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MATHEMATICS OF COMPUTATION Volume 73, Number 247, Pages 1041–1066 S 0025-5718(03)01578-3 Article electronically published on August 19, 2003

CONVERGENCE OF NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS FOR SECOND ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS SUSANNE C. BRENNER Abstract. The convergence of V -cycle and F -cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems.

1. Introduction Let Ω ⊂ R2 be a bounded polygonal domain. Consider the variational problem of finding u ∈ H01 (Ω) such that (1.1) where F ∈ H −1 (Ω) and (1.2)

∀ v ∈ H01 (Ω) ,

a(u, v) = F (v)

a(v, w) =

Z hX 2 Ω

i,j=1

aij (x)

i ∂v ∂w + r(x)vw dx . ∂xi ∂xj

¯ a12 = a21 , r ≥ 0 on Ω, ¯ and We assume that aij , r ∈ C (Ω), 1

(1.3)

2 X

aij (x)ξi ξj ≥ c(ξ12 + ξ22 )

¯ , ξ1 , ξ2 ∈ R , ∀x ∈ Ω

i,j=1

where c is a positive constant. Under these conditions the bilinear form a(·, ·) on H01 (Ω) × H01 (Ω) is bounded and coercive, and (1.1) has a unique solution. It is well known (cf. §5.C and §14.A of [32]) that there exists α ∈ ( 12 , 1] such that the solution u of (1.1) belongs to H 1+α (Ω) ∩ H01 (Ω) whenever F ∈ H −1+α (Ω) and (1.4)

kukH 1+α (Ω) ≤ CΩ kF kH −1+α (Ω) .

Approximate solutions of the variational problem (1.1) can be obtained by the finite element method (cf. [30, 25]). The resulting symmetric positive definite systems are sparse and can be solved efficiently by multigrid algorithms (cf. [36], [39], [7], [16], [49]). For the symmetric V -cycle algorithm with equal numbers of presmoothing and post-smoothing steps, the classical result by Braess and Hackbusch Received by the editor May 29, 2001 and, in revised form, January 10, 2003. 2000 Mathematics Subject Classification. Primary 65N55, 65N30. Key words and phrases. Multigrid, V -cycle, F -cycle, nonconforming finite elements. This work was supported in part by the National Science Foundation under Grant No. DMS00-74246. c

2003 American Mathematical Society

1041

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SUSANNE C. BRENNER

(cf. [35], [5], [3], [39], [9]) states that, in the case where α = 1 (i.e., when Ω is convex), C for m = 1, 2, . . . , γk ≤ C +m where γk is the contraction number of the k-th level V -cycle algorithm in the norm p (1.5) k · ka = a(·, ·) , m is the number of pre-smoothing and post-smoothing steps, and C (with or without subscripts) henceforth denotes a generic positive constant which is independent of k and m. The case where 12 < α < 1 (i.e., when Ω has re-entrant corners) is more subtle. It was not until the early nineties, after a multiplicative theory for multilevel methods (cf. [13, 12]) had been developed, that the following result was established (cf. [57, 53, 10, 56, 11, 34, 41]): (1.6)

γk ≤ δ ,

where δ ∈ (0, 1) is independent of k. The asymptotic behavior of γk with respect to m was studied in [22] by an additive convergence theory. It was shown that, for the P1 finite element and with the Richardson relaxation scheme as smoother, C for m ≥ m0 , (1.7) γk ≤ α m where the positive integer m0 is independent of k. It then follows easily from (1.6) and (1.7) that C for m = 1, 2, . . . . γk ≤ C + mα In other words, a complete generalization of the result of Braess and Hackbusch to the case of less than full elliptic regularity has been obtained. The results in [22] were generalized to include other smoothers in [23]. In this paper we extend the theory in [22] to nonconforming finite elements and establish the same estimate (1.7). Since the V -cycle algorithm for nonconforming finite elements in general does not converge uniformly for m = 1, the estimate (1.7) is the best possible general estimate for such methods. As a by-product we also obtain similar estimates for nonconforming F -cycle algorithms (cf. [17], [40], [51], [49]). As far as we know, this is the first convergence result for nonconforming F -cycle algorithms, even though it has been known for some time (cf. [50]) that these algorithms perform very well in practice. The theory developed in this paper can also be applied to fourth order problems (cf. [58]). We also note in passing that most of the general convergence results for nonconforming multigrid algorithms were obtained for the W -cycle algorithm and the variable V -cycle preconditioner (cf. [18], [14], [21] and the references therein, and also [44], [47] for the cascadic multigrid algorithm). The only existent nonconforming V -cycle convergence results involve either special elements (cf. [54], [45], [46]), conforming finite element spaces on coarse grids (cf. [52], [55]), or the suboptimal Galerkin approach (cf. [29]). The rest of the paper is organized as follows. In Section 2 we set up V -cycle and F -cycle algorithms for nonconforming finite elements in an abstract setting. The assumptions for the convergence theory are stated in Section 3. We then prove a strengthened Cauchy-Schwarz inequality in Section 4. The estimate (1.7) for V -cycle and F -cycle algorithms is established in Section 5. In Section 6 we show

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that the assumptions in Section 3 can be verified within an abstract framework for nonconforming multigrid methods. Applications to concrete nonconforming finite elements are then given in Section 7. For future reference we state here two simple inequalities: (1.8a)

2ab ≤ (θa)2 + (θ−1 b)2

(1.8b)

(a + b) ≤ (1 + θ )a + (1 + θ 2

2

∀ a, b ∈ R , θ ∈ (0, 1) ,

2

−2

)b

2

∀ a, b ∈ R , θ ∈ (0, 1) .

2. Nonconforming V -cycle and F -cycle multigrid algorithms Let T1 be a triangulation of Ω and let the triangulations Tk , for k = 2, 3, . . ., be obtained by successive regular subdivisions. The mesh size hk of Tk therefore satisfies the relation hk = 2hk+1

(2.1)

for k = 1, 2, . . . .

The discontinuous energy space H (Tk ) associated with Tk is defined by (2.2) H 1 (Tk ) = {v ∈ L2 (Ω) : v T ∈ H 1 (T ) ∀ T ∈ Tk } . 1

We define the nonconforming variational form ak (·, ·) on H 1 (Tk ) by 2 i X Z hX ∂v ∂w aij (x) + r(x)vw dx ∀ v, w ∈ H 1 (Tk ) , (2.3) ak (v, w) = ∂xi ∂xj T i.j=1 T ∈Tk

and the corresponding nonconforming energy (semi-)norm k · kak by p ∀ v ∈ H 1 (Tk ) . (2.4) kvkak = ak (v, v) Note that H 1 (T1 ) ⊂ H 1 (T2 ) ⊂ · · · and (2.5)

kvkak−1 = kvkak

∀ v ∈ H 1 (Tk−1 ) .

Moreover, it follows from the boundedness and coercivity of a(·, ·) and the Poincar´e inequality that (2.6)

kζkak = kζka ≈ kζkH 1 (Ω) ≈ |ζ|H 1 (Ω)

∀ ζ ∈ H01 (Ω) .

Let Vk ⊂ H 1 (Tk ) be a nonconforming finite element space associated with Tk such that ak (·, ·) is positive definite on Vk , and let (·, ·)k be a discrete inner product on Vk . We can then represent ak (·, ·) by the operator Ak : Vk −→ Vk defined by (2.7)

(Ak v1 , v2 )k = ak (v1 , v2 )

∀ v1 , v2 ∈ Vk .

Note that Ak is symmetric positive definite with respect to (·, ·)k and the following relation holds: (2.8)

ak (Ask v1 , v2 ) = ak (v1 , Ask v2 )

∀ v1 , v2 ∈ Vk , s ∈ R .

2.1. V -cycle and F -cycle multigrid algorithms. The k-th level multigrid V cycle and F -cycle algorithms are multilevel iterative methods for the equation (2.9)

Ak z = g .

We assume that the finite element space Vk−1 is connected to Vk by the (linear) k : Vk−1 −→ Vk , and denote its transpose with intergrid transfer operator Ik−1 respect to the discrete inner products by Ikk−1 : Vk −→ Vk−1 , i.e., (2.10)

k w)k (Ikk−1 v, w)k−1 = (v, Ik−1

∀ v ∈ Vk , w ∈ Vk−1 .

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SUSANNE C. BRENNER

k The operator Pkk−1 : Vk −→ Vk−1 is the transpose of Ik−1 with respect to the nonconforming variational forms, i.e.,

(2.11)

k w) ak−1 (Pkk−1 v, w) = ak (v, Ik−1

∀ v ∈ Vk , w ∈ Vk−1 .

These operators satisfy the well-known relation Ikk−1 Ak = Ak−1 Pkk−1 .

(2.12)

Finally we take Λk to be a number dominating the spectral radius ρ(Ak ) of Ak . Algorithm 2.1 (The V -cycle algorithm). The k-th level symmetric V -cycle algorithm produces M GV (k, g, z0 , m) as an approximate solution for (2.9) with initial guess z0 , where m denotes the number of pre-smoothing and post-smoothing steps. For k = 1 we define M GV (1, g, z0 , m) = A−1 1 g. For k ≥ 2 the approximate solution M GV (k, g, z0 , m) is computed recursively in three steps: Pre-smoothing. For j = 1, . . . , m, compute zj by zj = zj−1 + Λ−1 k (g − Ak zj−1 ) . Coarse grid correction. Let rk−1 = Ikk−1 (g − Ak zm ) and compute zm+1 by k M GV (k − 1, rk−1 , 0, m) . zm+1 = zm + Ik−1

Post-smoothing.

For j = m + 2, . . . , 2m + 1, compute zj by zj = zj−1 + Λ−1 k (g − Ak zj−1 ) .

We then define M GV (k, g, z0 , m) = z2m+1 . Algorithm 2.2. (The F -cycle algorithm) The k-th level F -cycle algorithm (associated with the symmetric V -cycle algorithm) produces M GF (k, g, z0 , m) as an approximate solution for (2.9). For k = 1, we define M GF (1, g, z0 , m) = A−1 1 g. For k ≥ 2, we define M GF (k, g, z0 , m) recursively in three steps: Pre-smoothing. For j = 1, . . . , m, compute zj by zj = zj−1 + Λ−1 k (g − Ak zj−1 ) . Coarse grid correction. Let rk−1 = Ikk−1 (g − Ak zm ) and compute zm+1 by zm+ 12 = M GF (k − 1, rk−1 , 0, m) , k M GV (k − 1, rk−1 , zm+ 12 , m) . zm+1 = zm + Ik−1

Post-smoothing.

For j = m + 1, . . . , 2m + 1, compute zj by zj = zj−1 + Λ−1 k (g − Ak zj−1 ) .

We then define M GF (k, g, z0 , m) = z2m+1 . Remark 2.3. We use the Richardson relaxation scheme as the smoother for simplicity. The theory in this paper can be applied to other smoothers if the definition of the mesh-dependent norms in Section 3 is modified appropriately (cf. [23]).

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2.2. Error representations. Let Ek,m : Vk −→ Vk be the operator relating the initial error and the final error of the multigrid V -cycle algorithm applied to the equation (2.9), i.e., Ek,m (z − z0 ) = z − M GV (k, g, z0 , m) . j and Pjj−1 The operator Ek,m can be described in terms of the operators Ij−1 (2 ≤ j ≤ k) and the operators Rj : Vj −→ Vj defined by

Rj = Idj − Λ−1 j Aj ,

(2.13)

where Idj : Vj −→ Vj is the identity operator. Clearly we have aj (Rj v, w) = aj (v, Rj w)

(2.14)

∀ v, w ∈ Vj .

The following relations (cf. [36], [7]) are well known: 
 k k (2.15) Pkk−1 + Ik−1 Ek−1,m Pkk−1 Rkm Ek,m = Rkm Idk − Ik−1 (2.16)

for k ≥ 2 ,

E1,m = 0 .

Using (2.15) and (2.16), we obtain an additive expression for Ek,m : 
 k k Pkk−1 + Ik−1 Ek−1,m Pkk−1 Rkm Ek,m = Rkm Idk − Ik−1

(2.17)

k = Rkm (Idk − Ik−1 Pkk−1 )Rkm h
k−1 k−2 k m + Rkm Ik−1 Rk−1 Pk−1 Idk−1 − Ik−2

i k−1 k−2 m Ek−2,m Pk−1 Pkk−1 Rkm Rk−1 + Ik−2

=

k X

j Tk,j,m Rjm (Idj − Ij−1 Pjj−1 )Rjm Tj,k,m ,

j=2

where Tk,k,m = Idk ,

(2.18)

and for j < k, Tj,k,m : Vk −→ Vj and Tk,j,m : Vj −→ Vk are defined by (2.19)

j m Rj+1 · · · Pkk−1 Rkm , Tj,k,m = Pj+1

(2.20)

k m · · · Rj+1 Ijj+1 . Tk,j,m = Rkm Ik−1

Note that for 1 ≤ j ≤ k ≤ ` the following relations are valid: (2.21)

Tj,`,m = Tj,k,m Tk,`,m

and T`,j,m = T`,k,m Tk,j,m .

Also (2.11) and (2.14) imply (2.22)

aj (Tj,k,m v, w) = ak (v, Tk,j,m w)

∀ v ∈ Vk , w ∈ Vj .

˜ k,m : Vk −→ Vk be the operator relating the initial error and the final error Let E of the F -cycle algorithm applied to the equation (2.9), i.e., ˜ k,m (z − z0 ) = z − M GF (k, g, z0 , m) . E The following relations (cf. [49]) are also well known: (2.23) (2.24)

˜ 1,m = 0 , E   ˜ k−1,m P k−1 Rm , k ≥ 2 . ˜ k,m = Rm (Idk − I k P k−1 ) + I k Ek−1,m E E k k−1 k k−1 k k

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SUSANNE C. BRENNER

3. Assumptions In this section we state the assumptions for the convergence theory and derive some of their immediate consequences. First we introduce a scale of mesh-dependent norms (cf. [4]) on each Vk : q ∀ s ∈ R , v ∈ Vk . Ask v, v)k (3.1) |||v|||s,k = It is clear from (2.4), (2.7) and (3.1) that p (3.2) |||v|||0,k = (v, v)k p (3.3) |||v|||1,k = ak (v, v) = kvkak (3.4)

|||Ask v|||t,k

∀ v ∈ Vk , ∀ v ∈ Vk ,

= |||v|||t+2s,k

∀ v ∈ Vk , s, t ∈ R .

Moreover the Cauchy-Schwarz inequality implies (3.5)

|||v|||1+t,k =

ak (v, w) |||w||| 1−t,k w∈Vk \{0}

∀ t ∈ R , v ∈ Vk .

sup

To avoid the proliferation of constants we henceforth use the notation A . B to represent the inequality A ≤ (constant) × B, where the constant is independent of both the mesh (i.e., independent of the mesh size and the mesh level) and the number of smoothing steps. The statement A ≈ B is equivalent to A . B and B . A. Assumptions on Vk . We assume that (3.6) (3.7)

(v, v)k ≈ kvk2L2 (Ω) kvkak .

∀ v ∈ Vk ,

h−1 k kvkL2 (Ω)

∀ v ∈ Vk .

k k and Pkk−1 . We assume that Ik−1 and Pkk−1 have the following Assumptions on Ik−1 properties:

(3.8)

k 2 v|||21,k ≤ (1 + θ2 )|||v|||21,k−1 + C1 θ−2 h2α |||Ik−1 k |||v|||1+α,k−1

∀ v ∈ Vk−1 , θ ∈ (0, 1) , (3.9)

k v|||21−α,k |||Ik−1

≤ (1 + θ

2

)|||v|||21−α,k−1

+

2 C2 θ−2 h2α k |||v|||1,k−1

∀ v ∈ Vk−1 , θ ∈ (0, 1) , (3.10)

2 |||Pkk−1 v|||21−α,k−1 ≤ (1 + θ2 )|||v|||21−α,k + C3 θ−2 h2α k |||v|||1,k

∀ v ∈ Vk , θ ∈ (0, 1) , where α is the index of elliptic regularity in (1.4) and the positive constants C1 , C2 and C3 are mesh-independent. Remark 3.1. The estimates in [22] corresponding to (3.9) and (3.10) involve an index β ∈ (0, 12 ) instead of α. Using the tools developed in Section 6 of this paper, one can also replace β by α in [22].

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k k Assumptions on Ik−1 Pkk−1 and Pkk−1 Ik−1 . We assume the nonconforming finite element spaces have the following approximation properties:
k (3.11) Pkk−1 v|||1−α,k . h2α ∀ v ∈ Vk , ||| Idk − Ik−1 k |||v|||1+α,k
k (3.12) ∀ v ∈ Vk−1 . v|||1−α,k−1 . hα ||| Idk−1 − Pkk−1 Ik−1 k |||v|||1,k−1

We now derive some simple consequences of the assumptions above. First we note that (3.6) and (3.7) imply ρ(Ak ) . h−2 k .

(3.13)

It follows easily (cf. [4], [36]) from (2.13), (3.1) and (3.13) that (3.14)

|||v|||s,k . ht−s k |||v|||t,k

(3.15)

|||Rk v|||s,k ≤ |||v|||s,k

(3.16)

|||Rkm v|||s,k

.

∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2 , ∀ v ∈ Vk , s ∈ R ,

(t−s)/2 ht−s |||v|||t,k k m

∀ v ∈ Vk , 0 ≤ t ≤ s ≤ 2 , m ≥ 1 .

The estimate (3.8) and (3.12) imply through (2.11), (3.5) and (3.14) three additional estimates: k v|||1,k . |||v|||1,k−1 |||Ik−1

(3.17) (3.18) (3.19)

||| Idk−1 −

|||Pkk−1 v|||1,k−1
k Pkk−1 Ik−1 v|||1,k−1

∀ v ∈ Vk−1 ,

. |||v|||1,k

∀ v ∈ Vk ,

. hα k |||v|||1+α,k−1

∀ v ∈ Vk−1 .

Again, to avoid the proliferation of constants, we henceforth say that an estimate holds for m sufficiently large if it is valid for m ≥ m∗ , where the positive integer m∗ is mesh-independent. Lemma 3.2. Given any number ω ∈ (0, 1), the following estimates hold for m sufficiently large : (3.20) (3.21) (3.22)

k m Rk−1 v|||1,k ≤ (1 + ω)|||v|||1,k−1 |||Ik−1

|||Pkk−1 Rkm v|||1−α,k−1 k v|||1+α,k |||Rkm Ik−1

∀ v ∈ Vk−1 ,

≤ (1 + ω)|||v|||1−α,k

∀ v ∈ Vk ,

≤ (1 + ω)|||v|||1+α,k−1

∀ v ∈ Vk−1 .

Proof. From (2.1), (3.8), (3.15) and (3.16) we have k m m m 2 Rk−1 v|||21,k ≤ (1 + ω 2 )|||Rk−1 v|||21,k−1 + C1 ω −2 h2α |||Ik−1 k |||Rk−1 v|||1+α,k−1

≤ (1 + ω 2 )|||v|||21,k−1 + C10 ω −2 m−α |||v|||21,k−1 , where the positive constant C10 is independent of the mesh and the number of smoothing steps. The estimate (3.20) then follows if mα ≥

C10 ω −3 . 2

Similarly we obtain (3.21) using (3.10), (3.15) and (3.16). The estimate (3.22) then follows from (2.11), (2.14), (3.5) and (3.21). 

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SUSANNE C. BRENNER

4. A strengthened Cauchy-Schwarz inequality In this section we derive a strengthened Cauchy-Schwarz inequality which takes into account the effect of smoothing. We begin by estimating the bounds of the operator Tj,k,m with respect to various mesh-dependent norms. For brevity we will sometimes suppress the parameter m and write Tj,k instead of Tj,k,m . From (2.18)–(2.20), (3.21) and (3.22) we immediately have the following lemma. Lemma 4.1. Let j ≤ k. Given any ω ∈ (0, 1), the following estimates hold for m sufficiently large : (4.1)

|||Tj,k,m v|||1−α,j ≤ (1 + ω)k−j |||v|||1−α,k

∀ v ∈ Vk ,

(4.2)

|||Tk,j,m v|||1+α,k ≤ (1 + ω)

∀ v ∈ Vj .

k−j

|||v|||1+α,j

The next three lemmas and one corollary are preparatory for the crucial estimate involving TK,k,m Tk,K,m for k ≤ K. Lemma 4.2. Let k ≤ K. Then the estimate |||TK,k,m v|||1,K . |||v|||1,k

(4.3)

∀ v ∈ Vk

holds for m sufficiently large. Proof. Let v ∈ Vk be arbitrary. Given any ω ∈ (0, 1), we have, from (1.8a), (2.11), (2.18), (2.20), (3.1), (3.3), (3.15), (3.19), (4.2) and the Cauchy-Schwarz inequality, that
|||TK,k v|||21,K = aK TK,k v, TK,k v
m K m K IK−1 TK−1,k v, RK IK−1 TK−1,k v = aK RK
K K TK−1,k v, IK−1 TK−1,k v ≤ aK IK−1
K−1 K IK−1 TK−1,k v, TK−1,k v = aK−1 PK
(4.4) = aK−1 TK−1,k v, TK−1,k v
K−1 K IK−1 − IdK−1 ]TK−1,k v, TK−1,k v + aK−1 [PK ≤ |||TK−1,k v|||21,K−1 K−1 K + |||(PK IK−1 − IdK−1 )TK−1,k v|||1,K−1 |||TK−1,k v|||1,K−1 −2 2α 2 ≤ (1 + θK )|||TK−1,k v|||21,K−1 + C♦ θK hK |||TK−1,k v|||21+α,K−1 −2 2 2 ≤ (1 + θK )|||TK−1,k v|||21,K−1 + C] θK (1 + ω)2(K−k) h2α K |||v|||1+α,k

for all v ∈ Vk , provided m is sufficiently large. Note that θK ∈ (0, 1) is arbitrary, and the positive constant C] is independent of the mesh, the number of smoothing steps and the parameters ω and θK . Iterating (4.4), we find  Y  2 2 (4.5) |||TK,k v|||1,K ≤ (1 + θq ) |||v|||21,k  + C]

k+1≤q≤K

X k+1≤p≤K



Y

(1 +

  −2 2(p−k) 2α θp (1 + ω) hp |||v|||21+α,k ,

θq2 )

p+1≤q≤K

where θk+1 , . . . , θK are arbitrary numbers in (0, 1).

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Note that hp = 2k−p hk by (2.1). We now choose  α/2 4 −1 ω= 3 so that (4.6)

  2 −α p−k 2α hk = 3−α(p−k) h2α (1 + ω)2(p−k) h2α p = (1 + ω) 4 k ,

and then we take

 α(q−k)/2 2 θq = 3

(4.7)

for k + 1 ≤ q ≤ K .

Combining (4.5)–(4.7), we have 2 |||TK,k v|||21,K ≤ ρ1 |||v|||21,k + C] ρ1 ρ2 h2α k |||v|||1+α,k ,

(4.8) where ρ1 =

"

∞ Y j=1

 α·j # ∞  α·j X 2 1 < ∞. 1+ < ∞ and ρ2 = 3 2 j=1 

The estimate (4.3) follows from (3.14) and (4.8). Lemma 4.2, (2.22) and (3.5) immediately imply the following corollary. Corollary 4.3. Let k ≤ K. Then the estimate (4.9)

|||Tk,K,m v|||1,k . |||v|||1,K

∀ v ∈ VK

holds for m sufficiently large. Lemma 4.4. Let k ≤ K. Then the estimate (4.10)

|||TK,k,m v|||1−α,K . |||v|||1−α,k

∀ v ∈ Vk

holds for m sufficiently large. Proof. Let v ∈ Vk be arbitrary. It follows from (2.18), (2.20), (3.9), (3.15) and (4.3) that, for m sufficiently large, m K IK−1 TK−1,k v|||21−α,K |||TK,k v|||21−α,K = |||RK K ≤ |||IK−1 TK−1,k v|||21−α,K

(4.11)

−2 2α 2 ≤ (1 + θK )|||TK−1,k v|||21−α,K−1 + C2 θK hK |||TK−1,k v|||21,K−1 −2 2α 2 )|||TK−1,k v|||21−α,K−1 + C† θK hK |||v|||21,k ≤ (1 + θK

for any θK ∈ (0, 1), where the positive constant C† is independent of the mesh, the number of smoothing steps and the parameter θK . Iterating (4.11), we find  Y  (4.12) (1 + θq2 ) |||v|||21−α,k |||TK,k v|||21−α,K ≤  + C†

k+1≤q≤K

X k+1≤p≤K



Y

(1 +

  −2 2α θp hp |||v|||21,k ,

θq2 )

p+1≤q≤K

where θk+1 , . . . , θK are arbitrary numbers in (0, 1). As in the proof of Lemma 4.2, by choosing  α(q−k)/2 1 for k + 1 ≤ q ≤ K , (4.13) θq = 2

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SUSANNE C. BRENNER

we can deduce from (2.1) and (4.12) that (4.14)

2 |||TK,k v|||21−α,K . |||v|||21−α,k + h2α k |||v|||1,k .



The estimate (4.10) now follows from (3.14) and (4.14). Lemma 4.5. Let k ≤ K. Then the estimate (4.15)

2 |||Tk,K,m v|||21−α,k . |||v|||21−α,K + h2α k |||v|||1,K

∀ v ∈ VK

holds for m sufficiently large. Proof. Let v ∈ VK be arbitrary. From (2.19), (3.10), (3.15) and (4.9) we have, for m sufficiently large, k m Rk+1 Tk+1,K v|||21−α,k |||Tk,K v|||21−α,k = |||Pk+1

(4.16)

−2 2α 2 ≤ (1 + θk+1 )|||Tk+1,K v|||21−α,k+1 + C3 θk+1 hk+1 |||Tk+1,K v|||21,k+1 −2 2α 2 )|||Tk+1,K v|||21−α,k+1 + C[ θk+1 hk+1 |||v|||21,K , ≤ (1 + θk+1

where θk+1 ∈ (0, 1) is arbitrary and the positive constant C[ is independent of the mesh, the number of smoothing steps and the parameter θk+1 . Iterating (4.16), we find  Y  (4.17) (1 + θq2 ) |||v|||21−α,K |||Tk,K v|||21−α,k ≤ k+1≤q≤K



X

+ C[

k+1≤p≤K



Y

  (1 + θq2 ) θp−2 h2α |||v|||21,K . p

k+1≤q≤p−1

Choosing θq by formula (4.13), we deduce (4.15) from (2.1) and (4.17) as in the proof of Lemma 4.2.  We can now establish a crucial estimate. Lemma 4.6. Let k ≤ K. Then the estimate (4.18)

|||Tk,K,m TK,k,m v|||1−α,k . |||v|||1−α,k

∀ v ∈ Vk

holds for m sufficiently large. Proof. It follows from (3.14), Lemma 4.2, Lemma 4.4 and Lemma 4.5 that 2 |||Tk,K,m TK,k,m v|||21−α,k . |||TK,k,m v|||21−α,K + h2α k |||TK,k,m v|||1,K 2 2 . |||v|||21−α,k + h2α k |||v|||1,k . |||v|||1−α,k ,

provided m is sufficiently large.



The following proposition is the main result of this section. Proposition 4.7 (Strengthened Cauchy-Schwarz inequality with smoothing). Let 1 ≤ j ≤ k ≤ K. Given any ω ∈ (0, 1), the estimate
(4.19) aK TK,j,m Rjq vj , TK,k,m Rkq vk  k−j
−α
1+ω −α h−α .q j |||vj |||1−α,j hk |||vk |||1−α,k 2α holds for all vj ∈ Vj and vk ∈ Vk , provided m is sufficiently large.

NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1051

Proof. Given any ω ∈ (0, 1), from (2.1), (2.21), (2.22), (3.5), (3.15), (3.16), (4.1) and Lemma 4.6, we obtain

aK TK,j Rjq vj , TK,k Rkq vk = aj Rjq vj , Tj,k Tk,K TK,k Rkq vk ≤ |||Rjq vj |||1+α,j |||Tj,k Tk,K TK,k Rkq vk |||1−α,j h−2α j |||vj |||1−α,j (1 + ω)k−j |||vk |||1−α,k qα  α
−α
(1 + ω)k−j hk = h−α j |||vj |||1−α,j hk |||vk |||1−α,k α q hj  k−j
−α
1+ω h−α = q −α j |||vj |||1−α,j hk |||vk |||1−α,k , α 2 .



provided m is sufficiently large. Corollary 4.8. Let vk ∈ Vk for 1 ≤ k ≤ K. Then the estimate K K K X  X X TK,k,m Rkq vk , TK,k,m Rkq vk . q −α h−2α |||vk |||21−α,k (4.20) aK k k=1

k=1

k=1

holds for m sufficiently large. Proof. It follows from Proposition 4.7 that, given any ω ∈ (0, 1), (4.21)

aK

K X

TK,k Rkq vk ,

k=1

K X

K  X
TK,k Rkq vk = aK TK,j Rjq vj , TK,k Rkq vk

k=1

. q −α

K X j,k=1



1+ω 2α

|k−j|

j,k=1

h−α j |||vj |||1−α,j




h−α k |||vk |||1−α,k




for m sufficiently large. We now choose ω so that (1 + ω)2−α < 1. The estimate (4.20) then follows from (4.21) and a discrete Young’s inequality (cf. [37]).  5. Convergence of V -cycle and F -cycle algorithms In this section we establish the asymptotic behavior of the contraction numbers ˜ K,m (K ≥ 2) with respect to m. of EK,m and E The analysis in [22] for conforming V -cycle multigrid methods uses the fact that k k Pkk−1 ]2 = Idk − Ik−1 Pkk−1 [Idk − Ik−1

(5.1)

in the case Vk−1 ⊂ Vk ⊂ H 1 (Ω). Since (5.1) does not hold for nonconforming finite element spaces, we need to consider first the contraction property of an auxiliary operator EK,m : VK −→ VK defined by (5.2)

EK,m =

K X

 −α k TK,k,m Rkm (Idk − Ik−1 Pkk−1 )h−2α k Ak

k=2

 k Pkk−1 )Rkm Tk,K,m . × (Idk − Ik−1

Lemma 5.1. The estimate (5.3)

|||EK,m v|||1,K . m−α |||v|||1,K

holds for m sufficiently large.

∀ v ∈ VK

1052

SUSANNE C. BRENNER

Proof. Let v ∈ VK be arbitrary and (5.4)

k−1 k k Pkk−1 )h−2α A−α )Rkm Tk,K,m v . vk = (Idk − Ik−1 k k (Idk − Ik−1 Pk

We have, by (5.2), (5.4) and Corollary 4.8, (5.5)

aK (EK,m v, Ek,m v) = aK

K X

TK,k,m Rkm vk ,

k=2

. m−α

K X

TK,k,m Rkm vk



k=2

K X

h−2α |||vk |||21−α,k k

k=2

for m sufficiently large. Moreover, (3.4), (3.11) and (5.4) imply k−1 k )Rkm Tk,K,m v|||1+α,k |||vk |||1−α,k . |||A−α k (Idk − Ik−1 Pk

(5.6)

−α/2

= |||Ak

k (Idk − Ik−1 Pkk−1 )Rkm Tk,K,m v|||1,k .

k Pkk−1 )Rkm Tk,K,m v. It follows from (2.8), (2.11), (2.14), (2.22), Let wk = (Idk − Ik−1 (3.1) and (5.2) that K X

−α/2

h−2α |||Ak k

k (Idk − Ik−1 Pkk−1 )Rkm Tk,K,m v|||21,k

k=2

(5.7)

=

K X

−α/2

h−2α ak Ak k

−α/2

wk , Ak

k (Idk − Ik−1 Pkk−1 )Rkm Tk,K,m v




k=2

=

K X

k aK TK,k,m Rkm (Idk − Ik−1 Pkk−1 )h−2α A−α k k wk , v




k=2

= aK (EK,m v, v) . Combining (3.3), (5.5)–(5.7), and the Cauchy-Schwarz inequality, we find
|||EK,m v|||21,K = aK (EK,m v, EK,m v . m−α aK (EK,m v, v) ≤ m−α |||Ek,m v|||1,K |||v|||1,K 

and (5.3) follows. We can now prove the convergence of the symmetric V -cycle algorithm.

Theorem 5.2 (Convergence of the symmetric V -cycle algorithm). There exist positive mesh-independent constants C and m∗ such that C ∀ v ∈ VK , K ≥ 1 , m ≥ m∗ . (5.8) |||EK,m v|||1,K ≤ α |||v|||1,K m Proof. The case where K = 1 is trivial. Let v ∈ VK (K ≥ 2) be arbitrary and k Pkk−1 )Rkm Tk,K,m v . vk = (Idk − Ik−1

(5.9)

Then (2.17), (5.9) and Corollary 4.8 imply that (5.10)

aK (EK,m v, EK,m v) = aK

K X

TK,k,m Rkm vk ,

k=2

. m−α

K X k=2

for m sufficiently large.

K X k=2

h−2α |||vk |||21−α,k k

TK,k,m Rkm vk



NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1053

From (3.4) and (5.9) we have (5.11)

−α/2

|||vk |||1−α,k = |||Ak

k (Idk − Ik−1 Pkk−1 )Rkm Tk,K,m v|||1,k .

Combining (3.3), (5.7), (5.10)–(5.11), Lemma 5.1 and the Cauchy-Schwarz inequality, we find |||EK,m v|||21,K = aK (EK,m v, EK,m v) . m−α aK (EK,m v, v) ≤ m−α |||Ek,m v|||1,K |||v|||1,K ≤ m−2α |||v|||21,K 

and (5.8) follows. Finally we prove the convergence of the F -cycle algorithm.

Theorem 5.3 (Convergence of the F -cycle algorithm). There exist positive meshindependent constants C and m∗ such that ˜ K,m v|||1,K ≤ C |||v|||1,K ∀ v ∈ VK , K ≥ 1 , m ≥ m∗ . (5.12) |||E mα Proof. Suppose that ˜ k−1,m v|||1,k−1 ≤ ηk−1 |||v|||1,k−1 ∀ v ∈ Vk−1 . (5.13) |||E From (2.24), (3.11), (3.15)–(3.18), (5.13) and Theorem 5.2, we have, for m sufficiently large, ˜ k−1,m P k−1 Rm v|||1,k ˜ k,m v|||1,k ≤ |||Rm (Idk − I k P k−1 )Rm v|||1,k + |||I k Ek−1,m E |||E . . .

k k−1 k k k−1 k k k−1 k−1 m −α/2 −α k m −α ˜ m hk |||(Idk − Ik−1 Pk )Rk v|||1−α,k + m |||Ek−1,m Pk Rk v|||1,k−1 −α/2 α m hk |||Rkm v|||1+α,k + m−α ηk−1 |||Pkk−1 Rkm v|||1,k−1 m−α (1 + ηk−1 )|||v|||1,k ∀ v ∈ Vk .

In other words we have, ˜ k,m v|||1,k ≤ ηk |||v|||1,k (5.14) |||E

∀ v ∈ Vk , m ≥ m† ,

where ηk = C‡ m−α (1 + ηk−1 )

(5.15)

and the positive constants m† and C‡ are mesh-independent. In view of (2.23) and (5.13)–(5.15), we can obtain by mathematical induction C‡ ˜ k,m v|||1,k ≤ |||v|||1,k ∀ v ∈ Vk , k ≥ 1 , m ≥ m∗ , (5.16) |||E α m − C‡ 1/α

provided m∗ ≥ max(m† , C‡ (5.16).

). The estimate (5.12) follows immediately from 

6. An abstract framework for nonconforming multigrid methods In order to apply the convergence results in Section 5 to a specific nonconforming multigrid method, one must verify the assumptions (3.6)–(3.12). This can be accomplished through the framework developed in [21]. Indeed the standard discrete estimate (3.6) and inverse estimate (3.7) (cf. [30], [25]) are the assumptions (P) and (I) in [21], and (3.11) is established in Lemma 4.2 there. The truly new ingredients among the assumptions in Section 3 are therefore the estimates (3.8)–(3.10) and (3.12). We will show in this section that they can be derived using the framework in [21] for second order problems and four additional

1054

SUSANNE C. BRENNER

conditions (cf. (6.10)–(6.13) below). In the following discussion we will rely heavily on the results in [21]. 6.1. Results from [21] and new conditions. A key ingredient of the theory in [21] is the relation between the nonconforming finite element space Vk and a conforming finite element space V˜k ⊂ H01 (Ω) (referred to as a conforming relative of Vk in [21]). These spaces are connected by the linear maps Ek : Vk −→ V˜k and Fk : V˜k −→ Vk . Two of the properties of these maps are (cf. (FE) and Lemma 3.1 in [21]): (6.1)

Fk ◦ Ek = Idk ,

(6.2)

v kL2 (Ω) kFk v˜kL2 (Ω) . k˜

and kFk v˜kak . k˜ v kH 1 (Ω)

∀ v˜ ∈ V˜k .

Let ζ ∈ H 1+α (Ω) ∩ H01 (Ω), ζk ∈ Vk and ζk−1 ∈ Vk−1 be related by ∀ v ∈ Vk ,

a(ζ, Ek v) = ak (ζk , v) a(ζ, Ek−1 v) = ak−1 (ζk−1 , v)

∀ v ∈ Vk−1 .

Then the following estimates are valid within the framework in [21] (cf. Theorem 3.5 and Lemma 3.7 there): kζ − ζk kak . hα k kζkH 1+α (Ω) ,

(6.3) (6.4)

|||Πk ζ − ζk |||1−α,k . h2α k kζkH 1+α (Ω) ,

(6.5)

|||ζk−1 − Pkk−1 ζk |||1−α,k−1 . h2α k kζkH 1+α (Ω) ,

where Πk : H01 (Ω) −→ Vk is an interpolation operator. k and Πk are also established within the The following estimates concerning Ik−1 framework in [21] (cf. Lemma 3.3, (Π-1), (Π-2) and (I-2) there): k v|||s,k . |||v|||s,k−1 |||Ik−1

(6.6) (6.7)

kζ − Πk ζkL2 (Ω) . hk |ζ|H 1 (Ω) kζ − Πk ζkak . hα k kζkH 1+α (Ω)

(6.8) (6.9)

k Πk−1 ζ|||1−α,k . h2α |||Πk ζ − Ik−1 k kζkH 1+α (Ω)

∀ v ∈ Vk−1 , 0 ≤ s ≤ 1 , ∀ ζ ∈ H01 (Ω) , ∀ ζ ∈ H 1+α (Ω) ∩ H01 (Ω) , ∀ ζ ∈ H 1+α (Ω) ∩ H01 (Ω) .

We will derive the estimates (3.8)–(3.10) and (3.12) using (6.1)–(6.9) and four k and the inadditional conditions imposed on the intergrid transfer operator Ik−1 terpolation operator Πk . Four additional conditions. We assume that, in addition to the conditions (I-1) and (I-2) in [21], (6.10)

k 2 v|||20,k ≤ (1 + θ2 )|||v|||20,k−1 + C0 θ−2 h2α |||Ik−1 k |||v|||α,k−1

for all v ∈ Vk−1 and θ ∈ (0, 1), where the positive constant C0 is mesh-independent. Furthermore, the interpolation operator Πk−1 : H01 (Ω) −→ Vk−1 actually maps the larger space H01 (Ω) + Vk into Vk−1 and satisfies, in addition to (Π-1) and (Π-2) in [21], the estimates (6.11) kΠk−1 vkak−1 . kvkak

∀ v ∈ H01 (Ω) + Vk ,

(6.12) kΠk−1 v − vkL2 (Ω) . hk kvkak (6.13)

|||Πk−1 v|||20,k−1

≤ (1 + θ

where the positive constant

2

C00

)|||v|||20,k

∀ v ∈ Vk , +

2 C00 θ−2 h2α k |||v|||α,k

is mesh-independent.

∀ v ∈ Vk , θ ∈ (0, 1) ,

NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1055

6.2. Derivation of (3.8)–(3.10) and (3.12). We begin by introducing an operator Jk . Let Qk : L2 (Ω) −→ V˜k be the L2 -orthogonal projection operator. Then we define Jk φ = Fk Qk φ

(6.14)

∀ φ ∈ L2 (Ω) .

Lemma 6.1. The operator Jk satisfies (6.15)

Jk Ek v = v |||Jk ζ|||1−α,k . kζkH 1−α (Ω)

(6.16)

∀ v ∈ Vk , ∀ ζ ∈ H01−α (Ω) .

Proof. The relation (6.15) follows immediately from (6.1) and (6.14). From (3.2), (3.3), (3.6), (6.2) and standard properties of Qk (cf. [15]) we have (6.17)

|||Jk ζ|||0,k ≈ kFk Qk ζkL2 (Ω) . kQk ζkL2 (Ω) . kζkL2 (Ω)

∀ ζ ∈ L2 (Ω) ,

(6.18)

|||Jk ζ|||1,k = kFk Qk ζkak . kQk ζkH 1 (Ω) . kζkH 1 (Ω)

∀ ζ ∈ H01 (Ω) .

The estimate (6.16) follows from (6.17), (6.18) and interpolation between Hilbert scales (cf. [48], [38], [16]).  Lemma 6.2. Given ζk ∈ Vk , let ζ ∈ H01 (Ω) be defined by a(ζ, φ) = ak (ζk , Jk φ)

(6.19)

∀ φ ∈ H01 (Ω) .

Then we have (6.20)

a(ζ, Ek v) = ak (ζk , v)

(6.21)

|ζ|H 1 (Ω) . |||ζk |||1,k ,

(6.22)

kζkH 1+α (Ω) . |||ζk |||1+α,k ,

(6.23)

kΠk ζkak . |||ζk |||1,k .

∀ v ∈ Vk ,

Proof. The relation (6.20) follows immediately from (6.15) and (6.19). From (3.3), (6.18), (6.19) and the Cauchy-Schwarz inequality we have a(ζ, ζ) = ak (ζk , Jk ζ) ≤ kζk kak kJk ζkak . |||ζk |||1,k kζkH 1 (Ω) . which implies (6.21) in view of (2.6). The estimate (6.23) then follows immediately from (2.6), (6.11) and (6.21). Using (3.5), (6.16) and (6.19), we find ak (ζk , Jk φ) ≤ |||ζk |||1+α,k |||Jk φ|||1−α,k . |||ζk |||1+α,k kφkH 1−α (Ω)

∀ φ ∈ H01 (Ω) .

Thus the right-hand side of (6.19) defines a linear functional F on H01 (Ω) which actually belongs to H −1+α (Ω) and (6.24)

kF kH −1+α (Ω) . |||ζk |||1+α,k .

The estimate (6.22) follows from (1.4) and (6.24).



We are now ready to derive the estimates (3.8)–(3.10) and (3.12). In the following derivations we use C to denote a generic mesh-independent positive constant that is also independent of the parameter θ. Lemma 6.3. The estimate (3.8) holds.

1056

SUSANNE C. BRENNER

Proof. Let ζk−1 ∈ Vk−1 be arbitrary and ζ ∈ H01 (Ω) be defined by (6.25)

a(ζ, φ) = ak−1 (ζk−1 , Jk−1 φ)

∀ φ ∈ H01 (Ω) .

Then it follows from (1.8b), (2.1), (2.5), (3.3), (3.14), (6.3), (6.6), (6.8), (6.9), (6.25) and Lemma 6.2 that
2 k k ζk−1 |||21,k ≤ kζk−1 kak + kζk−1 − Ik−1 ζk−1 kak |||Ik−1 ≤ (1 + θ2 )kζk−1 k2ak−1 + Cθ−2 kζk−1 − ζk2ak + kζ − Πk ζk2ak k k + kΠk ζ − Ik−1 Πk−1 ζk2ak + kIk−1 (Πk−1 ζ − ζk−1 )k2ak




2 ≤ (1 + θ2 )|||ζk−1 |||21,k−1 + Cθ−2 h2α k |||ζk−1 |||1+α,k−1 .

 Lemma 6.4. The estimate (3.9) holds. Proof. Let C∗ be a constant that is greater than or equal to the constants C0 and C1 in (6.10) and (3.8), and define, for any θ ∈ (0, 1), (6.26)

α hv1 , v2 ik−1,θ = (1 + θ2 )(v1 , v2 )k−1 + C∗ θ−2 h2α k (Ak−1 v1 , v2 )k−1

for all v1 , v2 ∈ Vk−1 . Note that Ak−1 is symmetric positive definite with respect to the inner product h·, ·ik−1,θ . It follows from (3.1), (3.8), (6.10) and (6.26) that k v|||20,k ≤ hA0k−1 v, vik−1,θ |||Ik−1

∀ v ∈ Vk−1 ,

≤ hAk−1 v, vik−1,θ

∀ v ∈ Vk−1 ,

k v|||21,k |||Ik−1

which imply through (3.1) and interpolation between Hilbert scales k 2 2 −2 2α v|||21−α,k ≤ hA1−α hk |||v|||21,k−1 . |||Ik−1 k−1 v, vik−1,θ = (1 + θ )|||v|||1−α,k−1 + C∗ θ

 Lemma 6.5. It holds that (6.27)

2 |||Πk−1 v|||21,k−1 ≤ (1 + θ2 )|||v|||21,k + C\ θ−2 h2α k |||v|||1+α,k

for all v ∈ Vk and θ ∈ (0, 1), where the positive constant C\ is mesh-independent. Proof. Let ζk ∈ Vk be arbitrary and define ζ ∈ H01 (Ω) by (6.19). From (1.8b), (2.1), (2.5), (3.1), (6.3), (6.8), (6.11) and Lemma 6.2 we have
2 |||Πk−1 ζk |||21,k−1 ≤ kζk kak + kζk − Πk−1 ζk kak ≤ (1 + θ2 )|||ζk |||21,k + Cθ−2 kζk − ζk2ak + kζ − Πk−1 ζk2ak + kΠk−1 (ζ − ζk )k2ak




2 ≤ (1 + θ2 )|||ζk |||21,k + Cθ−2 h2α k |||ζk |||1+α,k .

 The next lemma follows from (6.13), (6.27) and interpolation between Hilbert scales, as in the proof of Lemma 6.4. Lemma 6.6. It holds that (6.28)

2 |||Πk−1 v|||21−α,k−1 ≤ (1 + θ2 )|||v|||21−α,k + C θ−2 h2α k |||v|||1,k

for all v ∈ Vk and θ ∈ (0, 1), where the positive constant C is mesh-independent.

NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1057

We need one more estimate for the derivation of (3.10). Lemma 6.7. The following estimate holds: (6.29)

kΠk−1 ζ − Πk−1 Πk ζkL2 (Ω) . hk |ζ|H 1 (Ω)

∀ ζ ∈ H01 (Ω) .

Proof. From (2.1), (2.6), (6.7), (6.11) and (6.12) we have kΠk−1 ζ − Πk−1 Πk ζkL2 (Ω) ≤ kΠk−1 ζ − ζkL2 (Ω) + kζ − Πk ζkL2 (Ω) + kΠk ζ − Πk−1 Πk ζkL2 (Ω) . hk |ζ|H 1 (Ω) + hk kΠk ζkak . hk |ζ|H 1 (Ω) .  Lemma 6.8. The estimate (3.10) holds. Proof. Let ζk ∈ Vk be arbitrary. Define ζ ∈ H01 (Ω) by (6.19) and ζk−1 ∈ Vk−1 by ak−1 (ζk−1 , v) = a(ζ, Ek−1 v)

∀ v ∈ Vk−1 .

We have, by (1.8b), |||Pkk−1 ζk |||21−α,k−1 (6.30)

≤ |||Πk−1 Πk ζ|||1−α,k−1 + |||Πk−1 Πk ζ − Pkk−1 ζk |||1−α,k−1


2

≤ (1 + θ2 )|||Πk−1 Πk ζ|||21−α,k−1 + Cθ−2 |||Πk−1 Πk ζ − Pkk−1 ζk |||21−α,k−1 . The first term on the right-hand side of (6.30) can be estimated using (1.8b), (3.3), (3.14), (6.4), (6.28) and Lemma 6.2: 2 |||Πk−1 Πk ζ|||21−α,k−1 ≤ (1 + θ2 )|||Πk ζ|||21−α,k + Cθ−2 h2α k |||Πk ζ|||1,k
2 2 ≤ (1 + θ2 ) |||ζk |||1−α,k + |||Πk ζ − ζk |||1−α,k + Cθ−2 h2α k |||ζk |||1,k

(6.31)

2 −2 2α ≤ (1 + θ2 )2 |||ζk |||21−α,k + Cθ−2 h4α hk |||ζk |||21,k k kζkH 1+α (Ω) + Cθ −2 2α hk |||ζk |||21,k ≤ (1 + θ2 )2 |||ζk |||21−α,k + Cθ−2 h4α k |||ζk |||1+α,k + Cθ 2 ≤ (1 + θ2 )2 |||ζk |||21−α,k + Cθ−2 h2α k |||ζk |||1,k .

Similarly the second term on the right-hand side of (6.30) can be estimated using (2.1), (3.2), (3.6), (3.14), (6.4), (6.5), (6.29) and Lemma 6.2: |||Πk−1 Πk ζ − Pkk−1 ζk |||21−α,k−1 (6.32)

≤ |||Πk−1 (Πk ζ − ζ)|||1−α,k−1 + |||Πk−1 ζ − ζk−1 |||1−α,k−1
2 + |||ζk−1 − Pkk−1 ζk |||1−α,k−1
2(α−1) 2 |||Πk−1 (Πk ζ − ζ)|||20,k−1 + h2α ≤ C hk k |||ζk |||1,k 2 ≤ Ch2α k |||ζk |||1,k .

Combining (6.30)–(6.32), we have 2 |||Pkk−1 ζk |||21−α,k−1 ≤ (1 + θ2 )3 |||ζk |||21−α,k + Cθ−2 h2α k |||ζk |||1,k ,

which implies (3.10) since θ ∈ (0, 1) is arbitrary.



We now turn to the derivation of (3.12) by first noting that the estimate (6.33)

|||Pkk−1 v|||1−α,k−1 . |||v|||1−α,k

follows from (3.10) and (3.14).

∀ v ∈ Vk

1058

SUSANNE C. BRENNER

Lemma 6.9. The estimate (3.12) holds. Proof. Let ζk−1 ∈ Vk−1 be arbitrary and define ζ ∈ H01 (Ω) and ζk ∈ Vk by (6.25) and (6.20), respectively. Then it follows from (2.1), (3.1), (3.5), (3.14), (6.4)–(6.6), (6.9), (6.33) and Lemma 6.2 that
k )ζk−1 , w ak−1 (Idk−1 − Pkk−1 Ik−1
k ζk−1 − ζk ), w = ak−1 (ζk−1 − Pkk−1 ζk , w) − ak−1 Pkk−1 (Ik−1 ≤ |||ζk−1 − Pkk−1 ζk |||1−α,k−1 |||w|||1+α,k−1 k + |||Pkk−1 (Ik−1 ζk−1 − ζk )|||1−α,k−1 |||w|||1+α,k−1 k . h2α k kζkH 1+α (Ω) |||w|||1+α,k−1 + |||Ik−1 (ζk−1 − Πk−1 ζ)|||1−α,k
k + |||Ik−1 Πk−1 ζ − Πk ζ|||1−α,k + |||Πk ζ − ζk |||1−α,k |||w|||1+α,k−1

. h2α k kζkH 1+α (Ω) |||w|||1+α,k−1 . h2α k |||ζk−1 |||1+α,k−1 |||w|||1+α,k−1 . hα k |||ζk−1 |||1,k−1 |||w|||1+α,k−1 for all w ∈ Vk−1 , which implies (3.12) because of (3.5).



Remark 6.10. The proof of Lemma 6.9 actually establishes the stronger estimate k )v|||1−α,k−1 . h2α |||(Idk−1 − Pkk−1 Ik−1 k |||v|||1+α,k−1

∀ v ∈ Vk−1 .

7. Applications In this section we apply the theory developed in Sections 3–6 to two nonconforming finite element methods for the variational problem (1.1). As demonstrated in [21], both of these methods satisfy the assumptions of the framework developed in that paper. Therefore, from the discussion in Section 6, it only remains for us to check the additional conditions (6.10)–(6.13) for these examples. 7.1. The nonconforming P1 finite element method. Let e be (a segment of) an edge of T ∈ Tk . We define Z 1 v T ds for v ∈ H 1 (T` ) , ` ≥ 1 . (7.1) Me,T (v) = |e| e Remark 7.1. In the case where ` > k and e is an edge of T , the integral on the right-hand side of (7.1) should be interpreted as Z e

2X v T ds =

`−k

j=1

Z ej

v T ds j

where e1 , . . . , e2`−k are the edges from T` that form a partition of e and Tj ⊂ T is a triangle in T` with ej as an edge (cf. Figure 7.1 for the case ` − k = 2). The nonconforming P1 finite element space Vk is defined by (cf. [31]) Vk = {v ∈ L2 (Ω) : v T ∈ h1, x1 , x2 i ∀ T ∈ Tk , Me,T (v) = Me,T 0 (v) if e (7.2)

is the common edge of T and T 0 in Tk , and Me,T (v) = 0

if e ⊂ ∂Ω is an edge of T } .

NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1059

e4 T4 e3 T3 e2

T

T2 e1 T1

Figure 7.1. Interpretation of (7.1) when ` − k = 2 Henceforth we will denote Me,T (v) by Me (v) if there is no ambiguity about T or if v has an unambiguous mean value on e. Let Ek be the set of the internal edges of Tk . We define (·, ·)k by X Me (v1 )Me (v2 ) ∀ v1 , v2 ∈ Vk . (7.3) (v1 , v2 )k = h2k P∞

e∈Ek

Let `=1 H 1 (T` ) be the subspace of L2 (Ω) whose members are finite sums of functions the spaces H 1 (T1 ), H 1 (T2 ), . . .. The weak interpolation operator P∞ from 1 Πk : `=1 H (T` ) −→ Vk is defined by X
1 Me,T (v) ∀ e ∈ Ek . (7.4) Me Πk v = 2 e⊂∂T,T ∈Tk

Note that there are exactly two triangles in the sum and the mean value of Πk v on e is just the average of the mean values of v on e from these two triangles. The restriction of Πk to H01 (Ω) is precisely the weak interpolation operator used k : Vk−1 −→ Vk in in [21] (cf. also [19], [6]), and the intergrid transfer operator Ik−1 [21] is just the restriction of Πk to Vk−1 . The following lemma, a simple consequence of the Bramble-Hilbert lemma (cf. [8]) and scaling, is the key to the estimates (6.10)–(6.13). Lemma 7.2. Let e1 , e2 be two of the edges of T ∈ Tk . Then we have |Me1 (v) − Me2 (v)| . |v|H 1 (T )

∀ v ∈ H 1 (T ) .

Verification of (6.11). Let e ∈ Ek−1 be the common edge of two triangles T and T 0 in Tk−1 , and let e1 (resp. e2 ) be the half of the edge e neighboring the two triangles T1 and T2 (resp. T3 and T4 ) in Tk (cf. Figure 7.2).

T2 T1 e1

T

T3

e2 T4

T

Figure 7.2. Two neighboring triangles in Tk−1

1060

SUSANNE C. BRENNER

Then for v ∈ H01 (Ω) + Vk , we have  1 (7.5) Me,T (v) = Me1 (v) + Me2 (v) = Me,T 0 (v) . 2 In view of (7.4) and (7.5), the interpolation operator Πk−1 : H01 (Ω) + Vk −→ Vk−1 can be analyzed locally on each triangle of Tk−1 . Let T be a triangle in Tk−1 subdivided into four triangles T1 , . . . , T4 and let e1 , . . . , e6 be the edges from Tk that are on ∂T (cf. Figure 7.3). For any v ∈ H01 (Ω)+Vk we obtain from (7.4), (7.5) and scaling |Πk−1 v|2H 1 (T ) .

(7.6)

6 X

|Mei (v) − Mej (v)|2 .

i,j=1

Note that Lemma 7.2 implies (7.7)

|Mei (v) − Mej (v)|2 .

4 X

|v|2H 1 (T` )

∀ v ∈ H 1 (T ) + Vk , 1 ≤ i, j ≤ 9.

`=1

For example, we have (cf. Figure 7.3) |Me1 (v) − Me4 (v)|2 . |Me1 (v) − Me7 (v)|2 + |Me7 (v) − Me9 (v)|2 + |Me9 (v) − Me4 (v)|2 . |v|2H 1 (T1 ) + |v|2H 1 (T4 ) + |v|2H 1 (T3 ) . It follows from (7.6) and (7.7) that (7.8)

|Πk−1 v|2H 1 (T ) .

4 X

|v|2H 1 (T` )

∀ v ∈ H 1 (T ) + Vk .

`=1

We obtain by summing (7.8) over all the triangles in Tk−1 X Z X Z (7.9) |∇(Πk−1 v)|2 dx . |∇v|2 dx ∀ v ∈ H01 (Ω) + Vk . T ∈Tk−1

T

T ∈Tk

T

The estimate (6.11) then follows from (1.3), (2.3), (2.4), (7.9) and the Poincar´e inequality for nonconforming P1 finite element functions (cf. [24] and the references therein).  Let T be a generic triangle in Tk−1 subdivided into four triangles T1 , . . . , T4 ∈ Tk (cf. Figure 7.3). Henceforth we will denote by e1 , . . . , e9 the edges of T1 , . . . , T4 (cf.

e5

e4

T3 e9

e6 e7

T1 e1

T4

e8 T 2

e3

e2

Figure 7.3. A divided triangle in Tk−1

NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1061

Figure 7.3) and denote by e˜j (1 ≤ j ≤ 3) the edge of T that contains the edges e2j−1 and e2j from Tk . Verification of (6.12). Let T be a generic triangle in Tk−1 . For any v ∈ Vk , we have by (7.4) and (7.5) Me˜j (Πk−1 v) =

(7.10)

 1 Me2j−1 (v) + Me2j (v) . 2

Observe also that Me˜j (w) =

 1 Me2j−1 (w) + Me2j (w) 2

∀ w ∈ H 1 (T ) ,

which together with (7.7) implies (7.11)

|Me˜j (w) − Mei (w)| . |w|H 1 (T )

∀ w ∈ H 1 (T ) , 1 ≤ j ≤ 3 , 1 ≤ i ≤ 9 .

We have by scaling

(7.12)

kΠk−1 v − vk2L2 (T ) . h2k

9 X 2  Mej (Πk−1 v) − Mej (v) , j=1

and it follows from (7.7), (7.8), (7.10) and (7.11) that

(7.13)

9 X 

4 2 X |v|2H 1 (T` ) + |Πk−1 v|2H 1 (T ) Mej (Πk−1 v) − Mej (v) .

j=1

`=1

.

4 X

|v|2H 1 (T` ) .

`=1

Combining (7.12) and (7.13), we find kΠk−1 v − vk2L2 (T ) . h2k

4 X

|v|2H 1 (T` ) .

`=1

In view of (1.3), (2.3) and (2.4), we obtain the estimate (6.12) by summing up the  preceding estimate over all the triangles in Tk−1 . Verification of (6.13). Denote by ET the set of the three edges of T . Then we have by (7.2) and (7.3) (7.14)

(v, v)k =

h2k X X [Me (v)]2 2 T ∈Tk e∈ET

∀ v ∈ Vk .

1062

SUSANNE C. BRENNER

The relation (7.14) allows us to focus on a generic T ∈ Tk−1 in the derivation of (6.13). From (1.8), (7.7) and (7.10) we have 3 X j=1

(7.15)

1X [Mei (v)]2 2 i=1 6

[Me˜j (Πk−1 v)]2 ≤

=

6 3
2 1X 1 Xn [Mei (v)]2 + Mej+6 (v) − [Mej+6 (v) − Me2j−1 (v)] 4 i=1 4 j=1
2 o + Mej+6 (v) − [Mej+6 (v) − Me2j (v)]

X (1 + θ2 ) X 1X [Mei (v)]2 + [Mei (v)]2 + Cθ−2 |v|2H 1 (T` ) 4 i=1 2 i=7 6

=

9

4

`=1

for all v ∈ Vk and θ ∈ (0, 1), where the positive constant C is mesh-independent. Summing up the estimate (7.15) over all the triangles in Tk−1 and using (1.3), (2.1), (2.3), (2.4), (3.2), (3.3), (7.2), (7.3) and (7.14), we arrive at |||Πk−1 v|||20,k−1 = (7.16)

h2k−1 2

X

X

[Me˜(v)]2

T ∈Tk−1 e˜∈ET

≤ (1 + θ2 )

X h2k−1 X [Me (v)]2 + Cθ−2 h2k |v|2H 1 (T ) 4 T ∈Tk

e∈Ek

≤ (1 + θ2 )(v, v)k + Cθ−2 h2k kvk2ak = (1 + θ2 )|||v|||20,k + Cθ−2 h2k |||v|||21,k . 

The estimate (6.13) follows from (3.14) and (7.16).

Verification of (6.10). Let e be an edge of Tk which is on the common boundary of T, T 0 ∈ Tk−1 (cf. Figure 7.2). Then we have by (1.8a) and (7.4) (7.17)

k v)]2 = [Me (Ik−1

2 1 1 1 Me,T (v) + Me,T 0 (v) ≤ [Me,T (v)]2 + [Me,T 0 (v)]2 . 4 2 2

For T ∈ Tk−1 , let ETb (resp. ETi ) denote the set of the edges from Tk that are on ∂T (resp. interior to T ). We obtain from (7.3) and (7.17) the estimate (7.18)

k k v, Ik−1 v)k ≤ h2k (Ik−1

 X X 1 X [Me,T (v)]2 + [Me (v)]2 2 i b

T ∈Tk−1

e∈ET

e∈ET

for all v ∈ Vk−1 . Again the estimate (7.18) allows us to focus on a generic T ∈ Tk−1 . It follows from (1.8b) and (7.11) that (7.19)

6 9 3 X X 1X [Mei (v)]2 + [Mei (v)]2 ≤ 2(1 + θ2 ) [Me˜j (v)]2 + Cθ−2 |v|2H 1 (T ) 2 i=1 i=7 j=1

for all v ∈ H 1 (T ) and θ ∈ (0, 1), where the positive constant C is mesh-independent.

NONCONFORMING V -CYCLE AND F -CYCLE MULTIGRID ALGORITHMS

1063

Summing up the estimate (7.19) over all the triangles in Tk−1 , we find by (1.3), (2.1), (2.3), (2.4), (3.2), (3.3), (7.14) and (7.18) k k k v|||20,k = (Ik−1 v, Ik−1 v)k |||Ik−1  X 1 X X ≤ h2k [Me,T (v)]2 + [Me (v)]2 (7.20) 2 i b T ∈Tk−1

≤ (1 + θ )

e∈ET

e∈ET

2 2 hk−1

2

X

X

[Me˜(v)]2 + Cθ−2 h2k

T ∈Tk−1 e˜∈ET

X

|v|2H 1 (T )

T ∈Tk−1

≤ (1 + θ2 )|||v|||20,k−1 + Cθ−2 h2k |||v|||21,k−1 for all v ∈ Vk−1 and θ ∈ (0, 1). The estimate (6.10) follows from (3.14) and (7.20).



7.2. The rotated Q1 finite element method. We shall adopt the notation in subsection 7.1 with obvious modifications. For simplicity we assume that T1 (and hence any Tk ) is a triangulation of Ω by rectangles whose sides are parallel to the coordinate axes. The rotated Q1 finite element space (cf. [42]) is defined by Vk = {v ∈ L2 (Ω) : v ∈ h1, x1 , x2 , x21 − x22 i ∀ R ∈ Tk , Me,R (v) = R

(7.21)

Me,R0 (v) if e is the common edge of R and R0 in Tk , and Me,R (v) = 0 if e ⊂ ∂Ω is an edge of R} .

We define the inner product (·, ·)k by X Me (v1 )Me (v2 ) (7.22) (v1 , v2 )k = h2k

∀ v ∈ Vk .

e∈Ek

P 1 The weak interpolation operator Πk : ∞ `=1 H (T` ) −→ Vk is defined by X 1 Me (v) ∀ e ∈ Ek . (7.23) Me (Πk v) = 2 e⊂∂R,R∈Tk

H01 (Ω)

is precisely the weak interpolation operator used in The restriction of Πk to k : Vk−1 −→ Vk used in [21] is just the [21], and the intergrid transfer operator Ik−1 restriction of Πk to Vk−1 . The following analog of Lemma 7.2 is again a simple consequence of the BrambleHilbert lemma and scaling. Lemma 7.3. Let e1 and e2 be two edges of the rectangle R ∈ Tk . Then we have |Me1 (v) − Me2 (v)| . |v|H 1 (R)

∀ v ∈ H 1 (R) .

Using Lemma 7.3, the verification of (6.10)–(6.13) proceeds as in subsection 7.1 and we therefore omit the details. Remark 7.4. The results of this paper can also be applied to other nonconforming quadrilateral elements (cf. [33], [27], [28], [26]). Remark 7.5. The nonconforming P1 and the rotated Q1 finite elements are equivalent to the lowest order triangular and rectangular Raviart-Thomas mixed finite elements (cf. [43], [2], [1]). The results of this paper can therefore be applied to multigrid methods for the lowest order triangular and rectangular Raviart-Thomas mixed methods (cf. [20], [1]).

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