Smoothers, mesh dependent norms, interpolation and multigrid

Applied Numerical Mathematics 43 (2002) 45–56 www.elsevier.com/locate/apnum

Smoothers, mesh dependent norms, interpolation and multigrid ✩ Susanne C. Brenner Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

Abstract New estimates for the nodal interpolation operator for the P1 finite element are established with respect to mesh dependent norms that are defined in terms of the smoothers in multigrid algorithms. These estimates are useful in the additive approach to the convergence of V -cycle multigrid algorithms and lead to results on the asymptotic behavior of the contraction numbers with respect to the number of smoothing steps.  2002 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Multigrid; Richardson; Damped Jacobi; Symmetric Gauss–Seidel; Mesh dependent norms; Interpolation estimates; Finite element

1. Introduction An additive theory for the convergence of multigrid V -cycle algorithms was recently developed in [1]. This theory is effective in determining the asymptotic behavior of the contraction numbers of V -cycle algorithms with respect to the number of smoothing steps. By combining the results of the new theory with the results of Zhang [2] and Bramble and Pasciak [3,4] for V -cycle algorithms with one smoothing step, the classical theorem of Braess and Hackbusch (cf. [5–7]) was generalized to second order elliptic boundary value problems with less than full elliptic regularity in [1]. These new techniques were applied in [8] to establish the convergence of nonconforming V -cycle and F -cycle algorithms with a sufficiently large number of smoothing steps. The Richardson iteration was employed as the smoother in both [1,8] for simplicity. In this paper we extend the results in [1] to the damped Jacobi iteration and the symmetric Gauss–Seidel iteration by deriving new estimates for the nodal interpolation operator, using mesh dependent norms that are defined in terms of the smoother. The rest of the paper is organized as follows. In Section 2 we describe the V -cycle algorithm for computing the P1 finite element solution of a model second order elliptic boundary value problem. Mesh ✩

This work was supported in part by the National Science Foundation under Grant No. DMS-00-74246. E-mail address: [email protected] (S.C. Brenner).

0168-9274/02/$22.00  2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 0 2 ) 0 0 1 1 7 - 4

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S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

dependent norms related to the Richardson, damped Jacobi and symmetric Gauss–Seidel iterations are introduced in Section 3, where a unified treatment of the smoothing and approximation properties of the corresponding multigrid algorithms is also presented. New estimates for the nodal interpolation operator for the P1 finite element are then established in Sections 4 and 5. Applications to the convergence of multigrid V -cycle algorithms are discussed in Section 6. Throughout this paper we use C (with or without subscripts) to denote a generic positive constant which is independent of grid sizes and grid levels. To avoid the proliferation of constants, we also use the notation φ
ψ (ψ  φ) to represent the inequality φ  Cψ. The notation φ ≈ ψ is equivalent to φ
ψ and φ  ψ. 2. The V -cycle algorithm for a model problem Let Ω ⊂ R2 be a bounded polygonal domain and consider the Poisson problem of finding u ∈ H01 (Ω) such that a(u, v) = F (v), where F ∈ H

v ∈ H01 (Ω),

(1)

−1

(Ω) and
a(v, w) = ∇v · ∇w dx,

∀ v, w ∈ H 1 (Ω).

(2)

Ω

It is well-known (cf. [9]) that there exists a number α ∈ (1/2, 1] such that F ∈ H −1+α (Ω) implies u ∈ H 1+α (Ω) and the following elliptic regularity estimate holds: uH 1+α (Ω)
F H −1+α (Ω) .

(3)

Let T1 be a triangulation of Ω and the triangulations Tk (k = 2, 3, . . .) be obtained from T1 through regular subdivisions. Let Vk ⊂ H01 (Ω) be the P1 finite element space associated with Tk , Vk be its dual space, hk = maxT ∈Tk diam T and Vk be the set of internal vertices in Tk . In particular we have the relation hk−1 = 2hk

for k = 2, 3, . . . .

(4)

The kth level discrete problem for (1) is to find uk ∈ Vk such that a(uk , v) = F (v),

∀ v ∈ Vk .

(5)

Using the canonical bilinear form ·, · between a vector space and its dual, we can define the operator Ak : Vk → Vk by

Ak v, w = a(v, w),

∀ v ∈ Vk .

(6) φk ∈ Vk

is the restriction of F to Vk . Then we can rewrite (5) as Ak uk = φk , where Let γ ∈ Vk . The V -cycle algorithm (cf. [10–12]) produces MGV (k, γ , z0, m1 , m2 ) as an approximate solution to the equation Ak z = γ ,

(7)

where z0 is the initial guess and m1 (respectively m2 ) is the number of pre-smoothing (respectively post-smoothing) steps. It is described in terms of the operators Ak , the natural injection operators k : Vk−1 → Vk , the operators Ikk−1 : Vk → Vk−1 defined by Ik−1     k−1 k v , ∀ α ∈ Vk , v ∈ Vk−1 , (8) Ik α, v = α, Ik−1

S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

47

and the preconditioners Bk−1 : Vk → Vk . We assume that Bk is symmetric positive definite, i.e., the bilinear form Bk v, w is symmetric positive definite. For k = 1 we take MGV (1, γ , z0, m1 , m2 ) = A−1 1 γ.

(9)

For k  2, we compute MGV (k, γ , z0, m1 , m2 ) in three steps. In the pre-smoothing step, we compute z1 , . . . , zm1 by z = z−1 + Bk−1 (γ − Ak z−1 )

for  = 1, . . . , m1 .

(10)

In the coarse grid correction step, we compute zm1 +1 by   k MGV k − 1, Ikk−1 (γ − Ak zm1 ), 0, m1 , m2 . zm1 +1 = zm1 + Ik−1

(11)

In the post-smoothing step we compute zm1 +2 , . . . , zm1 +m2 +1 by z = z−1 + Bk−1 (γ − Ak z−1 )

for  = m1 + 2, . . . , m1 + m2 + 1.

(12)

The final output of the algorithm is given by MGV (k, γ , z0, m1 , m2 ) = zm1 +m2 +1 .

(13)

Clearly the V -cycle algorithm is determined by the preconditioner Bk−1 . The choice of  n(p)v(p)w(p), ∀ v, w ∈ Vk

Bk v, w = Λ

(14)

p∈Vk

yields the geometrically consistent Richardson iteration as the smoother, where n(p) denotes the number of triangles in Tk containing p as a vertex and Λ is a scaling factor. If n(p) is replaced by 1, then we have the standard Richardson iteration. The choice of  a(ϕp , ϕp )v(p)w(p), ∀ v, w ∈ Vk (15)

Bk v, w = Λ p∈Vk

gives the damped Jacobi iteration as the smoother. Here ϕp ∈ Vk is the natural nodal basis function which equals one at the vertex p and vanishes at all other vertices, and the positive constant Λ is a scaling factor. In both (14) and (15), we choose Λ so that the spectral radius of Bk−1 Ak is less than one. Let nk = dim Vk and ϕ1 , . . . , ϕnk be the natural nodal basis of Vk corresponding to the vertices p1 , . . . , pnk in Vk , which is ordered in some fashion. Let Ak be the nk × nk matrix whose (i, j ) component is given by (Ak )i,j = a(ϕi , ϕj ) and D k (respectively Lk and U k ) be the diagonal (respectively strictly lower-diagonal and strictly upper-diagonal) part of Ak . Then the choice of

Bk v, w =

nk  

(D k + Lk )D −1 k (D k + U k )

 i,j

v(pi )w(pj ),

∀ v, w ∈ Vk

(16)

i,j =1

leads to the symmetric Gauss–Seidel iteration as the smoother. Note that the matrix Ak is symmetric positive definite, Ak = Lk + D k + U k

and

U k = Ltk .

General properties of these iterative methods can be found in [13,14].

(17)

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S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

3. Smoothers and mesh dependent norms For the analysis of the V -cycle algorithm it is convenient to introduce an inner product on Vk in terms of (the inverse of) the preconditioner Bk−1 (cf. [15,10,11]). Lemma 1. Let Bk be defined by (14), (15), or (16), and (·, ·)k be defined by (v, w)k = h2k Bk v, w,

∀ v, w ∈ Vk .

(18)

Then we have (v, v)k ≈ v2L2 (Ω) ,

∀ v ∈ Vk .

(19)

Proof. By simple scaling arguments (cf. [16]) we have  2 v(p) ≈ v2L2 (Ω) , ∀ v ∈ Vk , h2k

(20)

p∈Vk

a(ϕp , ϕp ) ≈ 1,

∀ p ∈ Vk .

(21)

For Bk defined by (14) or (15), the estimate (19) follows from (20) and (21). Let v ∈ Vk be arbitrary, v t = (v(p1 ), . . . , v(pnk )) be the vector containing the nodal values of v and (·, ·) be the Euclidean inner product. For Bk defined by (16), we have, by (21), 2   2 (22) (v, v)k = h2k D −1 k (D k + U k )v, (D k + U k )v ≈ hk (D k + U k )v , where  ·  is the Euclidean norm. Since the number of nonzero components on any row of L or U is bounded by a mesh-independent constant, it follows from (20), (21), (22) and the Cauchy–Schwarz inequality that (v, v)k
v2L2 (Ω) ,

∀ v ∈ Vk .

(23)

On the other hand, a simple calculation using (17) and (21) shows that  1  1  (D k + U k )v, v = (D k v, v) + (Ak v, v)  (D k v, v)  v2 , 2 2 which implies v
(D k + U k )v .

(24)

Combining (20), (22) and (24) we find 2 v2L2 (Ω) ≈ h2k v2
h2k (D k + U k )v
(v, v)k ,

(25)

and the equivalence (19) follows from (23) and (25).

∀ v ∈ Vk ,

✷

Since the dual space Vk can be identified with Vk through the inner product (·, ·)k , the V -cycle algorithm in Section 2 can be described without referring to Vk . Let the operator Ak : Vk → Vk be defined by (Ak v, w)k = a(v, w),

∀ v, w ∈ Vk .

(26)

S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

49

−1 Then (6), (18) and (26) imply Ak = h−2 k (Bk Ak ), and (7) can be rewritten as Ak z = g, where g = 2 −1 (hk Bk ) γ satisfies (g, v)k = γ , v for all v ∈ Vk . We can now denote the output of the V -cycle algorithm by MGV (k, g, z0, m1 , m2 ), and the smoothing step in (10) and (12) becomes

z = z−1 + h2k (g − Ak z−1 ).

(27)

→ Vk−1 be defined by Let     k−1 k w k , ∀ v ∈ Vk , w ∈ Vk−1 . Ik v, w k−1 = v, Ik−1 Ikk−1 : Vk

Then (8) and (28) imply that  k−1    Ik (γ − Ak v), w = Ikk−1 (g − Ak v), w k−1 ,

(28)

∀ v ∈ Vk , w ∈ Vk−1 .

We can therefore rewrite the coarse grid correction step (11) as   k MGV k − 1, Ikk−1 (g − Ak zm1 ), 0, m1 , m2 . zm1 +1 = zm1 + Ik−1

(29)

k and Ikk−1 . Thus the V -cycle algorithm can be expressed completely in terms of the operators Ak , Ik−1 Note that in this formulation the preconditioner does not appear in the description of the V -cycle algorithm. It enters the picture only through (18). Consequently it is possible to give a unified treatment for the smoothing and approximation properties of the V -cycle algorithms that use these three types of smoothers. Henceforth we assume that (·, ·)k is defined by (18), where Bk is defined by (14), (15) or (16). We define the mesh-dependent norms | · |s,k (cf. [17,15]) by

  (30) |v|s,k = Ask v, v k , ∀ s ∈ R, v ∈ Vk , k  1.

The effect of a single smoothing step described by (27) is measured by the operator Rk = I dk − h2k Ak , where I dk : Vk → Vk is the identity operator. Lemma 2 (Smoothing Properties). The following estimates hold: |Rk v|s,k  |v|s,k ,  m  R v 
h(t −s)m(t −s)/2|v|t,k , k k s,k

(31) (32)

for all v ∈ Vk , 0  t  s  2 and k, m  1. Proof. Note that Ak is symmetric positive definite with respect to (·, ·)k . The proofs of the estimates (31) and (32) are standard (cf. [17,15,10]) once we know that the spectral radius of h2k Ak (denoted by ρ(h2k Ak )) satisfies   (33) ρ h2k Ak  1. For the Richardson and damped Jacobi iterations, the estimate (33) holds because of the scaling factor Λ in (14) and (15). Adopting the convention in the proof of Lemma 1, for Bk defined by (16), we have by (17) and (26),     −1 2 (v, v)k = h2k (D k + Lk )D −1 k (D k + U k )v, v = hk D k + Lk + U k + Lk D k U k v, v   2 = h2k (Ak v, v) + h2k D −1 k U k v, U k v  hk (Ak v, v)k , which implies (33).

✷

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S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

k The effect of the coarse grid correction step (29) is measured by the operator I dk − Ik−1 Pkk−1 , where k−1 k−1 k is defined by a(Pk v, w) = a(v, Ik−1 w) ∀ v ∈ Vk , w ∈ Vk−1 , or equivalently, Pk is the restriction to Vk of the Ritz projection operator that maps H01 (Ω) onto Vk−1 .

Pkk−1

Lemma 3 (Approximation Property). It holds that     I dk − I k P k−1 v 
hsk |v|H 1 (Ω) , ∀ v ∈ Vk , 0  s  α, k−1 k 1−s,k

(34)

where α is the index of elliptic regularity in (3). Proof. From (2), Lemma 1, (26) and (30), we have |v|0,k ≈ vL2 (Ω)

and

|v|1,k = |v|H 1 (Ω) ,

∀ v ∈ Vk .

On the other hand we have the following estimates (cf. [18]) for the L2 (Ω)-orthogonal projection operator Qk which maps L2 (Ω) onto Vk : Qk vL2 (Ω)  vL2 (Ω) ,

∀ v ∈ L2 (Ω),

|Qk v|H 1 (Ω)
|v|H 1 (Ω) ,

∀ v ∈ H01 (Ω).

It then follows from interpolation (cf. [19,17,20]) that |v|s,k ≈ vH s (Ω) ,

∀ v ∈ Vk , 0  s  1,

(35)

s (Ω) is the space of functions on Ω whose trivial extensions belong to H s (R2 ). The estimate where H (34) follows from (3), (35) and a standard duality argument (cf. [17,1]). ✷ The key ingredients in the additive multigrid theory besides (31), (32) and (34) are the following estimates for the nodal interpolation operator Πk :   ∀ v ∈ Vk−1 , (36) (Πk v, Πk v)k  1 + θ 2 (v, v)k−1 + Cθ −2 h2k |v|2H 1 (Ω) ,   (37) (Πk−1 v, Πk−1 v)k−1  1 + θ 2 (v, v)k + Cθ −2 h2k |v|2H 1 (Ω) , ∀ v ∈ Vk , where 0 < θ < 1 is arbitrary and the positive constant C is independent of both θ and the meshes. These estimates were proved in [1] for the inner product corresponding to the Bk defined by (14). We will derive them for the inner products corresponding to the damped Jacobi and the Gauss–Seidel iterations in the next two sections, where we frequently use the simple estimate 2ab  θ 2 a 2 + θ −2 b2 ,

∀ a, b, θ ∈ R.

(38)

4. Estimates for the damped Jacobi iteration First we note that for the inner product (·, ·)J,Λ,k corresponding to the Jacobi iteration we have, in view of (15) and (18),    2 |ϕp,T |2H 1 (T ) v(p) , (39) (v, v)J,Λ,k = Λh2k T ∈Tk p∈VT

S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

51

where VT is the set of the vertices of T and ϕp,T is the polynomial of degree one that equals 1 at p and vanishes at the other two vertices of T . Note that |ϕp,T |H 1 (T ) ≈ 1,

∀ T ∈ T k , p ∈ VT

(40)

by a standard scaling argument. For T ∈ Tk−1 , we will denote by T1 , . . . , T4 the four triangles in Tk obtained by connecting the midpoints of the edges of T . We have the following elementary estimate: 4   

v(p) − v¯T

2


|v|2H 1 (T ) ,

∀ T ∈ Tk−1 , v ∈ Vk ,

(41)

i=1 p∈VTi

where v¯T = ( T v dx)/|T | is the mean of v over T . It follows from (38), (40) and (41) that, for any v ∈ Vk and θ ∈ (0, 1), 4   i=1 p ∈VTi

4   2    2 |ϕp ,Ti |2H 1 (Ti ) v(p ) = |ϕp ,Ti |2H 1 (Ti ) v(p ) − v¯T + v¯T i=1 p ∈VTi

   4 1 + θ2 |ϕp,T |2H 1 (T ) v¯T2 + C1 θ −2 |v|2H 1 (T ) p∈VT

 2   2  4 1 + θ2 |ϕp,T |2H 1 (T ) v(p) + C2 θ −2 |v|2H 1 (T ) .

(42)

p∈VT

In the derivation of (42) we have also used the fact that the triangles T1 , . . . , T4 are similar to T and therefore 4    |ϕp ,Ti |2H 1 (Ti ) = 4 |ϕp,T |2H 1 (T ) . (43) i=1 p ∈VTi

p∈VT

Combining (4), (39) and (42), we find, for arbitrary v ∈ Vk−1 and θ ∈ (0, 1),    2 |ϕp,T |2H 1 (T ) v(p) (Πk v, Πk v)J,Λ,k = Λh2k T ∈Tk p∈VT

= Λh2k

4    T ∈Tk−1 i=1 p ∈VTi

2  |ϕp ,T |2H 1 (Ti ) v(p )

    2  2  Λ 1 + θ 2 4h2k |ϕp,T |2H 1 (T ) v(p) + C2 Λθ −2 h2k |v|2H 1 (T ) T ∈Tk−1 p∈VT

 2 = 1 + θ 2 (v, v)J,Λ,k−1 + C2 Λθ −2 h2k |v|2H 1 (Ω) ,

T ∈Tk−1

(44)

which implies

  (Πk v, Πk v)J,Λ,k  1 + θ 2 (v, v)J,Λ,k−1 + Cθ −2 h2k |v|2H 1 (Ω)

(45)

for all v ∈ Vk−1 and 0 < θ < 1. Using (39)–(41) and (43), we can similarly prove the estimate   (Πk−1 v, Πk−1 v)J,Λ,k−1  1 + θ 2 (v, v)J,Λ,k + Cθ −2 h2k |v|2H 1 (Ω)

(46)

for all v ∈ Vk and 0 < θ < 1.

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S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

5. Estimates for the symmetric Gauss–Seidel iteration Throughout this section we will adopt the convention in the proof of Lemma 1, and we will use Ek (v) to denote a generic expression satisfying the estimate     −2 2 2 Ek (v)  C θ 2 v2 (47) ∀ v ∈ Vk , θ ∈ (0, 1). L2 (Ω) + θ hk |v|H 1 (Ω) , Since the symmetric Gauss–Seidel iteration depends on the ordering of the natural nodal basis of Vk , our discussion will be restricted to some standard orderings that are usually associated with uniform rectangular grids. Therefore we assume in this section that the triangles of T1 (and hence those of Tk ) are uniform isosceles right-angled triangles, and the natural nodal basis of Vk is ordered by either (i) the lexicographical ordering, or (ii) the checker-board (red–black) ordering. In the lexicographical ordering, the nodes in Tk are ordered from left to right and from bottom to top. In the checker-board ordering, the nodes are first divided into two groups in a checker-board pattern and then within each group the nodes are ordered lexicographically. These two orderings are illustrated in Fig. 1 for a square. For the inner product (·, ·)GS,k corresponding to the symmetric Gauss–Seidel iteration we have, in view of (16) and (18),   ∀ v ∈ Vk . (48) (v, v)GS,k = h2k D −1 k (D k + U k )v, (D k + U k )v , Our strategy is to relate (v, v)GS,k to (v, v)J,1,k , which can be written as   (v, v)J,1,k = h2k D k v, v .

(49)

First we note that, by (17) and (48),     −1 2 (v, v)GS,k = h2k D −1 k (D k + U k + Lk )v, (D k + U k )v − hk D k Lk v, (D k + U k )v     −1 2 2 = h2k Ak v, D −1 k (D k + U k )v − hk (Lk v, v) − hk D k Lk v, U k v  1 2    −1 1 2 2 = h2k Ak v, D −1 k (D k + U k )v − hk (Ak v, v) + hk (D k v, v) − hk D k Lk v, U k v . (50) 2 2 From the definition of Ak , we have (Ak v, v) = a(v, v) = |v|2H 1 (Ω) .

Fig. 1. The lexicographical (left) and checker-board (right) orderings.

(51)

S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

53

The sparseness of Ak and (21) also imply ρ(Ak )
1.

(52)

Using Lemma 1, (21), (22), (38), (51) and (52) we obtain the estimate −1   2 h2k Ak v, D −1 k (D k + U k )v  hk Ak v D k (D k + U k )v  C† h2k (Ak v, v)1/2 (D k + U k )v    C†† θ 2 v2L2 (Ω) + θ −2 h2k |v|2H 1 (Ω) ,

∀ v ∈ Vk , θ ∈ (0, 1).

(53)

Let p1 , . . . , pnk be the vertices in Vk and denote the corresponding nodal basis functions by ϕ1 , . . . , ϕnk . We can write    nk    −1 −1 a(ϕj , ϕj ) a(ϕi , ϕj )v(pi ) a(ϕi , ϕj )v(pi ) . (54) D k Lk v, U k v = j =1

i>j

We also have the elementary estimate   2 v(p) − v(q) ≈ |v|2H 1 (T ) ,

i
∀ v ∈ Vk , T ∈ Tk .

(55)

p,q∈VT

Since a(ϕpi , ϕpj ) = 0 unless pi and pj are vertices of a triangle in Tk , we obtain from (20), (21), (38), (54) and (55)    nk   2  −1  2 2 −1 a(ϕi , ϕj ) a(ϕi , ϕj ) v(pj ) a(ϕj , ϕj ) hk D k Lk v, U k v = hk j =1

+ h2k

nk 

a(ϕj , ϕj )−1

j =1

+ h2k

nk 

=

h2k



i>j

i
     a(ϕi , ϕj ) v(pi ) − v(pj ) a(ϕi , ϕj )v(pi )

i>j

a(ϕj , ϕj )

j =1 nk 



−1



a(ϕi , ϕj )v(pj )

i>j

2

v(pj ) a(ϕj , ϕj )

j =1

−1

 i>j



i
   a(ϕi , ϕj ) v(pi ) − v(pj )

i
a(ϕi , ϕj )



 a(ϕi , ϕj ) + Ek (v).

(56)

i
Remark 4. The relations (50), (51) and (54) and the estimates (53) and (56) are valid for general grids. On uniform rectangular grids, we have a(ϕp , ϕp ) = 4,

∀ p ∈ Vk ,

and for two distinct vertices p, q ∈ Vk ,  −2, if p and q are vertical or horizontal neighbors, a(ϕp , ϕq ) = 0, otherwise. It follows from (58) that, for the checker-board ordering (cf. Fig. 1), we have    a(ϕi , ϕj ) a(ϕi , ϕj ) = 0 for j = 1, . . . , nk . i>j

i
(57)

(58)

(59)

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S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

Combining (49)–(51), (53), (54), (56) and (59), we find in this case 1 (60) (v, v)GS,k = (v, v)J,1,k + Ek (v), ∀ v ∈ Vk . 2 Let p ∈ Vk . We will denote by Sp,k the union of all the (closed) triangles in Tk that contain p as a vertex and define Vkb = {p ∈ Vk : Sp.k ∩ ∂Ω = ∅} to be the set of the internal vertices of Tk that are next to ∂Ω. Then Vki = Vk \ Vkb is the set of the internal vertices of Tk that stay away from ∂Ω. In the case of the lexicographical ordering (cf. Fig. 1), we have, by (58),     a(ϕi , ϕj ) = −2 = a(ϕi , ϕj ) , ∀ pj ∈ Vki , i>j

i
which together with (57) yield     2  2 1  a(ϕi , ϕj ) a(ϕi , ϕj ) = a(ϕj , ϕj ) v(pj ) . v(pj ) a(ϕj , ϕj )−1 4 i i i>j i
(61)

pj ∈ V k

Moreover, since (55) implies  2 v(p)
|v|2H 1 (Ω) ,

∀ v ∈ Vk ,

p∈Vkb

the following estimates hold for all v ∈ Vk :       2  −1  a(ϕi , ϕj ) a(ϕi , ϕj ) 
|v|2H 1 (Ω) , v(pj ) a(ϕj , ϕj )  pj ∈Vkb



i>j

(62)

i
 2 a(ϕj , ϕj ) v(pj )
|v|2H 1 (Ω) .

(63)

pj ∈Vkb

Combining (56) and (61)–(63), we find  1 2   h2k D −1 k Lk v, U k v = hk D k v, v + Ek (v), 4 which together with (49)–(51) and (53) imply 1 (v, v)GS,k = (v, v)J,1,k + Ek (v), ∀ v ∈ Vk 4 in the case of the lexicographical ordering. We conclude from (60) and (64) that a relation of the form (v, v)GS,k = c(v, v)J,1,k + Ek (v),

∀ v ∈ Vk

(64)

(65)

holds for either ordering, where c is a positive constant. Using (4), Lemma 1, (45), (47) and (65), we find, for arbitrary v ∈ Vk and θ ∈ (0, 1), (Πk v, Πk v)GS,k = c(Πk v, Πk v)J,1,k + Ek (Πk v)    c 1 + θ 2 (v, v)J,1,k−1 + C∗ θ −2 h2k |v|2H 1 (Ω)   + C θ 2 Πk v2L2 (Ω) + θ −2 h2k |Πk v|2H 1 (Ω)

S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

     c 1 + θ 2 (v, v)J,1,k−1 + C7 θ 2 v2L2 (Ω) + θ −2 h2k |v|2H 1 (Ω)      = 1 + θ 2 (v, v)GS,k−1 − Ek−1 (v) + C7 θ 2 v2L2 (Ω) + θ −2 h2k |v|2H 1 (Ω)      1 + θ 2 (v, v)GS,k−1 + C8 θ 2 v2L2 (Ω) + θ −2 h2k |v|2H 1 (Ω)    1 + c∗ θ 2 (v, v)GS,k−1 + C8 θ −2 h2k |v|2H 1 (Ω) .

55

(66)

Here we have used the fact that the nodal interpolation Πk : Vk−1 → Vk is just the natural injection and hence Πk vL2 (Ω) = vL2 (Ω)

and

|Πk v|H 1 (Ω) = |v|H 1 (Ω) ,

∀ v ∈ Vk−1 .

It follows immediately from (66) that   (Πk v, Πk v)GS,k  1 + θ 2 (v, v)GS,k−1 + Cθ −2 h2k |v|2H 1 (Ω)

(67)

for all v ∈ Vk−1 and 0 < θ < 1. For the nodal interpolation operator Πk−1 : Vk → Vk−1 , the estimates Πk−1 vL2 (Ω)
vL2 (Ω)

and

|Πk−1 v|H 1 (Ω)
|v|H 1 (Ω) ,

∀ v ∈ Vk

(68)

follow easily from (20) and (55). Using (4), Lemma 1, (46), (47), (65) and (68) we can similarly derive the estimate   (69) (Πk−1 v, Πk−1 v)GS,k−1  1 + θ 2 (v, v)GS,k + Cθ −2 h2k |v|2H 1 (Ω) for all v ∈ Vk and 0 < θ < 1. Remark 5. It is not difficult to check that the relation (65) also holds for the zebra-line ordering (where c = 5/16) and the four-color ordering (where c = 1/2). The estimates (67) and (69) are therefore also valid for these orderings (cf. [12] for a description).

6. Applications Let γk be the contraction number of the kth level multigrid V -cycle algorithm using the damped Jacobi iteration (on general grids) or the symmetric Gauss–Seidel iteration (on uniform rectangular grids and with the lexicographical, the checker-board, the zebra-line or the four-color ordering) as the smoother. As mentioned in Section 3, the theory developed in [1,8] can be applied once we have the estimates (31), (32), (34), (36) and (37). Therefore, for the model problem in Section 2, the results in Sections 3, 5 and [1,8] yield the estimate C for k = 1, 2, . . . , (70) γk  [max(m1 , 1) max(m2 , 1)]α/2 for m1 + m2 sufficiently large. On the other hand, it follows from the results of Zhang [2] and Bramble and Pasciak [3,4,21] that γk  δ < 1

for k = 1, 2, . . . .

Combining (70) and (71) we have C γk  C + [max(m1 , 1) max(m2 , 1)]α/2

(71)

for m1 + m2  1, k = 1, 2, . . . .

56

S.C. Brenner / Applied Numerical Mathematics 43 (2002) 45–56

Remark 6. The results in this paper can be extended to the more general variational form
  p(x)(∇v · ∇w) + r(x)(vw) dx, a(v, w) =

(72)

Ω

where p(x) and r(x) are C 1 functions and p(x) > 0, r(x)  0 for all x ∈ Ω. The key observation is that locally the difference between (72) and a multiple of (2) involves expressions that satisfy estimates of the form given in (47). Remark 7. The results in this paper can also be extended to nonconforming finite elements. It follows that the results in [8] are valid for nonconforming V -cycle and F -cycle algorithms using the damped Jacobi iteration or the symmetric Gauss–Seidel iteration as the smoother.

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