 This wor has been suorted in art by the National Science Foundation under Grant NSF-CCR , in art by Polish Scientific Grant , and in art by the Center for Comutational Sciences of the University of Kentucy at Lexington This wor has been suorted in art by a Brazilian graduate student fellowshi from Conselho Nacional de Desenvolvimento Cientifico e Tecnologico CNP, in art by a Dean s Dissertation New Yor University Fellowshi, and in art by the National Science Foundation under Grant NSF- CCR and the U.S. Deartment of Energy under contract DE-FG2-92ER25127 This wor has been suorted in art by the National Science Foundation under Grant NSF- CCR and, in art, by the U.S. Deartment of Energy under contracts DE-FG2-92ER25127 and DE-FG2-88ER2553

2 314 M. Dryja et al. 1. Introduction The urose of this aer is to develo multilevel Schwarz methods for a conforming finite element aroximation of second order ellitic artial differential euations. A secial emhasis is laced on roblems in three dimensions with ossibly large jums in the coefficients across the interface searating the subregions. To simlify the resentation only iecewise linear finite elements are considered. Our goal is to design and analyze methods with a rate of convergence which is indeendent of the jums of the coefficients, the number of substructures, and the number of levels. We consider two classes of the methods, additive and multilicative. The multilicative methods are variants of the multigrid V-cycle method. In our design and analysis, we use a general Schwarz method framewor develoed in Dryja and Widlund [13, 14, 17], and Dryja et al. [12] for the additive variant, and in Bramble et al. [4] for the multilicative one. Among the articular cases, discussed here, are the BPX algorithm, cf. Bramble et al. [5], and Xu [3], and the multilevel Schwarz method with one-dimensional subsaces considered by Zhang [33, 34]; see also Dryja and Widlund [15, 16]. It is well nown that these methods are otimal when the coefficients are regular. The roblems become uite challenging for roblems with highly discontinuous coefficients. An earlier study was carried out in Dryja and Widlund [16], where the BPX method was modified and alied to a Schur comlement system obtained after that the unnowns of the interior nodal oints of the substructures had been eliminated. In that case, the condition number of the reconditioned system was shown to be bounded from above by C (1+l) 2, where l is the number of level of the refinement; see further Sect. 9. The main uestion for roblems with discontinuous coefficients is the choice of a coarse sace. We introduce a coarse triangulation given by the substructures and assume that the coefficients can have large variations only across the interfaces of these substructures. We then design methods with several coarse saces, sometimes nown as exotic coarse saces; cf. Widlund [28]. Some are new and others have reviously been discussed; see Dryja et al. [12], Dryja and Widlund [17], and Saris [22]. One of our main results is that the condition number of the resulting systems can be estimated from above by C (1 + l) 2 with C indeendent of the jums of coefficients, of the number of substructures, and also of l; see Sect. 4. For multilicative variants such as the V-cycle multigrid, the convergence rate is bounded from above by 1 C (1 + l) 2, C > ; see Sect. 6. In Sect. 5, we study in detail the weighted L 2 rojection with weights given by the discontinuous coefficients of the ellitic roblem. Bramble and Xu [6], and Xu [29] have considered this roblem and established that the weighted L 2 rojection is not always stable in the resence of interior cross oints. Here, we introduce a new concet called uasi-monotone distribution of the coefficients which characterizes the cases for which certain otimal estimates for the weighted L 2 rojection are ossible. For roblems with uasi-monotone coefficients, the

3 Multilevel Schwarz methods 315 standard iecewise linear functions can be used as the coarse sace and otimal multilevel algorithms are obtained. In Sect. 7, we introduce aroximate discrete harmonic extensions and define new coarse saces by modifying reviously nown exotic coarse saces; see Saris [22] for a case of nonconforming elements. Using these extensions, we can avoid solving a local Dirichlet roblem for each substructure when using exotic coarse saces [28]. We show that the convergence rate estimate of our new iterative methods, with aroximate discrete harmonic extensions, are comarable to those using exact discrete harmonic extensions. The use of aroximate discrete harmonic extensions results in algorithms where the wor er iteration is linear in the number of degrees of freedom with the ossible excetion of the cost of solving the coarse roblem. Ellitic roblems with discontinuous coefficients have solutions with singular behavior. Therefore, in Sect. 8, we consider nonuniform refinements. We begin with a coarse triangulation that is shae regular and ossibly nonuniform and then refine it using a local refinement scheme analyzed by Bornemann and Yserentant [1]. We establish a condition number estimate for the iteration oerator which is bounded from above by C (1 + l) 2. For uasi-monotone coefficients, we obtain an otimal multilevel reconditioner. 2. Differential and finite element model roblems We consider the following selfadjoint second order roblem: Find u H 1 (Ω), such that (1) a(u,v)=f(v) v H 1 (Ω), where a(u,v)= Ω ρ(x) u vdx and f (v) = Ω fvdx for f L 2 (Ω). For simlicity, let Ω be a bounded olyhedral region in R 3 with a diameter of order 1. A triangulation of Ω is introduced by dividing the region into nonoverlaing shae regular simlices {Ω i } N i=1, with diameters of order H, which are called substructures or subdomains. This artitioning induces a coarse triangulation associated with the arameter H. In Sect. 8, we consider a case in which the coarse triangulation is shae regular but ossibly nonuniform. We assume that ρ(x) > is constant, in each substructure, with ossibly large jums occurring only across substructure boundaries. Therefore, ρ(x) = ρ i = const in each substructure Ω i. The analysis of our methods can easily be extended to the case when ρ(x) varies moderately in each subregion. We define a seuence of uasi-uniform nested triangulations {T } l = as follows. We start with a coarse triangulation T = {Ω i } N i=1 and set h = H. A triangulation T = {τj }N j =1 on level is obtained by subdividing each individual element τ 1 j in the set T 1 into several elements denoted by τj.

4 316 M. Dryja et al. We assume that all the triangulations are shae regular and uasi-uniform. Let hj = diameter(τj ), h = max j hj, and h = h l, where l is the number of refinement levels. We also assume that there exist constants γ<1, c >, and C, such that if an element τj n+ of level n + is contained in an element τj of level, then n+ cγ n diam(τj ) diam(τj ) C γ n. In Sect. 8, we consider a case in which the refinement is nonuniform. For each level of triangulation, we define a finite element sace V (Ω) which is the sace of continuous iecewise linear functions associated with the triangulation T. Let V (Ω) be the subsace of V (Ω) of functions which vanish on Ω, the boundary of Ω. We also use the notation V h (Ω) =Vl (Ω). The discrete roblem associated with (1) is given by: Find u V h (Ω), such that (2) a(u,v)=f(v) v V h (Ω). The bilinear form a(u,v) is directly related to a weighted Sobolev sace H 1 ρ (Ω) defined by the seminorm u 2 Hρ 1 (Ω) = a(u, u). We also define a weighted L 2 norm by: (3) u 2 L 2 ρ (Ω) = ρ(x) u(x) 2 dx Ω for u L 2 (Ω). Let Σ be a region contained in Ω such that Σ does not cut through any element τj T. We denote by V (Σ) the restriction of V (Ω) to Σ, and by V (Σ) the subsace of V (Σ) of functions which vanish on Σ. We also define Hρ 1 (Σ) and L 2 ρ(σ) by restricting the domain of integration of the weighted norms to Σ. To avoid unnecessary notations, we dro the arameter ρ when ρ = 1, and Σ when the domain of integration is Ω. In the case of a region Σ of diameter of order h, such as an element τj or the union of few elements, we use a weighted norm, (4) u 2 H 1 ρ (Σ) = u 2 H 1 ρ (Σ) + 1 h 2 u 2 L 2 ρ (Σ). We introduce the following notations: u v, w x, and y z meaning that there are ositive constants C and c such that u C v, w cx and cz y Cz, resectively. Here C and c are indeendent of the variables aearing in the ineualities and the arameters related to meshes, saces and, esecially, the weight ρ. Sometimes, we will use to stress that C =1.

5 Multilevel Schwarz methods Multilevel additive Schwarz method Any Schwarz method can be defined by a slitting of the sace V h into a sum of subsaces, and by bilinear forms associated with each of these subsaces. We first consider certain multilevel methods based on the MDS-multilevel diagonal scaling introduced by Zhang [34], enriched with a coarse sace as in Dryja [11], Dryja and Widlund [17], Dryja et al. [12], or Saris [22, 23]. Let N and N be the set of nodes associated with the sace V and V, resectively. Let φ j be a standard nodal basis function of V, and let Vj = san{φ h j }. We decomose V as V h = V X 1 + l = V = V X 1 + l = j N V j. We note that this decomosition is not a direct sum and that dim(vj ) = 1. Four different tyes of coarse saces V 1 X, and associated bilinear forms bx 1 (u, u) : V 1 X V 1 X R, with X = F, E, NN, and W, will be considered; see Sect. 4. We also consider variants of these four coarse saces using saces of aroximate discrete harmonic extensions given by simle exlicit formulas; see Sect. 7. The case when the coarse sace is V = V H is considered in Sect. 5. We introduce oerators P j : V h V a(p ju,v)=a(u,v) j,by v V j, and an oerator T 1 X : V h V 1 X, by an inner roduct bx 1 (, ) and the formula (5) b X 1(T X 1u,v)=a(u,v) v V X 1. The analysis can easily be extended to the case when we use aroximate solvers for the other one-dimensional subsaces Vj. Thus, we do not necessarily need to save, in memory, or recomute, all the values of a(φ j,φ j ), for =1,,l, and j N. Let l (6) T X = T 1 X + Pj. We now relace (2) by = j N (7) T X u = g, g = T X 1u + l = j N P j u. By construction, (2) and (7) have the same solution. We notice that T 1 X u (and similarly the Pj u) can be comuted, without the nowledge of u, the solution of (2), since we can find T 1 X u by solving (8) b X 1(T X 1u,v)=a(u,v)=f(v) v V X 1.

6 318 M. Dryja et al. Euation (7) is tyically solved by a conjugate gradient method. In order to estimate its rate of convergence, we need to obtain uer and lower bounds for the sectrum of T X. The bounds are obtained by using the following theorem; cf. Dryja and Widlund [17], Zhang [33, 34]. Theorem 1. Suose the following three assumtions hold: i) There exists a constant C such that for all u V h there exists a decomosition u = u 1 + l = j N u j, with u 1 V 1 X, u j Vj, such that l b 1(u X 1, X u 1)+ X a(uj, uj ) C 2 a(u, u). = j N ii) There exists a constant ω such that a(u, u) ω b X 1(u, u) u V X 1. iii) There exist constants ɛ mn ij, m, n =,,l and i N m, j N n such that a(u m i, u n j ) ɛ mn ij a(u m i, u m i ) 1/2 a(u n j, u n j ) 1/2 ui m Vi m uj n Vj n. Then, T X is invertible, a(t X u,v)=a(u,t X v), and (9) C 2 a(u, u) a(t X u, u) (ρ(e )+ω)a(u,u) u V h. Here ρ(e ) is the sectral radius of the tensor E = {ɛ mn ij } l i,j,m,n=. 4. Exotic coarse saces and condition numbers We now introduce certain geometrical objects in rearation for the descrition of our exotic coarse saces V 1 X. Let F ij reresent the oen face which is shared by two substructures Ω i and Ω j. Let E l reresent an oen edge, and V m a vertex of the substructure Ω i. Let W i denote the wire baset of the subdomain Ω i, i.e. the union of the closures of the edges of Ω i. We denote the interface between the subdomains by Γ = Ω i \ Ω, and the wire baset by W = W i \ Ω. The sets of nodes belonging to Ω, Ω i, Ω i, F ij, E l, W i, and Γ are denoted by Ω h, Ω i,h, Ω i,h, F ij,h, E l,h, W i,h, and Γ h, resectively. We now roceed to discuss several alternative coarse saces and to establish bounds for the condition numbers of the corresonding additive multilevel methods.

7 Multilevel Schwarz methods Neumann-Neumann coarse saces We first consider the Neumann-Neumann coarse saces which have been analyzed in Dryja and Widlund [17], Mandel and Brezina [18], and Saris [22]. An interesting feature of these coarse saces is that they are of minimal dimension with only one degree of freedom er substructure, even in the case when the substructures are not simlices. For any β 1/2, we introduce the weighted counting functions µ i,β, for all i =1,,N, defined by µ i,β (x) = j ρ β j, x Ω i,h\ Ω h, µ i,β (x) =, x (Γ h \ Ω i,h ) Ω h. For each x Ω i,h \ Ω h, the sum is taen over the values of j for which x Ω j,h. The seudo inverse µ + i,β of µ i,β is defined by and µ + i,β(x) =(µ i,β (x)) 1, x Ω i,h \ Ω h, µ + i,β(x) =, x (Γ h \ Ω i,h ) Ω h. We extend µ + i,β elsewhere in Ω as a minimal energy, discrete harmonic function using the values on Γ h Ω h as boundary values. The resulting functions belong to V h(ω) and are also denoted by µ+ i,β. We can now define the coarse sace V 1 NN V h by (1) V NN 1 = san{ρ β i µ + i,β}, i.e. we use one basis functions for each substructure Ω i. We remar that we can even define a Neumann-Neumann coarse sace for β = by considering, the limit of the sace V 1 NN when β aroaches, i.e. ρ i µ + i, = lim β ρβ i µ + i,β. For instance, for x F ij,h, we obtain ρ i µ + i, (x) =1, if ρ i >ρ j, and We note that V 1 NN given by (11) ρ i µ + i, (x)=, if ρ i <ρ j, ρ i µ + i, (x) =1/2, if ρ i = ρ j. is also the range of an interolator I NN h u 1 = Ih NN u(x) = i u (i) 1 = i ū h i ρ β i µ+ i,β. : V h V NN 1, Here, ū h i is the discrete average value of u over Ω i,h.

8 32 M. Dryja et al. We note the coarse saces defined with β =1/2, β = 1, and β 1/2 have been used by Dryja and Widlund [17], Mandel and Brezina [18], and Saris [22], resectively. Recently, Wang and Xie [27] introduced another coarse sace which is similar to ours with β =. However, their basis functions only tae the value or 1 on Γ h. We introduce the bilinear form b 1 NN(u,v):VNN 1 V 1 NN R, defined by (12) b NN 1 (u,v)=a(u,v). Theorem 2. Let T NN be defined by (6), and let 1/2 β. Then for any u V h (Ω), we have (1 + l) 2 a(u, u) a(t NN u, u) a(u, u). The bounds are also indeendent of β. Proof. We use Theorem 1: Assumtion i). For notational convenience, we introduce, for =,,l, the bilinear forms b (u, u ): V V R, defined by (13) b (u, u )= u(x 2 j )a(φ j,φ j)= a(uj,u j). j N j N We use a level decomosition given by u = j u j, with u j x j is the osition of the node j N. We decomose u V h (Ω) as u=hu+pu in Ω, = u (x j )φ j. Here, where (14) u = H (i) u + P (i) u in Ω i. Here, H (i) u is the discrete harmonic art of u, i.e. ( H (i) u, v) L 2 (Ω i ) = v V(Ω h i ), H (i) u=u on Ω i, and P (i) u V h(ω i )istheh 1 rojection associated with the sace V h(ω i ), i.e. ( (P (i) u), φ) L2 (Ω i ) =( u, φ) L2 (Ω i) φ V(Ω h i ). We decomose P (i) u and H (i) u searately. We start by decomosing v (i) = P (i) u in Ω i as where P (i) v (i) = P (i) v (i) +(P (i) 1 P (i) )v (i) + +(P (i) l P (i) l 1 )v(i), : V h (Ω i ) V (Ω i ), is the H 1 rojection defined by ( (P (i) v (i) ), φ) L 2 (Ω i ) =( v(i), φ) L2 (Ω i) φ V (Ω i ).

9 Multilevel Schwarz methods 321 P (i) We extend P (i) v (i) by zero to Ω\ Ω i, and also denote this extension by v (i). Thus, P (i) v (i) V (Ω). Let v (i) = P (i) v (i), v (i) =(P (i) P (i) 1 )v(i), =1,,l. Hence (15) v (i) = v (i) + v(i) 1 + +v(i) + +v (i) l 1 +v(i) l. We use the decomosition (15) for all i =1,,N.The global decomosition of v is eual to v (i) in Ω i, and is defined by (16) v = l v, v = = N i=1 v (i). We now decomose H u. Let w = H u u 1, where u 1 is defined in (11). We decomose w as where w = N, i=1 = I h (u ρ β i µ+ i,β) ū h i ρ β i µ+ i,β = I h (ρ β i µ+ i,β (u ū h i )) on Γ Ω and extended as a discrete harmonic function in each Ω j, j =1,,N. Here, I h is the standard linear interolant based on the h-triangulation of Γ.Itis easy to show that w(x) = i w(i) (x) x Ω. Note that the suort of is the union of the Ω j which have a vertex, edge, or face in common with Ω i. We decomose further as (17) = F ij + E l + V m, F ij Ω i E l Ω i V m Ω i where F ij, E l, and V m are the faces, edges, and vertices of Ω i. Here, F ij, E l, and V m are the discrete harmonic extensions into Ω with ossibly nonzero interface values only on F ij,h, E l,h, and V m, resectively. The suort of each function on the right hand side of (17) is the union of a few Ω j and its interior is denoted by Ω Fij, Ω El,orΩ Vm, resectively. We can assume that the regions Ω Fij, Ω El, and Ω Vm are convex. If not, we can extend them to convex regions and use a tric develoed in Lemma 3.6. of Zhang [34]. We now decomose F ij, E l, and V m in the same way as the v (i). Let us first consider F ij. We obtain (18) F ij =,F ij + 1,F ij + +,F ij + + l 1,F ij + l,f ij.

10 322 M. Dryja et al. Here,F ij = P (i),f ij F ij,,f ij =(P (i),f ij P (i) 1,F ij ) F ij, =1,,l. P (i),f ij : V h (Ω F ij ) V (Ω F ij ), is the H 1 -rojection. As before, we extend P (i),f ij F ij by zero outside Ω Fij. (19) where and (2) Here, We decomose E l E l and V m in the same way, and obtain =,E l + 1,E l + +,E l + + l 1,E l + l,e l,,e l = P (i),e l E l,,e l =(P (i),e l P (i) 1,E l ) E l, =1,,l, V m =,V m + 1,V m + +,V m + + l 1,V m + l,v m.,v m = P (i),v m V m,,v m =(P (i),v m P (i) 1,V m ) V m, =1,,l. We can now define a global decomosition of the function w as (21) w = w + w 1 + +w + +w l 1 +w l, where w = (,F ij +,E l +,V m ). i F ij Ω i E l Ω i V m Ω i We chec straightforwardly that (21) is valid x Ω. Using (21) and (16), we define a decomosition for u V h as (22) u = u 1 + u + +u + +u l 1 +u l, where u = v + w, =,,l. We obtain the desired decomosition in Assumtion i) by decomosing u as in (13). We will now rove that (23) l b (u, u ) (1 + log(h /h)) 2 a(u, u). = 1 Therefore, in view of (13), we obtain C 2 = C (1+l)2 in Assumtion i) of Theorem 1. We start with the decomosition of v. Lemma 1. For the decomosition of v given by (16), we have (24) l b (v,v ) a(u,u). =

11 Multilevel Schwarz methods 323 Proof. We first show that (25) l = b (v (i),v(i) ) ρ i u 2 H 1 (Ω i). Note that for 1, v (i) Hence, =(P (i) b (v (i),v(i) ) 1 h 2 P (i) 1 )v(i) =(I P (i) 1 )v(i). ρ i v (i) 2 L 2 (Ω i) ρ i v (i) 2 H 1 (Ω i). The last ineuality follows from the well nown error estimate for H 1 rojections on convex domains; see Ciarlet [9]. For =, b (v (i),v(i) ) 1 h 2 ρ i v (i) 2 L 2 (Ω ρ i) i v (i) 2 H 1 (Ω, i) using Friedrichs ineuality. Adding the above ineualities, we obtain Thus l = ρ i {( P (i) v (i), v (i) ) L 2 (Ω i) + b (v (i),v(i) ) l =1 ( (P (i) P (i) 1 ) v(i), v (i) ) L 2 (Ω i )} = ρ i v (i) 2 L 2 (Ω i ) = ρ i P (i) u 2 L 2 (Ω i ) ρ i u 2 L 2 (Ω i ). N i=1 l = b (v (i),v(i) ) a(u,u). Lemma 2. For the decomosition of F ij, given by (18), we have (26) Proof. Note that l = b (,F ij,,f ij ) (1 + l) 2 ρ i u 2 H 1 (Ω i ). b (,F ij,,f ij ) 1 h 2 (ρ i + ρ j ),F ij L 2 (Ω Fij ). Here, Ω Fij = Ω i Ω j F ij. Note that,f ij =inω\ω Fij. Using the same arguments as in the roof of Lemma 1, we obtain (27) Note that l = b (,F ij,,f ij ) (ρ i + ρ j ) F ij 2 H 1 (Ω Fij ).

12 324 M. Dryja et al. (ρ i + ρ j ) F ij 2 H 1 (Ω Fij ) (ρ i + ρ j ) F ij 2 H 1/2 (F ij ) =(ρ i +ρ j ) (I h (ρ β i µ+ i,β(u ū h i ))) Fij,h 2 H 1/2 (F ij ) (1 + ρj ρ = ρ i ) i (1+( ρj ρ i ) β ) 2 (u ūh i ) Fij,h 2 ρ H 1/2 (F ij ) i (u ūi h ) Fij,h 2. H 1/2 (F ij ) The first ineuality above follows from an extension theorem for finite element functions given in [1]. A simle roof for this extension theorem is given in Lemma 14. Here, (v) Fij,h is the iecewise linear function on F ij T h such that (v) Fij,h = v at the nodal oints F ij,h and (v) Fij =on F ij,h. For the last ineuality, we have also used the fact that β 1/2. For β =, we use a limiting rocess. Let θ Fij be the discrete harmonic function, defined in Ω i, which euals 1 on F ij,h and zero on Ω i,h \F ij,h. Using Lemma 3 of Dryja and Widlund [17], which is an ineuality for H 1/2 (F ij ), and Poincaré s ineuality, we obtain ρ i (u ūi h ) Fij,h 2 ρ H 1/2 (F ij ) i (I h (θ Fij (u ūi h ))) H 1/2 ( Ω i) ρ i (I h (θ Fij (u ū h i ))) H 1 (Ω i ) and using Lemma 4.5 of Dryja et al. [12] and a Poincaré Friedrichs ineuality, we obtain ρ i (1 + log H /h) 2 u ū h i 2 H 1 (Ω i ) ρ i (1 + l) 2 u 2 H 1 (Ω i ). Combining this ineuality with (27), we obtain (26). The next two lemmas are roved in the same way as Lemma 2; see a similar argument in the roof of Theorem 4 Lemma 3. For the decomosition of E l, given in (19), we have (28) l = b (,E l,,e l ) (1 + l) ρ i u 2 H 1 (Ω i ). Lemma 4. For the decomosition of V m, given in (2), we have (29) l = b (,V m,,v m ) ρ i u 2 H 1 (Ω i). Corollary 1. For the decomosition of w, given in (21), we have (3) l b (w,w ) (1 + l) 2 a(u, u). = The roof of Corollary 1 follows from Lemmas 2, 3, and 4. We now estimate a(u 1, u 1 ).

13 Multilevel Schwarz methods 325 Lemma 5. For u 1 = i u(i) 1, u(i) 1 =ūh i ρ β i µ+ i,β, (31) a(u 1, u 1 ) (1 + log H /h)a(u, u) (1 + l)a(u, u) The roof of this result, for β =1/2, can be found in Dryja and Widlund [17]. For different values of β, we use an argument similar to that of the roof of Lemma 2; cf. also Saris [22]. Returning to the roof of Theorem 2, we find that (23) follows from Corollary 1, and Lemmas 1 and 5. The bound for C has then been established. Assumtion ii). Trivially, we have ω = 1. Assumtion iii) We need to show that ρ(e ) const. This has been established in Remar 3.3 in Zhang [34]. Remar 1. The analysis and results in this aer can also be extended to the cases in which we relace the bilinear forms b 1 NN (, ) by the inexact solvers introduced in Dryja and Widlund [17] A face based coarse sace The next exotic coarse sace is denoted by V 1 F V h, and is based on values on the wire baset W h and averages over the faces F ij. This coarse sace can conveniently be defined as the range of an interolation oerator Ih F : V h V 1 F, defined by Ih F u(x) Ω i = u(x )ϕ (x)+ ūf h ij θ Fij (x). W i,h F ij Ω i Here, ϕ (x) is the discrete harmonic extension, into Ω i, of the standard nodal basis function associated with a node. ūf h ij is the discrete average value of u over F ij,h. We define the bilinear form, in the standard way, from the norm given by b F 1(u, u) = i + H(1 + l) ρ i { W i,h h(u(x ) ū h i ) 2 F ij Ω i (ū h F ij ū h i ) 2 }, where ū h i is the discrete average value of u over Ω i,h. Theorem 3. Let T F be defined by (6). Then for any u V h (Ω), we have (1 + l) 2 a(u, u) a(t F u, u) a(u, u).

14 326 M. Dryja et al. Proof. Assumtion i). The decomosition here is the same as before, excet that we now choose = I h (ρ 1/2 i µ + i,1/2 (u I h F u)) on Γ Ω, and extend these boundary values as a discrete harmonic function elsewhere. We note that this decomosition is simler since vanishes on the wire baset. Therefore, E l = and V m =.A counterart of Lemma 2 holds since (ρ i + ρ j ) F ij 2 H 1 (Ω Fij ) (ρ i + ρ j ) (I h (ρ 1/2 i µ + i,1/2 (u I F h u))) Fij,h 2 H 1/2 (F ij ) ρ i (u ū h F ij ) Fij,h 2 H 1/2 (F ij ) ρ i (1 + log H /h) 2 u ū h F ij 2 H 1 (Ω i ) ρ i (1 + log H /h) 2 u 2 H 1 (Ω i ) ρ i (1 + l) 2 u 2 H 1 (Ω i ). Here we have used the same arguments as in the roof of Lemma 2. Finally, Lemma 5 is relaced by Lemma 6. For u V h (Ω) b F 1(I F h u, I F h u) (1 + log H /h)a(u, u) (1 + l)a(u, u). The roof of this result can be found in the roof of Theorem 6.7 of Dryja et al. [12]. Assumtion ii). We have ω 1; see Dryja et al. [12]. Assumtion iii). As in Subsect Remar 2. Another ossible decomosition for w is given by w = w Fij. F ij Γ Here, w Fij is the discrete harmonic function on Ω with ossibly nonzero interface values only on F ij,h. We note that the suort of w Fij is Ω Fij. We can decomose as in (18), and obtain w Fij b (w,fij,w,fij ) (ρ i + ρ j ) w Fij 2 H 1/2 (F ij ) =(ρ i +ρ j ) (u ū h F ij ) Fij,h 2 H 1/2 (F ij ) (1 + log H /h)2 u ū h F ij 2 H 1 ρ (Ω F ij ) (1 + log H /h) 2 u 2 H 1 ρ (Ω) (1 + l)2 u 2 H 1 ρ (Ω).

15 Multilevel Schwarz methods An edge based coarse sace We can decrease the dimension of V 1 F and define another coarse sace. Rather than using the values at all the nodes on the edges as degrees of freedom, only one degree of freedom er edge, an average value, is used. The resulting sace, denoted by V 1 E V h, is the range of the interolation oerator I h E : V h V 1 E, defined by Ih E u(x) Ω i = u(v m )ϕ m (x) V m Ω i + ū h E l θ El (x)+ ū h F ij θ Fij (x). E l W i F ij Ω i Here, ūe h l is the discrete average value of u over E l,h, and θ El the discrete harmonic function which euals 1 on E l,h and is zero on Ω i,h \ E l,h. ϕ m (x) is the discrete harmonic extension into Ω i of the boundary values of standard nodal basis function associated with the vertex V m. We define a bilinear form by b 1(u, E u) = ρ i {h (u(v m ) ūi h ) 2 i V m Ω i +H (ūe h m ūi h ) 2 +H(1 + l) (ūf h ij ūi h ) 2 }. E m Ω i F ij Ω i Theorem 4. Let T E be defined by (6). Then, for any u V h (Ω), we have (1 + l) 2 a(u, u) a(t E u, u) a(u, u). Proof. Assumtion i). Here, we choose = I h (ρ 1/2 i µ + i,1/2 (u I h E u)) on Γ Ω and extend these boundary values as a discrete harmonic function elsewhere. Note also that, we have V m =.The roof of a counterart of Lemma 2 is similar to that given in the roof of Theorem 3. The roof of a variant of Lemma 3 for this case roceeds as follows. Let E l be the edge comonent of, given similarly as in (17), and let,e l be the decomosition of E l given as in (19). Using similar arguments as in Lemma 2, we have l = b (,E l,,e l ) m ρ m E l 2 H 1 (Ω El ), where the sum m is taen over all substructures, which share the oen edge E l. We now use the fact that µ + i,1/2 =( m ρ1/2 m ) 1 on E l,h, and an inverse ineuality, and obtain ρ m E l 2 H 1 (Ω El ) ρ i h (ūe h l u(x )) 2. m E l,h We next use a Sobolev tye ineuality to obtain

16 328 M. Dryja et al. (32) ρ i E l,h h (ū h E l u(x )) 2 ρ i (1 + log H /h) u 2 H 1 (Ω i ). Results very similar to (32) can be found in Bramble et al. [3], Bramble and Xu [6], Dryja [1], Dryja et al. [12], and Dryja and Widlund [17]. Therefore, (33) ρ m E l 2 H 1 (Ω El ) ρ i (1 + l) u 2 H 1 (Ω i ). m Finally, we use Lemma 7. For u V h b E 1(I E h u, I E h u) (1 + l) a(u, u). The roof of this result can be found in the roof of Theorem 6.1 of Dryja et al. [12]. Assumtion ii). We have ω 1; see Dryja et al. [12]. Assumtion iii). As in Subsect Remar 3. We can also simlify the roof by decomosing w as w = + w El. F ij Γ w Fij E l Γ Here, w Fij is chosen as in Remar 2 and w El is the iecewise discrete harmonic function with ossibly nonzero interface values only on E l,h. We note that the suort of w El is in Ω El. We decomose w El as in (19) and obtain ρ m w El 2 H 1 (Ω El ) ρ m h (ūe h l u(x )) 2 m m E l,h (1 + l) u 2 H 1 ρ (Ω E l ) A wire baset based coarse sace Finally, we consider a coarse sace V 1 W V h, due to Smith [24]. It is based only on the values on the wire baset W h. The interolation oerator Ih W : V h V 1 W, and is defined by Ih W u(x) Ω i = u(x )ϕ (x)+ ū F h ij θ Fij (x). W i,h F ij Ω i Here, ū F h ij is the discrete average value of u on F ij,h. Let ūw h i be the discrete average value of u on W i,h. We define the bilinear form by b 1(u, W u) =(1+l) ρ i h(u() ūw h i ) 2. i W i,h

17 Multilevel Schwarz methods 329 Theorem 5. Let T W be defined by (6). Then, for any u V h (Ω), we have (1 + l) 2 a(u, u) a(t W u, u) a(u, u). Proof. Assumtion i). Let = I h (ρ 1/2 i µ + i,1/2 (u I h W E l = and V m u)). Therefore, we have =.The roof of the counterart of Lemma 2 is as follows: (ρ i + ρ j ) F ij 2 H 1 (Ω Fij ) ρ i ((u ū h F ij ) θ Fij ) Fij,h 2 H 1/2 (F ij ) ρ i { (u θ Fij ) Fij,h 2 H 1/2 (F ij ) +(ūh F ij ) 2 (θ Fij ) Fij,h 2 } H 1/2 (F ij ) ρ i {(1 + log H /h) 2 u 2 H 1 (Ω i ) +(1+logH/h) u 2 L 2 ( F ij ) } (34) ρ i (1 + log H /h) 2 u 2 H 1 (Ω i ) ρ i (1 + l) 2 u 2 H 1 (Ω i ) Here we have used ideas of the roof of Lemma 2, and the following results: (35) (36) θ Fij 2 H 1 (Ω i ) H (1 + log H /h), (ū h F ij ) 2 1 H u 2 L 2 ( F ij ), and (37) u 2 L 2 ( F ij ) (1 + log H /h) u 2 H 1 (Ω i ). For the roof of (35), see Lemma 4.4 of [12]. The roof of (36) is a direct conseuence of Schwarz ineuality. The roof of (37) is related to the roof of (32). To get the seminorm bound in (34), we use the same arguments as for (33). Finally, we also use Lemma 8. For u V h(ω) b 1(I W h W u, Ih W u) (1 + log H /h) 2 a(u, u) (1 + l) 2 a(u, u). The roof of this result can be found in the roof of Theorem 6.4 of Dryja et al. [12]. Assumtion ii). We have ω 1; see Dryja et al. [12]. Assumtion iii). As in Subsect Remar 2 also alies in this case. 5. Secial coefficients and an otimal algorithm In this section, we show that if the coefficients ρ i satisfy certain assumtions, the L 2 ρ rojection is H 1 ρ -stable and we can use the sace of iecewise linear functions V H (Ω) as a coarse sace and obtain an otimal multilevel reconditioner. It should be ointed out that the L 2 ρ rojection is not H 1 ρ -stable in general; see the counterexamle given in Xu [29].

18 33 M. Dryja et al Quasi-monotone coefficients Let V m, m =1,,L, be the set of substructure vertices including those on Ω. We denote by Ω mi, i =1,,s(m), the substructures that have the vertex V m in common, and let ρ mi denote their coefficients. Let Ω Vm be the interior of the closure of the union of the substructures Ω mi, i.e. the interior of s(m) i=1 Ω mi. By using the fact that all substructures are simlices, we see that each Ω mi has a whole face in common with Ω Vm. Two-dimensional illustrations of Ω Vm = i Ω mi are given by Fig. 1 and Fig. 2. Ω Ω m2 m1 V Ω m3 m Ω m6 Ω m4 Ωm5 Fig. 1 V Ω m3 m Ω Ω m2 m1 Fig. 2 Definition 1. For each Ω Vm, order its substructures such that ρ m1 = max i:1,,s(m) ρ mi. We say that a distribution of the ρ mi is uasi-monotone in Ω Vm if for every i =1,,s(m), there exists a seuence i j, j =1,,R, with (38) ρ mi = ρ mir ρ mij+1 ρ mij ρ mi1 =ρ m1, where the substructures Ω mij and Ω mij +1 have a face in common. If the vertex V m Ω, then we additionally assume that Ω m1 Ω contains a face for which V m is a vertex. A distribution ρ i on Ω is uasi-monotone with resect to the coarse triangulation T if it is uasi-monotone for each Ω Vm.

19 Multilevel Schwarz methods 331 We also define uasi-monotonicity with resect to a triangulation T,as before, by relacing the Ω i and the substructure vertices V m by elements τj and nodes in N, resectively. Remar 4. The analysis and results can easily be extended to the case in which ρ mi = ρ mir ρ mij+1 ρ mij ρ mi1 =ρ m1. In two dimensions, uasi-monotonicity with resect to T imlies, for the same distribution of the ρ i, uasi-monotonicity with resect to T. We can show this as follows. The nodes of N divide into three sets: i) those which coincide with vertices of the substructures (nodes of the coarse triangulation), ii) those which belong to edges of the substructures, and iii) those which belong to the interior of the substructures. By examing the three cases, it is now easy to see that a distribution of the coefficients is uasi-monotone with resect to T if it is uasi-monotone with resect to T. In three dimensions there are cases in which a distribution of ρ i is uasimonotone with resect to T but not uasi-monotone with resect to T.In this case, the nodes of N are divided in four sets: those at vertices, edges, faces, and interiors of the substructures. There are no roblems for those of the vertices, faces and interior sets but there can be comlications with the edge set of nodes. Quasi-monotonicity of the coefficients for nodes belonging to the edge set does not follow from the uasi-monotonicity with resect to the coarse triangulation. To see that, let E l be an edge of a substructure Ω i, and let V m be a vertex of Ω i and an end oint of E l. There are more substructures sharing V m than substructures sharing the whole edge E l. From this observation, it easy to distribute coefficients in such a way that they are uasi-monotone with resect to the coarse triangulation but not uasi-monotone with resect to a finer triangulation. We note that in Theorem 6 only the uasi-monotonicity of the coefficients ρ i with resect to T is needed. We have introduced uasi-monotonicity with resect to T only to obtain a comlete theory for the stability of the L 2 ρ- rojection on finer levels; see Lemma 9 and Corollary A new interolator We define an interolation oerator I M : V h (Ω) V (Ω), as follows. This oerator is central in our study of the roerties of the weighted L 2 -rojection. Definition 2. Given u V h (Ω), define u = I M u V (Ω) by the values of u at two sets of nodes of N : i) For a nodal oint P N, let u (P) be the average of u over an element τj P T. ii) For a nodal oint P N Ω, let u (P) be the average of u over τ j P Ω.

20 332 M. Dryja et al. Here, τj P is the element, or one of the elements, with the vertex P with the largest coefficient ρ i. N Ω is the set of nodes of N which belong to Ω, and N = N \N Ω. It is easy to see that, for any constant c, I M (u c) =IM (u) c, and also that u vanishes on Ω whenever u vanishes on Ω. We note that τ j P Ω is a face of τj P for a uasi-monotone distribution of coefficients ρ i with resect to the triangulation T, but that τ j P Ω might be just an edge or a vertex for coefficients that are not uasi-monotone with resect to T. Lemma 9. For a uasi-monotone distribution of coefficients ρ i with resect to the triangulation T, we have u V h (Ω) (39) and (4) Here, τ,m j τ,ext j of this lemma. τ,ext j common with τ j. Furthermore, Proof. We have (I I M )u L 2 ρ (τ j ) h u H 1 ρ ( τ,m j ) τ j T, I M u H 1 ρ (τ j ) u H 1 ρ ( τ,m j ) τ j T. is a connected Lischitz region given exlicitly in the roof is the union of the τ i I M u V (Ω) if u V h (Ω). u I M u 2 L 2 ρ (τ j ) = ρ(τ j ) u I M u 2 L 2 (τ j ) which have a vertex, edge, or face in ρ(τj )( I M u c 2 L 2 (τj ) + u c 2 L 2 (τj )). Here ρ(τj ) is the value of ρ in the element τj. Using the definition and roerties of I M, we obtain I M u c 2 L 2 (τj ) = I M (u c) 2 L 2 (τj ) h 3 (I M (u c))(p) 2. P τj Here, each P is a vertex of the element τj. For a case in which P N, I M u(p), the average value of u over an element τj P, can be bounded from above in terms of the L 2 norm of u in τj P, i.e. h 3 (I M (u c))(p) 2 u c 2 L 2 (τ j P ). Here, τj P is the element given in Definition 2. For a case in which P N Ω, I M u(p), the average value of u over a triangle τ j P Ω, can be bounded from above in terms of the energy norm (4) of u in τj P. Indeed,

21 Multilevel Schwarz methods 333 h 3 (I M (u c))(p) 2 h u c 2 L 2 ( τ j P Ω) h2 u c 2 H 1 (τ j P ). From the definition of uasi-monotonicity with resect to the triangulation T, there exists for each P a seuence of elements τ j i, i =1,,n, with (41) ρ(τ j )=ρ(τ j n ) ρ(τ j 2 ) ρ(τ j 1 )=ρ(τ j P ). Let τ,m j P = n i=1 τ j n and τ,m j = P τ j τ,m j P. Then, u I M u 2 L 2 ρ (τ j ) ρ(τ j ) h 2 u c 2 H 1 ( τ,m j ). Note that τ,m j is a connected Lischtz region with a diameter of order h. Thus, we can use Poincaré s ineuality to obtain inf ρ(τ c and use (41) to obtain To obtain (4), we use j ) h 2 u c 2 H 1 ( τ,m j ) ρ(τ j )h 2 u 2 H 1 ( τ,m j ), ρ(τj ) h 2 u 2 H 1 ( τ,m ) h2 u 2. H j ρ 1( τ,m ) j I M u 2 H 1 ρ (τ j ) 4 i=1 ρ(τ j ) h (I M (u c))(p i ) 2. Here, the P i are the vertices of the element τj. For the rest of the roof, we use the same arguments as before. Corollary 2. For a uasi-monotone distribution of coefficients ρ i with resect to T, we have (42) (I Q ρ)u L 2 ρ (Ω) h u H 1 ρ (Ω) u V h (Ω), and (43) Q ρu H 1 ρ (Ω) u H 1 ρ (Ω) u V h (Ω). Here, Qρ is the weighted L 2 -rojection from V h (Ω) to V (Ω). Proof. We obtain (42) from (39), since Qρ gives the best aroximation with resect to L 2 ρ(ω). Finally, we note that (43) follows from (42); see Theorem 3.4 in Bramble and Xu [6]. Remar 5. Lemma 9 and Corollary 2 can easily be extended to functions which do not vanish on the whole boundary Ω. Using Lemma 9, we can also establish otimal multilevel algorithms for roblems with Neumann or mixed boundary conditions, and uasi-monotone coefficients with resect to T.

22 334 M. Dryja et al An otimal algorithm We rove that the MDS algorithm, using the sace of iecewise linear functions, V, as a coarse sace, is otimal if the coefficient is uasi-monotone with resect to the coarse mesh T. It is imortant to note that to rove our next theorem, we do not need to have uasi-monotonicity with resect to the fine meshes T. Theorem 6. Let T be defined by (6) with V 1 = V = V H,b 1(, )=a(, ). For a uasi-monotone distribution of the coefficients ρ i with resect to T,we have a(t u, u) a(u, u) u V h (Ω). Proof. We only need to consider Assumtion i); Assumtions ii) and iii) have been checed in the roofs of the revious theorems. Let the {θ m } be a artition of unity over Ω with θ m C (Ω V m ). Because of the size of the overla of the subregions Ω Vm, these functions can be chosen such that θ m is bounded by C /H. We decomose w = u I Mu as (44) L w = w m, m=1 where w m = I h (θ m w). Here, I h is the standard linear interolant with resect to the triangulation T l. We note that w m =, on and outside of Ω Vm, m =1,,L. Using standard arguments, cf. Dryja and Widlund [13], we can show that w m 2 Hρ 1(Ω Vm ) w 2 Hρ 1(Ω )+ 1 Vm H 2 w 2 L 2 ρ (Ω ), Vm and by using Lemma 9, we obtain w m 2 Hρ 1(Ω Vm ) u 2 Hρ 1 ( Ω ext ). Vm Here, Ω ext V m is the closure of the union of Ω Vm and the Ω i which have a vertex, edge, or face in common with Ω Vm. By assumtion, we have uasi-monotone coefficients with resect to T. We now remove the substructure Ω m1 from Ω Vm obtaining Ω c m 1 = Ω Vm \ Ω m1. We decomose w m as w m = H (m) w m +(w m H (m) w m ). Here, H (m) w m is the iecewise discrete harmonic function on Ω m1 and Ω c m 1 that euals w m on Ω m1 Ω Vm. We stress that we use the weight ρ = 1 in obtaining this iecewise discrete harmonic function. We decomose H (m) w m as in (2), and obtain (45) H (m) w m =(H (m) w m ) l + +(H (m) w m ) 1 +(H (m) w m ). Therefore, by using that ρ m1 is maximal and the arguments in the roof of Lemma 2, we obtain

23 Multilevel Schwarz methods 335 l b ((H (m) w m ), (H (m) w m ) ) ρ m1 H (m) w m 2 H 1 (Ω Vm ) = (46) ρ m1 H (m) w m 2 H 1/2 ( Ω m1 Ω c m 1 ) ρ m 1 H (m) w m 2 H 1 (Ω m1 ) ρ m1 w m 2 H 1 (Ω m1 ) w m 2 Hρ 1(Ω ). Vm Let w m = w m H (m) w m. Using the triangular ineuality, we obtain w m H 1 ρ (Ω Vm ) w m H 1 ρ (Ω Vm ). Note that w m vanishes on Ω m1 Ω Vm. Therefore, we can decomose w m in Ω m1 and Ωm c 1, indeendently. For the decomosition in Ω m1, we have no difficulties, since we have constant coefficients. For the decomosition in Ωm c 1, we can try to remove the substructure with largest coefficient ρ mi in Ωm c 1, and reeat the analysis just described. It is easy to show that we can remove all substructures, recursively, if we have a uasi-monotone distribution with resect to T. We note that the argument in (46) is invalid if we do not have uasimonotone coefficients. Remar 6. In the roof of Theorem 6, we just need to carry out the analysis locally for each Ω Vm. In a case of uasi-monotone coefficients with resect to T and with the coarse saces V 1 NN, V 1 F or V 1 E, we can derive a bound on the condition number of the multilevel additive Schwarz algorithm that is linear with resect to the number of levels l. The analysis also wors if we use different coarse saces in different arts of the domain Ω. We can also use the coarse sace V and an exotic sace V 1 X simultaneously. The resulting multilevel algorithm is otimal if the coefficient is uasi-monotone, and is almost otimal with a condition number bounded in terms of (1 + l) 2 otherwise. The same arguments can also be used to rove that we also have an otimal multilevel algorithm with Neumann or mixed boundary condition and uasi-monotone coefficients. 6. Multilicative versions In this section, we discuss some multilicative versions of the multilevel additive Schwarz methods; they corresond to certain multigrid methods. Let X = NN,F,E, or W. Following Zhang [34], we consider two algorithms defined by their error roagation oerators (47) and (48) E J = = 1 E G =( l = j N (I P j ))(I ηt X 1), l l (I T )=( (I η = j N P j ))(I ηt X 1),

24 336 M. Dryja et al. where η is a daming factor chosen such that T H 1 ρ w<2. The roducts in the above exressions can be arranged in any order; different orders result in different schemes; see Zhang [34]. When the roduct is arranged in an aroriate order, the oerators E G and E J corresond to the error roagation oerators of V-cycle multigrid methods using Gauss-Seidel and damed Jacobi method as smoothers for the refined saces, resectively. By alying techniues develoed in Zhang [34], and Dryja and Widlund [17], we can show that the norm of the error roagation oerators E G H 1 ρ and E J H 1 ρ can be estimated from above by 1 C (1 + l) 2. In a case in which we have uasi-monotone coefficients and use the standard coarse sace V H,wecan establish that the V-cycle multigrid methods, given by (47) and (48), are otimal. 7. Aroximate discrete harmonic extensions A disadvantage of using the coarse saces V 1 X, with X = F, E, NN, and W, is that we have to solve a local Dirichlet roblem exactly for each substructure to obtain the discrete harmonic extensions. However, we can define new exotic coarse saces, denoted by V X 1, with X = F,Ẽ,ÑN, and W by introducing aroximate discrete harmonic extensions. They are given by simle exlicit formulas [12, 22] and have the same Hρ 1 stability estimates as the discrete harmonic extensions. Here we use strongly the fact that our exotic saces V X 1 have constant values at the nodal oints of the faces of the substructures. We rove that the MDS, with these new coarse saces, have condition number estimate roortional to (1+l) 2. Let C, =1,,4, be the barycenters of the faces F i of Ω i, and let V be the vertex of Ω i that is oosite to C. Let C be the centroid of Ω i, i.e. the intersection of the line segments connecting the V to the C. Let E l, l =1,2,3, be the oen edges of F i ; see Fig. 3. To aroximate the discrete harmonic function θ Fij in Ω i, we use the finite element function ϑ Fij introduced in the roof of Lemma 4.4 of Dryja et al. [12] (see also Extension 3 in Saris [22]), given by (see Fig. 3). Definition 3. The finite element function ϑ Fij V h (Ω i ) is given by the following stes: i) Let u ij (C )= 1 4. ii) For a oint Q that belongs to a line segment connecting C to C, = 1,,4, define u ij (Q) by linear interolation between the values u ij (C )= 1/4 and u ij (C )=δ j, i.e. by u ij (Q) =λ(q) 1 4 +(1 λ(q))δ j. Here λ(q) =distance(q, C )/distance(c, C ) and δ j δ j =, otherwise. =1ifj =and

25 Multilevel Schwarz methods 337 V C Q E 3 S E 2 C E 1 Fig. 3 iii) For a oint S that belongs to a triangle defined by the revious Q as a vertex, and the edge E l as a side oosite to Q, l =1,,3, let u ij (S )=u ij (Q). iv) Let ϑ Fij = Ih u ij, where Ih is the interolation oerator into the sace V h (Ω i ) that reserves the values of a function u ij at the nodal oints of Ω i,h \W i,h and set them to zero on W i,h. v) In Ω j, which has a common face F ij with Ω i, ϑ Fij is defined as in Ω i. Finally, ϑ Fij is extended by zero outside Ω Fij. Note that ϑ Fij V h, and that ϑ Fij =1 on Ω i,h \W i,h. F ij Ω i We remar that other extensions are also ossible; see, e.g. Extension 2 of [22]. Using ideas in the roof of Lemma 4.4 in [12], we obtain Lemma 1. θ Fij 2 H 1 (Ω i ) ϑ F ij 2 H 1 (Ω i ) H (1 + log H /h). For the wire baset contributions, we relace the iecewise discrete harmonic function W i,h u(x )ϕ,by W i,h u(x )φ l, where φ l is the standard nodal basis function associated with a node. Using the definition of φ l and a Sobolev tye ineuality (see Lemma 4.3 of [12]), we obtain

26 338 M. Dryja et al. Lemma 11. W i,h (u(x ) ū h i ) φ l 2 H 1 (Ω i ) u ūh i 2 L 2 (W i) (1 + l) u 2 H 1 (Ω i ). New exotic coarse saces V X 1, with X = F, Ẽ,ÑN, and W are introduced by combining the aroximate discrete harmonic functions ϑ Fij and W i,h u(x )φ l. We define V X 1 as the range of the following interolators I X h : V h V X 1 : Modified Neumann-Neumann coarse saces Ih ÑN u =ũ 1 = i ũ i 1= i ū h i ρ β i µ + i,β. Here, µ + i,µ = µ+ i,β on Γ h Ω and is extended elsewhere in Ω as an aroximate discrete harmonic function given by: µ + i,β(x) = µ + i,β(x )φ l (x)+ µ + i,β(f ij ) ϑ Fij (x) x. W i,h F ij Ω i A modified face based coarse sace I F h u(x) Ω i = W i,h u(x )φ l (x)+ F ij Ω i ū h F ij ϑ Fij (x). A modified edge based coarse sace I Ẽh u(x) Ω i = u(v m )φ l V m (x) V m Ω i + ū El φ l (x)+ ū Fij ϑ Fij (x). E l W i E l,h F ij Ω i A modified wire baset based coarse sace I W h u(x) Ω i = u(x )φ l (x)+ ū Fij ϑ Fij (x). W i,h F ij Ω i It is imortant to note that our aroximate discrete harmonic extensions recover constant functions, because φ l (x)+ ϑ Fij (x) =1 x Ω i. W i,h F ij Ω i We define the bilinear forms exactly as before, i.e. b X 1 = b X 1,

27 Multilevel Schwarz methods 339 and oerators T X 1 : V h V X 1,by Let b X 1 (T X 1 u,v)=a(u,v) v V X 1. T X = T X 1 + l = Theorem 7. For any u V h (Ω), we have j N P j. (1 + l) 2 a(u, u) a(t X u, u) a(u, u). Proof. Let us first consider a case with V F 1 as the coarse sace. Assumtion ii) Using the triangle ineuality, the exlicit formulas for the aroximate discrete harmonic functions, and Lemma 1, we obtain u 2 Hρ 1(Ωi ) ρ i { h (u(x )) 2 i W i,h + H (1 + l) (ūf h ij ) 2 } u V F 1. F ij Ω i We now use that the aroximate discrete harmonic extension recovers constant functions to obtain (49) a(u, u) b F 1 (u, u). Assumtion i) Note that the bilinear form b F 1 (u, u) deends only on the values of u on Γ h, and let ũ 1 = I F h u and u 1 = Ih F u. We can therefore use Lemma 6 to obtain (5) b F 1 (ũ 1,ũ 1 )=b 1(u F 1,u 1 ) (1 + l) a(u, u) u V h. We now modify the decomosition in the roof of Theorem 3 u = u 1 + uj j and construct a decomosition for the current theorem by u =ũ 1 +(u 1 ũ 1 )+ uj j =ũ 1 + ũj + uj =ũ 1 + vj. j j j Here we use that (u 1 ũ 1 ) vanishes on Γ h Ω h, and then decomose (u 1 ũ 1 ) as in Lemma 1 to obtain a(ũj,ũj ) a(u 1 ũ 1,u 1 ũ 1 ) a(ũ 1,ũ 1 ). j

28 34 M. Dryja et al. We now use (49) and (5) and obtain a(ũ 1,ũ 1 ) b F 1 (ũ 1,ũ 1 ) (1 + l)a(u, u). Finally, we use the roof of Theorem 3 to obtain a(uj, uj ) (1 + l) 2 a(u, u). j The roof of this theorem for V Ẽ 1 or V W 1 as the coarse sace is uite similar. Let us finally consider the case with V ÑN 1 as the coarse sace. For Assumtion ii), we trivially have ω = 1. Assumtion i) is handled exactly as before. The only nontrivial art is to show that (51) a(ih ÑN u, Ih ÑN u) (1 + l)a(u, u) u V h (Ω). The idea of the roof of (51) is the same as in Dryja and Widlund [17]. We reduce the estimates to bounds related to the vertices, edges, and faces and use Lemma 4.4 in [12], and Lemmas 1 and Nonuniform refinements We now consider finite element aroximation with locally nested refinement. Such refinements can be used to imrove the accuracy of the solutions of roblems with singular behavior which arise in ellitic roblems with discontinuous coefficients, nonconvex domains, or singular data. We note, in articular, that solutions of ellitic roblems with highly discontinuous coefficients are very liely to become increasingly singular when we aroach the wire baset. Nested local refinements have reviously been analyzed by Bornemann and Yserentant [1], Bramble and Pascia [2], Cheng [7, 8], Oswald [2, 21], and Yserentant [31, 32]. By nested local refinement we mean that an element, which is not refined at level j, cannot be a candidate for further refinement. Under certain assumtions on the local refinement, otimal multilevel reconditioners have reviously been obtained for roblems with nearly constant coefficients in two and three dimensions. For roblems in two dimensions with highly discontinuous coefficients, the standard iecewise linear function can be used as a coarse sace to design multilevel reconditioners. A bound on the condition number can be derived, which is indeendent of the coefficients, and which grows at most as the suare of the number of levels; see, e.g. Yserentant [32]. Here, we extend the analysis to the case where the coefficients are uasi-monotone with resect to the coarse triangulation or are highly discontinuous in two or in three dimensions. Let us begin with a shae regular but ossibly nonuniform coarse triangulation T = T, which defines substructures Ω i with diameters H i. It follows from shae regularity that neighboring substructures are of comarable size. We introduce the following refinement rocedure: For = 1,,l, subdivide all the tetrahedra τ 1 j T 1 into eight tetrahedra (see, e.g., Ong [19]);