Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36

Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. Bl. 25-A, 1113 Sofia, Bulgaria Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36

CONTENTS 1. Introduction 2. Non-conforming Crouzeix-Raviart FEs 3. Two-level hierarchical decompositions 4. Optimal preconditioner for the block B 11 (FR) 5. Optimal preconditioner for the block Ã11 (DA) 6. Concluding remarks Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 2/36

1. Introduction We study strategies to construct hierarchical basis functions (HB) multilevel preconditioners to solve algebraic systems arising from second order elliptic problems, discretized by the non-conforming Crouzeix-Raviart finite elements. To this end we follow the known framework for constructing HB preconditioners for conforming FEM. However, applying the latter framework and the existing theoretical results to non-conforming FEM is not straightforward as the classical construction of hierarchical preconditioner relies on a nested sequence of finite element spaces, in most cases related to nested grids, while: The non-conforming FEM on nested grids produces non-nested FE spaces. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 3/36

2. Crouzeix-Raviart FEs Consider the selfadjoint elliptic boundary value problem Lu (a(x) u(x)) = f(x) in Ω, u(x) = 0 on Γ D, (a(x) u(x)) n = 0 on Γ N, where Ω is a polygonal domain in R 2, f(x) is a given function in L 2 (Ω), a(x) = [a ij (x)] 2 i,j=1is a symmetric and uniformly positive definite matrix in Ω, n is the unit vector of outward normal to the boundary Γ = Ω, and Γ = Γ D Γ N. We assume that the entries a ij (x) are piece-wise smooth functions on Ω = Ω Ω. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 4/36

The weak formulation of the above problem reads as follows: for some given f, find u V H 1 D (Ω) = {v H1 (Ω) : v = 0 on Γ D }, which satisfies A(u, v) = (f, v) v H 1 D(Ω), where A(u, v) = Ω a(x) u(x) v(x)dx. Consider two partitionings of Ω: a coarse triangulation T H and a fine one T h, which is obtained by a regular refinement of T H. The partitioning T H is assumed to be aligned with the discontinuities of the coefficient a(x) so that over each element E T H the function a(x) is smooth. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 5/36

course nodes coarse nodes fine nodes fine nodes Crouzeix-Raviart non-conforming FEs We discretize the variational problem using the Crouzeix-Raviart FEs, i.e., we seek the solution in the finite dimensional space V h = {v L 2 (Ω) : v e is linear e T h, v is continuous at the midpoints of the edges ofe T h and v is zero at the midpoints on Γ D }. The nodal FE basis φ h i (i = 1,..., n h) in V h is naturally defined as φ h i being equal to unity at one midpoint m k in e and zero at the other two midpoints. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 6/36

The discrete formulation becomes: find u h V h, which satisfies A : 0 h (u h, v h ) = (f, v h ) v h V h, where A h (u h, v h ) = e T h e a(e) u h v h dx. Here a(e) is a piece-wise constant coefficient matrix, defined by the integral averaged values of a(x) over each triangle from the coarser triangulation T H. Let us note that in this way arbitrary large coefficient jumps across the boundaries between adjacent finite elements from T H are allowed. Using the nodal basis, the FEM problem reads as A h u h = b h, u h, b h R n h where A h = (a ij ), is the global stiffness matrix with entries a ij = A h (φ h i, φh j ). Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 7/36

We pose no restrictions on the mesh and/or coefficient anisotropy. As it is known, to derive estimates for the CBS constant γ, it suffices to consider an isotropic problem in an arbitrary shaped triangle, T. Let us denote the angles in T by θ 1, θ 2 and θ 3 = π (θ 1 + θ 2 ), where a = cot θ 1, b = cot θ 2 and c = cot θ 3. Without loss of generality, for each triangle T, we assume that θ 1 θ 2 θ 3, and then denote α = a/c and β = b/c. A simple computation shows that the standard nodal basis element stiffness matrix for Crouzeix-Raviart non-conforming linear elements A CR e coincides with that for the conforming linear elements A cl e, up to a scalar factor 4: A CR e = 2 b + c c b 1 + β 1 β c a + c a = 2c 1 1 + α α. b a a + b β α α + β Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 8/36

3. Two-level decompositions Consider now the two consecutive mesh refinements, T H and T h. To build a hierarchical preconditioner, we need a suitable decomposition of V h, V h = V 1 V 2. In the case of conforming FEM, one of the spaces (V 2 ) is naturally induced by the coarse mesh. For non-conforming FEM, V H V h and the direct construction with, say, V 2 V H is impossible. To overcome the latter difficulty, we consider two constructions of suitable hierarchical decompositions of the Crouzeix-Raviart spaces were introduced. Following the introduced terminology, Blaheta, Margenov, Neytcheva (2004), we refer them as: Two-level first reduce decomposition (FR); Two-level decomposition with differences and aggregates (DA). Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 9/36

3.1. Two-level FR decomposition 7 II θ 3 3 4 I 6 1 2 5 θ 1 θ 2 8 III 9 Crouzeix-Raviart macro-element On macroelement level, we have the space V (E) = span {φ 1,..., φ 9 }. Let the basis functions φ i correspond to the midpoints m i, ordered as shown in the Figure. The FR splitting V (E) = V 1 (E) V 2 (E) is defined as follows V 1 (E) = span {φ 1, φ 2, φ 3, φ 4 φ 5, φ 6 φ 7, φ 8 φ 9 }, V 2 (E) = span {φ 4 + φ 5, φ 6 + φ 7, φ 8 + φ 9 }. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 10/36

Using the transformation matrix J F R, J F R = 1 2 2I J 1 J 2, J 1 = 1 1 1 1, J 2 = 1 1 1 1 1 1, 1 1 we introduce a corresponding hierarchical basis ϕ E = { φ i } 9 i=1 = J F R ϕ E. The hierarchical macroelement stiffness matrix is then computed as  E = J F R A E J T F R, and the related global stiffness matrix is obtained as Âh = E T ÂE. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 11/36

The hierarchical stiffness matrix Âh admits the 2 2 block structure  h = Â11  12  21  22 } V 1 } V 2, where V 1, V 2 are associated with the locally introduced red FR splitting. The matrix  h can be also seen as having a block 3 3 structure:  h = Ā 11 Ā (0) 12 Ā (0) 21 Ā (0) 31 Ā (0) 22 Ā (0) 32 Ā (0) 13 Ā (0) 23 Ā (0) 33 } interior basis functions } half-difference basis functions } half-sum basis functions ( V 2 ) Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 12/36

For our purposes, however,  h is first decomposed into a 2 2 block form  h = Ā11 Ā 12 Ā 21 Ā 22 } interior basis functions, } rest. As a first step of the FR algorithm, the interior unknowns are eliminated and  h is reduced to its Schur complement B = Ā22 Ā21Ā 1 11 Ā12. Next we consider a two-level splitting of the matrix B, again in a block 2 2 form B = B 11 B 12 B 21 B 22 where the first block corresponds to the half-difference and the second block corresponds to the half-sum basis functions. The 2 2 block decomposition of B can now be used to construct two-level preconditioners, since the matrix B 22 is associated with the coarse grid.,. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 13/36

3.2. CBS constant of FR algorithm Let us consider the local eigenvalue problem S E v = λb E,22 v, v const = (c, c, c) T, c 0, where S E = B E,22 B E,21 B 1 E,11 B E,12. The minimum eigenvalue of B 1 E,22 S E is found to be of the form λ min (B 1 E,22 S E) = 5σ σ(σ 8αβ), σ = (α + 1)(β + 1)(α + β). 8σ The local problem is associated with the angles of the current triangle T, namely θ 1, θ 2 and θ 3 = π (θ 1 + θ 2 ), where a = cot θ 1, b = cot θ 2 and c = cot θ 3, assuming that θ 1 θ 2 θ 3, and then α = a/c and β = b/c. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 14/36

To estimate γ 2 E, we substitute σ, α, β by c i = cos θ i, using meanwhile s i = sin θ i, i = {1, 2, 3}. Applying a = (1 bc)/(b + c) in the form of ab + bc + ca = 1 we get (a + c)(b + c) = 1 + c 2 and therefore σc 3 = (1 + c 2 )(a + b). By definition, a + b = c 1 s 1 + c 2 s 2 = c 1s 1 +c 2 s 1 s 1 s 2 = s 3 s 1 s 2 1 + c 2 = 1 + c2 3 s 2. 3 Then σc 3 = 1 s 1 s 2 s 3 αβ σ = ab c 2 σ = abc c 3 σ = c 1c 2 c 3 s 1 s 2 s 3 c 3 σ =c 1c 2 c 3, that is γ 2 E= 3 8 + 1 8 1 8c1 c 2 c 3. Since c 1 c 2 c 3 > 1, 1 8c 1 c 2 c 3 < 9 which simply leads to γ 2 E < 3 4. Theorem 3.1. (Blaheta, Margenov, Neytcheva (2004)) The related FR constant in the strengthened CBS inequality is uniformly bounded with respect to both coefficient and mesh anisotropy, i.e., γ 2 < 3 4. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 15/36

3.2. Two-level DA decomposition This decomposition is referred to as (D)ifferentiation and (A)ggregation splitting. If φ 1,..., φ 9 are the standard nodal non-conforming linear finite element basis functions on the macroelement, then we define V (E) = span {φ 1,..., φ 9 } = V 1 (E) V 2 (E), V 1 (E) = span {φ 1, φ 2, φ 3, φ 4 φ 5, φ 6 φ 7, φ 8 φ 9 }, V 2 (E) = span {φ 1 + φ 4 + φ 5, φ 2 + φ 6 + φ 7, φ 3 + φ 8 + φ 9 }. II 7 θ 3 4 3 6 1 2 5 θ 1 θ 2 8 III I 9 Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 16/36

The related matrix J transforms the macroelement stiffness matrix into a hierarchical form à E = J E A E J T E = ÃE,11 à E,12 à E,21 à E,22 φ i V 1 (E) φ i V 2 (E). For the whole finite element space V h with the standard nodal finite element basis ϕ = {φ (i) h : i = 1,..., N h}, we can similarly construct a new hierarchical basis ϕ = ϕ 1 ϕ 2 ϕ 3 and a corresponding splitting V h = V 1 V 2, (i) V 1 = span{ φ h ϕ (i) 1 ϕ 2 }, V 2 = span{ φ h ϕ 3}, and respectively à h = JA h J T = Ã11 à 12 à 21 à 22 φ i V 1 φ i V 2. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 17/36

Again, the analysis of the related two-level method is performed locally, by considering the corresponding problems on macroelements. The obtained result is summarized in the following theorem, which is analogous to the estimate in the case of FR decomposition. Theorem 3.2. (Blaheta, Margenov, Neytcheva (2004)) Let us consider the two level DA splitting. Then the CBS constant is uniformly bounded with respect to both coefficients and mesh anisotropy, γ 2 3/4. The latter estimate is independent on the discretization (mesh) parameter h and possible coefficient jumps aligned with the finite element partitioning T H. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 18/36

Proof. 6 4 2 1 T 3 3 7 1 T 4 2 5 1 2 3 8 9 The reference coarse grid triangle and the macroelement Ê. Let V 1 (Ê), V 2 (Ê) be the two-level splitting for the reference macroelement and for u V 1 (Ê), v V 2 (Ê) denote d (k) = u Tk, δ (k) = v Tk. Then the relations between the 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 1 ) = v(p 4 ) = v(p 5 ), v(p 2 ) = v(p 6 ) = v(p 7 ), v(p 3 ) = v(p 8 ) = v(p 9 ), imply that d (1) + d (2) + d (3) + d (4) = 0, δ (1) = δ (2) = δ (3) = δ (4) = δ. T T Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 19/36

Hence, A h, Ê (u, v) = 4 k=1 T k a u vdx = 4 k=1 = aδ, d (1) + d (2) + d (3) d (4) aδ (k), d (k) = 2 aδ, d (4) 2 δ a d (4) a where = area(t k ), x, y = x T y, and x a = ax, x. Further, 3 d (4) 2 a = d (1) + d (2) + d (3) 2 a 3 d (k) 2 a leads to and Thus, 4 A h,ê (u, u) = k=1 A h,ê (u,v) 2 3 4 A h,ê (u,u) d (k) 2 a ( 1 + 1 3 A h, Ê (v, v) = 4 δ 2 a. k=1 1 3 4 A h,ê (v,v)= 4 ) d (4) 2 a A h,ê (u,u) A h,ê (v,v). Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 20/36

The following theorem is useful for extending the two-level to multilevel case. Theorem 3.3. (Blaheta, Margenov, Neytcheva (2004)) Let Ã22 be the stiffness matrix corresponding to the space V 2 with the basis ϕ 3 from the splitting DA and let A H be the stiffness matrix corresponding to the coarse discretization T H FE space V H, equipped with the standard nodal finite element basis {φ (k) H Proof. : k = 1,..., N H}. Then à 22 = 4 A H. Consider the nodal basis function φ (i) H V H and DA basis function Let both basis functions be equal to unity in the nodes belonging to the C-edges. Then for any macroelement E = 4 k=1 T k we get φ (i) h φ 3. d (1) i = d (2) i = d (3) i = d (4) i = 2 φ (i) H, where d(k) i = φ (i) h T k. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 21/36

4. Preconditioning of B 11 (FR) Recall that the top-left block of Ā 11 in FR algorithm is block-diagonal. After the elimination, one obtains a (6 6) element matrix B E, which constitutes the macroelement contribution to the matrix B. Next we split B E as: B E = B E,11 B E,12 B E,21 B E,22 }two-level half-difference basis functions }two-level half-sum basis functions The matrix block B E,11 in is found explicitly, namely, B E,11 = 2p q 3 q + 2(1 + β + β 2 ) q + 2 q 2 β 2 q + 2 3 q + 2(1 + α + α 2 ) q 2 α 2, q 2 β 2 q 2 α 2 3 q + 2 (α 2 + αβ + β 2 ) q = α + αβ + β, and p = 3(α + αβ + β) + 3(α 2 + αβ + β 2 ) + αβ(3α + 3β + 1). Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 22/36

The following relations are known to hold or can be derived from Lemma 2.1. (r1) α + αβ + β = α(β + 1) + β = 1 c 2 0, c = cot θ 3, i.e., αβ α + β; (r2) α 2 + αβ + β 2 = α 2 + β(α + β) 0. Consider the off-diagonal elements of B E,11. It is easy to observe that 1. B E,11 (1, 2) = α + αβ + β + 2 > 0 for all valid values of α and β 2. B E,11 (1, 3) = α αβ β 2 β 2 < 0 3. B E,11 (2, 3) = α αβ β 2 α 2 < 0 and 4. B E,11 (2, 3) B E,11 (1, 3) B E,11 (1, 2), and B E,11 (1, 3) = B E,11 (1, 2) only for β = 1. Remark 3.1. Note that the direction of the strongest off-diagonal coupling of the macroelement matrix B E,11 is the same as of the related conforming FE. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 23/36

4.1. Additive preconditioner θ 3 2 e θ 1 3 θ 2 1 2 E 3 1 (a) (b) Q (c) Figure: Dominating off-diagonal couplings for (a) the element stiffness matrix corresponding to e T h, (b) the macroelement matrix B E,11,and (c) the macroelement matrix B Q,11. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 24/36

We construct the additive preconditioner C E,11 to B E,11 by deleting the weakest off-diagonal couplings in B E,11, i.e., we let C E,11 = 2p q 3 q + 2(1 + β + β 2 ) q + 2 0 q + 2 3 q + 2(1 + α + α 2 ) 0. 0 0 3 q + 2 (α 2 + αβ + β 2 ) Then C 11 is obtained by assembling the modified element matrices C E,11. Lemma 3.1. For any element size and shape and for any anisotropy a(x), there holds that ( 1 ) 7/15 C E,11 B E,11 ( 1 + ) 7/15 C E,11. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 25/36

Proof. Consider the generalized eigenvalue problem B E,11 v = λc E,11 v and the corresponding characteristic equation for λ, det (B E,11 λc E,11 ) = 0. The determinant is found to be (1 λ)(3 q + 2(1 + β + β 2 )) (1 λ)(q + 2) q 2 β 2 (1 λ)(q + 2) (1 λ)(3 q + 2(1 + α + α 2 )) q 2 α 2. q 2 β 2 q 2 α 2 (1 λ)(3 q + 2 (α 2 + αβ + β 2 )) Straightforward computation shows that µ i = 1 λ i, i = 1, 2, 3 satisfy µ 1 = 0, i.e., λ 1 = 1 µ 2 2,3 = (α + β)(α + αβ + β + 2(2α2 αβ) + β 2 ) (α + β + 2)[2(α 2 + αβ + β 2 ) + 3(α + αβ + β)]. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 26/36

We show below that which expanded form becomes µ 2 2,3 7 15, E(α, β) 16α 3 +16β 3 34α 2 34β 2 34α 2 β 34αβ 2 82αβ 42α 42β 0 (α, β) D = { (α, β) : 1 2 < α 1, 0 < β 1, α + β > 0, α β}. Case 1: Let α = 0. In this case E(0, β) 16β 2 34β 2 42β = 18β 2 42β 0. Case 2: Let α > 0. Then 16α 3 + 16β 3 34β 2 16β 2 + 16β 2 34β 2 0 and the remaining terms in E are negative, so E 0. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 27/36

Case 3: Let 1 2 < α < 0. We use the fact that β 1, i.e., β2 β and that αβ α + β. E = 16α 3 + 16β 3 34α 2 34β 2 34αβ(α + β) 82αβ 42(α + β) 16α 3 + 16β 3 14αβ 42(α + β) 16(α 3 + β 3 ) + 14(α + β) 42(α + β) = 16(α + β)(α 2 αβ + β 2 ) 28(α + β) It remains to prove that 16(α 2 αβ + β 2 ) 28 or α 2 αβ + β 2 7/4. The latter is true, since sup α,β (α 2 αβ + β 2 ) = sup (α 2 α + 1) = 7/4. α ( 1/2,0) E achieves its maximum value 0 for (α, β) equal to (0, 0) and ( 1/2, 1). Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 28/36

The assembled global preconditioner C 11 inherits the properties of C E,11. We collect the results in the following theorem. Theorem 3.1. By deleting the two smallest off-diagonal elements in the local matrix B E,11 and assembling the modified element matrix, we construct a preconditioner C 11 to B 11 such that ( 1 ) 7/15 C 11 B 11 ( 1 + ) 7/15 C 11, κ(c 1 11 B 11) < 1 4 ( 11 + ) 105 5.31. The spectral equivalence and the related condition number estimate hold independently of element size and shape, and problem anisotropy as well. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 29/36

4.2. Multiplicative preconditioner Following the construction first proposed by Margenov, Vassilevski (1994) (see Axelsson, Margenov (2003) for the complete analysis in the case of conforming FE), we partition the nodes in the block B 11 into two groups, where the first one contains the centers of parallelogram superelements Q which are weakly connected in the sense that the off-diagonal couplings are relatively small. With respect to this partitioning, B 11 admits the two-by-two block-factored form B 11 = D 11 F 11 T F 11 E 11 = D 11 0 T F 11 S 11 I D 11 1 F 11 0 I. Then, the multiplicative preconditioner C 11 is defined as C 11 = D 11 0 T F 11 E 11 I D 11 1 F 11 0 I. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 30/36

Theorem 3.2. The multiplicative preconditioner C 11 of B 11 has an optimal order convergence rate with a relative condition number uniformly bounded by Proof. κ ( C 11 1 B 11 ) < 15 8 = 1.875. We again consider the generalized eigenvalue problem S 11:Q v Q = λ Q E 11:Q v Q. As it is seen from the analysis in the case of conforming FEM, λ (2) Q = λ(3) Q = λ(4) Q = 1, and λ(1) Q = 1 (µ (2,3)) 2. Here µ (2,3) = 1 λ (2,3) stand for the eigenvalues introduced in the analysis of the additive preconditioner to B 11. This immediately gives ( µ (2,3)) 2 7 < 13 λ (1) Q > 8 15 after what the proof of the theorem follows straightforwardly. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 31/36

4.3. Solvers for C 11 The computational complexity of solving systems with C 11 is determined by their connectivity pattern only. This means, that the optimal order direct solver considered in the case of conforming FEM are directly applicable here. For the additive algorithm, the matrix C 11 has a generalized tridiagonal structure, that is, it is tridiagonal under a proper ordering of the elements. For the multiplicative preconditioner of B 11, the optimal order direct solver incorporates the nested dissection (ND) algorithm for the reduced system. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 32/36

5. Preconditioner of Ã11 (DA) The block à 11 can be further decomposed by splitting the unknowns into two groups - associated with the interior (I), and associated with the sides (S) of the macroelements, à 11 = Ã11,II à 11,SI à 11,IS à 11,SS Elimination of the block Ã11,II corresponding to the inner nodes gives rise to the Schur complement. S = à 11,SS à 11,SI à 1 11,IIÃ11,IS S = B 11, where B 11 is the block appearing in the FR decomposition. Therefore, the optimal preconditioners C 11 of B 11 are directly applicable to construct C 11, namely C 11 = I 0 à 11,SI à 1 11,II I Ã11,II à 11,IS 0 C 11. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 33/36

Theorem 3.3. For any element size and shape and any problem anisotropy it holds that Proof. κ ( ) C 11 1 Ã11 < 1 4 (11 + 105), if C 11 is the preconditioner to B 11, based on a modified element matrix; κ ( C 1 11 Ã11 form; ) < 15 8, if C 11 is the preconditioner to B 11 of factorized the cost of the application of preconditioners in both cases is proportional to the number of unknowns. The spectral equivalence is readily seen from the following expressions C 11 = Ã11,II Ã 11,SI Ã 11,IS, Ã 11 = Ã 11,SI Ã 1 11,IIÃ11,IS+C 11 Ã11,II Ã 11,SI Ã 11,IS. Ã 11,SI Ã 1 11,IIÃ11,IS+B 11 Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 34/36

6. Concluding remarks To be able to construct multilevel hierarchical preconditioners for the non-conforming discretization we have to answer the question if the two-level splitting of the finite element spaces is recursively applicable, or in other words, how V (k) 2 relates to V (k 1). In the FR case, it turns out that the block B 22 approximates A H. The spectral relations of B 22 and A H can be studied locally. For the reference macroelement, we find numerically that c 1 v T B 22 v v T A H v c 2 v T B 22 v for all v, where c 1 = 0.5, c 2 = 0.75 for isotropic problems, 0.25 c 1 and c 2 1 for all considered anisotropies. In the DA case, Theorem 3.2. shows that Ã22 = 4A H, which enables the recursive extension of the two-level hierarchical construction to the multilevel version straightforwardly. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 35/36

References 1. Blaheta R, Margenov S, Neytcheva M. Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems, Numerical Linear Algebra with Applications 2004; 11:309-326. 2. Blaheta R, Margenov S, Neytcheva M. Robust optimal multilevel preconditioners for non-conforming finite element systems, Numerical Linear Algebra with Applications 2005; 12: 495-514. 3. Blaheta R, Margenov S, M. Neytcheva. Multilevel methods and preconditioners: An overview. Technical Report, Center of Excellence BIS-21 grant ICA1-2000-70016, June 2003. 4. Margenov S, Vassilevski PS. Two-level preconditioning of non-conforming FEM systems, Griebel, Iliev, Margenov, Vassilevski, eds., Large-Scale Scientific Computations of Engineering and Environmental Problems, Notes on Numerical Fluid Mechanics VIEWEG 1998: V 62:239-248. Optimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 36/36