Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems

Transcription

1 Uniform estimate of te constant in te strengtened CBS inequality for anisotropic non-conforming FEM systems R. Blaeta S. Margenov M. Neytceva Version of November 0, 00 Abstract Preconditioners based on various multilevel extensions of two-level finite element metods (FEM) lead to iterative metods wic ave an optimal order computational complexity wit respect to te size of te system. Suc metods were first presented in [0, ], and are based on (recursive) two-level splittings of te finite element space. Te key role in te derivation of optimal convergence rate estimates plays te constant γ in te so-called Caucy-Bunyakowski-Scwarz (CBS) inequality, associated wit te angle between te two subspaces of te splitting. It turns out tat only existence of uniform estimates for tis constant is not enoug and accurate quantitative bounds for γ ave to be found as well. More precisely, te value of te upper bound for γ (0, ) is a part of te construction of various multilevel extensions of te related two-level metods. In tis paper an algebraic two-level preconditioning algoritm for second order elliptic boundary value problems is constructed, were te discretization is done using Crouzeix- Raviart non-conforming linear finite elements on triangles. An important point to make is tat in tis case te finite element spaces corresponding to two successive levels of mes refinements are not nested. To andle tis, a proper two-level basis is considered, wic enables us to fit te general framework for te construction of two-level preconditioners for conforming finite elements and to generalize te metod to te multilevel case. Te major contribution of tis paper is te derived estimates of te related constant γ in te strengtened CBS inequality. Tese estimates are uniform wit respect to bot coefficient and mes anisotropy. Up to our knowledge, te results presented in te paper are te first for non-conforming FEM systems. Institute of Geonics, Czec Academy of Sciences, Studentska 768, Ostrava-Poruba, Te Czec Republic, blaeta@ugn.cas.cz Central Laboratory for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Boncev str, bl. 5A, Sofia, Bulgaria, margenov@parallel.bas.bg Department of Scientific Computing, Institute of Information Tecnology, Uppsala University, Box 0, SE Uppsala, Sweden, maya@tdb.uu.se

2 KEY WORDS: multilevel preconditioners, ierarcical basis, CBS constant. Introduction. Two-level preconditioners for conforming finite element discretizations In tis paper we consider te elliptic boundary value problem Lu (a(x) u(x)) = f(x) in Ω, u = 0 on Γ D, (a(x) u(x)) n = 0 on Γ N, () were Ω is a convex polygonal domain in R, f(x) is a given function in L (Ω), a(x) = [a ij (x)] i,j= is a symmetric and uniformly positive definite matrix in Ω, n is te outward unit vector normal to te boundary Γ = Ω, and Γ = Γ D Γ N. We assume tat te entries a ij (x) are piece-wise smoot functions on Ω. Te weak formulation of te above problem reads as follows: given f L (Ω) find u V HD (Ω) = {v H (Ω) : v = 0 on Γ D }, satisfying A(u, v) = (f, v) v HD(Ω), were A(u, v) = a(x) u(x) v(x)dx. () We assume tat te domain Ω is discretized using triangular elements. Te partition is denoted by T and is assumed to be obtained by a proper refinement of a given coarser triangulation T H. Te partition T H is aligned wit te discontinuities of te coefficient a(x) so tat over eac element E T H te function a(x) is smoot. Te variational problem () is ten discretized using te finite element metod, i.e., te continuous space V is replaced by a finite dimensional subspace V. Ten te finite element formulation is: find u V, satisfying A (u, v ) = (f, v ) v V, were A (u, v ) = a(e) u v dx. () e T Here a(e) is a piece-wise constant coefficient matrix, defined by te integral averaged values of a(x) over eac triangle from te coarser triangulation T H. We note tat in tis way strong coefficient jumps across te boundaries between adjacent finite elements from T H are allowed. Piece-wise linear finite elements are considered in te paper. For te standard conforming FEM, te nodal basis is associated wit te vertices of te triangles wile for te case of nonconforming Crouzeix-Raviart finite elements, te interpolation nodes are te mid points of te sides. Te resulting discrete problem to be solved is ten a linear system of equations Ω A u = F, (4) e

3 θ 6 θ 4 θ θ 5 θ θ (a) Refined mes (b) One macro-element (c) One element Figure : Conforming linear finite elements. wit A being te corresponding global stiffness matrix and being te discretization (messize) parameter for te underlying triangulation T of Ω. Te aim of tis paper is to investigate two-level preconditioners for solving te system (4). Te general setting and some well-known results for te case of conforming finite elements are summarized in te rest of tis section. Te next two sections are devoted to te study of two-level preconditioners for te case of non-conforming Crouzeix-Raviart finite elements. In Section we consider a uniform estimate of te strengtened CBS constant for te twolevel splitting, firstly introduced in [0]. Section analyses anoter two-level splitting, introduced recently in [4], wic allows for an easier extension to te multi-level case. Some concluding remarks are given at te end.. Te two-level setting We are concerned wit te construction of a two-level preconditioner M for A, suc tat te spectral condition number κ(m A ) of te preconditioned matrix M A is uniformly bounded wit respect to te messize parameter, te sape of triangular finite elements and arbitrary coefficient anisotropy. Te classical teory for constructing optimal order two-level preconditioners was first developed in [7, ], see also []. Te general framework requires to define two nested finite element spaces V H V, tat correspond to two consecutive (regular) mes refinements, as illustrated in Figure (c) and (b). Let T H and T be two successive mes refinements of te domain Ω, wic correspond to V H and V. Let {φ (k) H, k =,,, N H} and {φ (k), k =,,, N } be te corresponding standard finite element basis functions. We split te mespoints from T into two groups, {N H } T H and {N \H } T \T H, (5) wic latter are te newly added node-points. Next we define te so-called ierarcical basis functions { φ (k), k =,,, N } = {φ (l) H on T H} {φ (m) on T \T H }. (6) Let ten à be te corresponding ierarcical stiffness matrix.

4 Under te splitting (5) bot matrices A and à admit in a natural way a two-by-two block structure [ ] ] A A A = }N\H [à Ã, à A A }N = }N \H. (7) H à à }N H [ ] I 0 As is well-known, tere exists a transformation matrix J =, wic relates te nodal J I point vectors for te standard and te ierarcical basis functions as follows, ] [ ] [ṽ v ṽ ṽ = = J, = v. ṽ = J v + v ṽ v Remark. Clearly, te ierarcical stiffness matrix à is more dense tan A and terefore its action on a vector is computationally more expensive. Te transformation matrix J, owever, enables us in practical implementations to work wit A, since à = JA J T.. Two-level preconditioners and te strengtened Caucy-Bunyakowski- Scwarz inequality Consider a general matrix A, wic is assumed to be symmetric positive definite and partitioned as in (7). Te quality of tis partitioning is caracterized by te corresponding CBS inequality constant: γ = sup v R n n, v R n were n = N and n = N H. Consider now two preconditioners to A. v T A v, (8) (v T A v ) / (v T / A v ) (a) A preconditioner of block-diagonal (additive) form. Consider te following symmetric block-diagonal matrix as a preconditioner to A, [ ] B 0 M B = 0 B under te additional assumptions α A B α A and β A B β A for some positive constants α i, β i, i =,. Te latter inequalities are in a positive semidefinite sense. Under te above assumptions, tere olds te following general estimate of te spectral condition number of M B A : κ(m B A) α α ( γ ) [ ] (+θ ) + 4 ( θ ) + θ γ [ ] (+θ ) + 4 ( θ ) + θ γ, 4 (9)

5 were γ is te CBS constant in (8) and θ i = α i β i, i =,. Wen B = A and B = A, ten estimate (9) reduces to κ(m B A) + γ γ. (0) (b) A full block-matrix factorization preconditioner (multiplicative, or of block Gauss-Seidel form). Tis type of preconditioning is based on te exact block-matrix factorization of A, [ ] [ ] A 0 I A A A =, A S 0 I were S = A A A A is te exact Scur complement of A. Te multiplicative preconditioner is ten of te form [ ] [ ] A 0 I A A M F =, () A M S 0 I were M S is properly cosen. Assuming tat tere olds δ 0 A M S δ A, were γ < δ 0 δ, ten κ(m F A) δ γ δ 0 γ δ δ 0 γ. () Detailed proofs of (9), (0) and (), and analysis of oter versions of constructing M B and M F are found, for instance, in []. In te ierarcical bases context V and V are subspaces of te finite element space V spanned, respectively, by te basis functions at te new nodes {N \H } and by te basis functions at te old nodes {N H }. For te strengtened CBS inequality constant, tere olds tat γ = cos(v, V ) = A(u, v) sup () u V, v V A(u, u)a(v, v) were A(, ) is te bilinear form wic appears in te variational formulation of te original problem. Wen V V = {0}, te constant γ is strictly less tan one. As sown in [7], te constant γ can be estimated locally over eac finite element E T H, wic means tat γ = max E γ E, were γ E = sup u V (E), v V (E) A E (u, v) AE (u, u)a E (v, v), v const. Te spaces V k (E) above contain te functions from V k restricted to E and A E (u, v) corresponds to A(u, v) restricted over te element E of T H (see also [6]). 5

6 (a) Refined mes II 7 θ 6 5 θ θ 8 III 9 4 (b) One macro-element I θ θ θ (c) One element Figure : Crouzeix-Raviart non-conforming linear finite elements. Using te local estimates, it is possible to sow tat te value of γ depends on te type of te basis functions cosen for V and V but is independent of (for te -version of te ierarcical bases metod). Furter, it does not depend on te geometry of te domain Ω. It is also easily seen tat γ is independent of any discontinuities of te coefficients of te bilinear form A(, ), as long as tey do not occur witin any element of te coarse triangulation used. Te -independence means tat if we ave a ierarcy of refinements of te domain wic preserve te properties of te initial triangulation (refinement by congruent triangles, for example), ten γ is independent of te level of te refinement as well. For certain implementations, it is sown tat γ is independent of anisotropy (see [, 9, 9, 5, 5]). Hence, as long as te rate of convergence is bounded by some function of γ, it is independent of various problem and discretization parameters, suc as te ones mentioned above. We stress ere, tat te above tecnique is developed and straigtforwardly applicable for conforming finite elements and nested finite element spaces, i.e., wen V H V. First reduce (FR) two-level preconditioning for Crouzeix- Raviart systems. Te FR algoritm We consider now a finite element discretization, done using Crouzeix-Raviart non-conforming linear finite elements on triangles. Figure (b) illustrates a macro-element obtained after one regular mes-refinement step. We see tat in tis case NH E = {I, II, III} and N E = {,,, 9}, tus, obviously V H and V are not nested. Let us recall now, tat, as noted in [4], see also [8], it suffices to provide a local analysis for a reference (isosceles rigt-angled) triangle and arbitrary coefficients in te bilinear form, or, equivalently, for te model Laplace problem on arbitrary saped triangles. Te latter strategy is adopted in tis section, wile te alternative approac is used in Section. For te rest, te analysis will old for a general sape triangle wit angles θ, θ and θ = π (θ + θ ). Let a = cot θ, b = cot θ and c = cot θ. Witout loss of generality we can assume tat 6

7 θ θ θ. Remarkably enoug, te standard nodal basis element stiffness matrix for Crouzeix-Raviart non-conforming linear elements (see Figure (c)) A CR e coincides wit tat for te conforming linear elements A cl e (see Figure (c)), up to a scalar factor 4, namely, A CR e = 4A cl e. Terefore, we will furter work wit te following element stiffness matrix for an arbitrary saped triangle, b + c c b c a + c a = c + β β α + α b a a + b β α α + β were α = a/c and β = b/c. It is sown (cf. Lemma in [8]), tat te following relations old. (i) a = ( bc)/(b + c); (ii) if θ θ θ ten a b c; (iii) a + b > 0 (iv) Te true region of definition for α = a/c and β = b/c is D = {(α, β) R : } < α, 0 < β, α + β > 0, α β. β D 0.5 α=β α 0.5 α+β=0 Figure : Te true region of definition (α, β) D. Next, we follow [0] to define an algebraic two-level preconditioner. Let ϕ E = {φ i (x, y)} 9 i= be te macro-element vector of te nodal basis functions and A E be te macroelement stiffness matrix corresponding to E T. Te global stiffness matrix A can be written as A = E T A E. 7

8 were te summation is understood as te FEM assembly procedure. Let us introduce te following macroelement level transformation matrix J E =, (4) wic defines locally a two level ierarcical basis ϕ E, namely, ϕ E = J E ϕ E. Te ierarcical two-level macro-element stiffness matrix is ten obtained as and te related global stiffness matrix reads as à E = J E A E J T E, à = E T à E. We split now te two-level stiffness matrix à into block form à = à à à Ã, (5) were à corresponds to interior nodal unknowns wit respect to te macro-elements E T. Te first step of First Reduce (FR) algoritm is to eliminate tese unknowns. For tis purpose we factor Ã, i.e., à = à 0 à B à à I 0 I, (6) were B = à Ãà à stands for te Scur complement of tis elimination step. Next we consider a two level splitting of te matrix B in te block form B = B B B B, (7) were te first block corresponds to te alf-difference basis functions. Now it is easily seen tat te matrix B can be associated wit te coarse grid. It is important to note tat kerb E; = kera e = span{,, } wic allows us to apply a local analysis to estimate te constant γ corresponding te splitting defined by te block partition (7). For our analysis we proceed as follows: 8

9 Step : We observe tat te top-left block of à is block-diagonal (it corresponds to te interior points,,, cf. Figure (b) wic are not connected to nodes in oter macroelements), tus, it can be eliminated exactly and tis can be done locally. Terefore, we first eliminate tat block and obtain te (6 6) matrix B E. Next we split B E as [ ] BE, B B E = E, }two-level alf-difference basis functions B E, B E, }two-level alf-sum basis functions written again in two-by-two block form wit blocks of order ( ). Step : We are now in a position to estimate te CBS constant γ locally. We will utilize te leftand side inequality of te following result (given, for instance, in Lemma 9. (b) from []), namely, γ vt Sv v T B v for all v, were S = B B B B. To estimate γ, it suffices to find an upper bound for λ min (B E, S E).. Uniform estimate of te constant in te strengtened CBS inequality For te estimation of γ we ave used symbolic computations wit MAPLE. Wit te latter, all matrices A E, ÃE, B E and S E can be manipulated as functions of te parameters α and β. Let us consider te local eigenvalue problem S E v = λb E, v, v const = (c, c, c) T. (8) Te minimum eigenvalue of B E, S E is found to be of te form λ min (B E, S E) = 5σ σ(σ 8αβ), were σ = (α + )(β + )(α + β). (9) 8σ It is easily seen (cf. (i)-(iv)) tat σ > 0. Using (9), we will next prove tat λ min (B E, S E) 4, (0) wic was inted by corresponding numerical experiments in MATLAB. Indeed, it is readily seen tat inequality (0) is equivalent to αβ σ. () Relation () is obviously true for α 0. To sow tat same olds for α < 0, we introduce te auxiliary function Ψ(α, β) = (α + )(β + )(α + β) + αβ. () 9

10 Figure 4: Surface of te MATLAB computed values of γ varying (α, β) D. If Ψ as an extremum in a point ( α, β) in te interior of D, ten Ψ/ α = 0 at tis point and α = β + β +. (β + ) Consider now Ψ( α, β), 0 < β. After some formula manipulations we find tat Ψ ( α, β) = (β + β + )(β + 5β + ) 4 (β + ) wic is negative. Tus, Ψ( α, β) is a strictly decreasing function of β and attends its minimum on te boundary of D, at β =. Now, a simple analysis of Ψ(α, ) = α +α+ sows tat te minimum of Ψ for α ( /, 0) as to be taken for α = /. We find ten Ψ( /, ) = 0, tus, our proposition is confirmed. We find tus, tat We collect te above results in a teorem. γ λ min (B E, S E) 4. Teorem. Let te assumptions of te introduced FR two-level algoritm are fulfilled. Ten, te related constant in te strengtened CBS inequality is uniformly bounded wit respect to bot coefficient and mes anisotropy by /4, namely, γ 4. () Tis result is also independent of te size of te problem (or te mes parameter ) and of possible coefficient jumps aligned wit te finite element partition T H. 0

11 Te obtained upper bound for γ is exact wic is also seen from te plot of te surface given in Figure 4. Having sown estimate (), from (9), (0) and () we can ten estimate te condition numbers of te preconditioned systems using te preconditioner M B and M F. For instance, estimate (0) becomes κ(m B A ) 7. (4) Two-level splitting by differentiation and aggregation (DA) for te Crouzeix - Raviart finite elements Splitting (4) is not te only one possible for discretizations wit Crouzeix - Raviart finite elements. Oter variants of two-level splittings are discussed in [4], referred to as differentiation and aggregation (DA) splittings. Te most important case of tese is defined and analysed in tis section. Te splitting is easily described for one macroelement E, see Figure (b). If φ,..., φ 9 are te standard nodal nonconforming linear finite element basis functions on te macroelement, ten we define V (E) = span {φ,..., φ 9 } = V (E) V (E), (5) V (E) = span {φ, φ, φ, φ 4 φ 5, φ 6 φ 7, φ 8 φ 9 }, (6) V (E) = span {φ + φ 4 + φ 5, φ + φ 6 + φ 7, φ + φ 8 + φ 9 }. (7) Using te transformation matrix J E, J E =, (8) te vector of te macroelement basis functions ϕ E = {φ i } 9 i= is transformed to a new ierarcical basis ϕ E = { φ i } 9 i = J E ϕ E. Accordingly, J transforms te macroelement stiffness matrix into a ierarcical form à E = J E A E J T E = [ ÃE, à E, à E, à E, ] φi V (E) φ i V (E). (9) For te wole finite element space V wit te standard nodal finite element basis ϕ = {φ (i) : i =,..., N }, we can similarly construct a new ierarcical basis ϕ = ϕ ϕ ϕ and a

12 corresponding splitting V = V V, (i) V = span{ φ ϕ (i) ϕ }, V = span{ φ ϕ } (0) Te new basis can be described in te following way. Denote te edges of te coarse triangular elements from T H as C edges and tese edges of te fine triangles from T, wic lie outside C edges, as F edges. Ten ϕ consists of te basis functions φ (i) corresponding to te nodes on F edges, ϕ consists from te differences φ (i) φ(j) corresponding to te pairs of nodes on C edges and ϕ consists from te aggregates φ (i) + φ(j) + φ(k) or φ (i) + φ(j) + φ(k) + φ (e) corresponding to te pairs of nodes on C edges and nodes on te F edges inside te coarse triangles adjacent to te given C edge, see Figure 5. Te transformation J suc tat ϕ = Jϕ, can be used for transformation of te stiffness matrix A to ierarcical form à = JA J T, wic allows preconditioning by te two-level preconditioners based on te splitting (0) ϕ = {φ (k) : k = 5, 6, 8, 9,, }, ϕ = {φ (i) φ(j) : (i, j) = (, ), (7, 4), (4, ), (, 0), (5, 6)}, ϕ = {φ (i) + φ(j) + φ(k) : (i, j, k) = (,, 9), (7, 4, 5), (, 0, ), (5, 6, 8)} {φ (4) + φ() + φ (6) + φ() }. Figure 5: Te ierarcical basis wit aggregations Now, we are in a position to analyze te constant γ = cos(v, V ) for te splitting (0). Again, as in te previous section, tis analysis is performed locally, by considering te corresponding problems on macroelements. We follow a procedure (cf. [5], [5]), wic sligtly differs from tat one used in te previous section in te sense tat we consider first te case of te reference rigt angle macroelement, see Figure 6. Let V (Ê), V(Ê) be te two-level splitting (5) - (7) for te reference macroelement Ê and for u V (Ê), v V(Ê) denote d(k) = d (k) (u) = u Tk, δ (k) = δ (k) (v) = v Tk. Ten te

13 6 4 T 7 T T T Figure 6: Te reference coarse grid triangle and te macroelement Ê. relations between te function values in some nodal points, namely u(p 4 ) = u(p 5 ), u(p 6 ) = u(p 7 ), u(p 8 ) = u(p 9 ) and v(p ) = v(p 4 ) = v(p 5 ), v(p ) = v(p 6 ) = v(p 7 ), v(p ) = v(p 8 ) = v(p 9 ), imply tat d () + d () + d () + d (4) = 0, () δ () = δ () = δ () = δ (4) = δ. () Hence, 4 4 A, Ê (u, v) = a u vdx = aδ (k), d (k) k= T k= k = aδ, d () + d () + d () d (4) () = aδ, d (4) δ a d (4) a were = area(t k ), x, y = x T y denotes te inner product in R, x a = ax, x. Furter, leads to and d (4) a = d () + d () + d () a 4 A, Ê (u, u) = k= d (k) a d (k) a k= ( + ) d (4) a (4) A, Ê (v, v) = 4 δ a. (5)

14 Tus, A, Ê (u, v) A 4,Ê(u, u) = 4 A 4,Ê(v, v) A (u, u),ê A, Ê (v, v). (6) In te case of arbitrary saped macroelement E we can use te affine mapping F : Ê E for transformation of te problem to te reference macroelement, for more details see e.g. [5], [5] and also te proof of te following teorem. Tis transformation canges te anisotropy will still old since te result (6) for te reference 4 of te problem, but te estimate γ E macroelement does not depend on anisotropy. Te obtained results are summarized in te following teorem, wic is analogous to Teorem.. Teorem. Let us consider te two level splitting wit aggregations (0). Ten te corresponding strengtened C.B.S. inequality constant γ is uniformly bounded wit respect to bot coefficients and mes anisotropy, γ /4. Te latter estimate is independent on te discretization (mes) parameter and possible coefficient jumps aligned wit te finite element partitioning T H. Te following teorem is useful for extending te two-level to multilevel preconditioners. Teorem. Let à be te stiffness matrix corresponding to te space V wit te basis ϕ from te splitting (0) and let A H be te stiffness matrix corresponding to te finite element space V H, corresponding to te coarse discretization T H, equipped wit te standard nodal finite element basis {φ (k) H : k =,..., N H}. Ten Proof Let x, y R N H. Ten were u H A H x, y = = i x i φ (i) H, v H à = 4 A H. (7) E T H A E (u H, v H ), Ãx, y = = i y i φ (i) H, u = Now, we sall sow tat 4A E (u H, v H ) = φ (i) ϕ e T, e E E T H x i φ(i), v = A e (u, v). e T, e E φ (i) ϕ y i φ(i). A e (u, v), 4

15 For te reference macroelement Ê, we get AÊ(u H, v H ) = 4 aδ(u H ), δ(v H ) 4 A Tk (u, v) = ad (k) (u), d (k) (v) T k were = area(t k ), δ(u H ) = u H, δ(v H ) = v H, d (k) (u) = u Tk, d (k) (v) = v Tk. Here T k are obtained by subdividing of Ê into four congruent triangles. It is easy to sow tat k= d () (u) = d () (u) = d () (u) = d (4) (u) = δ(u H ), d () (v) = d () (v) = d () (v) = d (4) (v) = δ(v H ). Tis gives 4AÊ(u H, v H ) = T k Ê A Tk (u, v). (8) For an arbitrary saped macroelement E, we exploit te existing affine mapping F : Ê E wit constant Jacobian DF (x) = G for a transformation of te problem. A E (u H, v H ) = ÂÊ (û H, ˆv H ) = â û H ˆv H dx were â = det(g) G ag T, û H = u H F, ˆv H = v H F are from V H (Ê), Ê A e (u, v) = A Tk (û, ˆv) = 4 â û ˆv dx, e T, e E T k Ê k= T k were â is te same as above, û = u F, ˆv = v F are from V (Ê). As (8) olds for any anisotropy coefficients on Ê, a similar identity olds for any macroelement E. 4 Concluding remarks Tis study is strongly motivated by te expanding interest in nonconforming finite elements, wic are very elpful for solving problems, were te standard conforming elements suffer from te so-called locking effects. Te success of te Crouzeix-Raviart and oter non-conforming finite elements can be explained e.g. by te fact tat tey produce algebraic systems tat are equivalent to te Scur complement system for te Lagrange multipliers arising from te mixed finite element metod for Raviart-Tomas elements (see [6]). Tere are also oter advantages of te non-conforming Crouzeix-Raviart finite elements, as less density of te stiffness matrix 5

16 etc. In our study, te class of robust AMLI algoritms (see e.g. [, 9, 9]) for conforming linear elements forms te background for generalizations addressed to tis important non-conforming case. In tis paper, we presented new estimates for te constant in te strengtened CBS inequality for Crouzeix-Raviart FEM systems. Te obtained uniform estimates wit respect to bot mes and coefficient anisotropy are te first results of tis kind for non-conforming finite elements. Note tat te obtained value of te strengtened CBS inequality constant is te same as in te case of conforming linear triangular finite elements. At tis stage of te study we underline te following issues. Te DA algoritm allows for a direct extension of te γ estimate to te multi-level case, see also Teorem.. For te FR algoritm, te obtained estimates of te CBS constant γ for te considered two-level algoritms are not directly applicable to te multilevel case. Te estimate of te CBS constant γ guarantees uniform convergence rate of te related PCG algoritm but tis is not enoug to ave a robust algoritm. Te next important step is to construct optimal order preconditioners for te systems wit te first diagonal block matrix corresponding to te current two level splitting. Tis block is well-conditioned for model type problems but becomes increasingly ill-conditioned wen te problem coefficients become strongly anisotropic or, equivalently, wen te mes aspect ratio increases. Crouzeix-Raviart non-conforming finite elements are also known as a stable discretization tool for pure displacement elasticity problem in te almost incompressible case. In te conforming case, te same estimates for γ old for bot te scalar elliptic and te elasticity problem (see e.g. [, 8, 5, 5]). Tis is not generally true in te non-conforming case. As it was sown in [7], for te model case of isosceles triangles and Crouzeix-Raviart nonconforming linear basis functions γ < 4 uniformly wit respect to te Poisson ratio ν [0, /). Tis result was obtained for FR two-level algoritm but its extention to te case of a general triangulation seems to be muc arder from tecnical point of view. We expect tat te ere presented DA algoritm based on a direct aggregation and differentiation will be better suited for generalizations to te case of pure displacement and oter elasticity problems, wic is to be addressed in a fortcoming work. 5 Acknowledgments Tis work as been supported by te Center of Excellence BIS- grant ICA Te second autor was also supported in part by te Bulgarian NSF grant I-00/000. 6

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