A Cascade Algorithm for the Stokes Equations Dietrich Braess
Wolfgang Dahmen
Abstract A variant of multigrid schemes for the Stokes problem is discussed. In particular, we propose and analyse a cascadic version for the Stokes problem. The analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms.
Key words: Saddle point problems, Stokes problem, cascadic iteration, smoother, projection method
1 Introduction Multigrid methods without coarse grid corrections have been defined and applied to elliptic problems of second order by Bornemann and Deuflhard [1]. They have called it a cascadic algorithm and showed that an optimal iteration method with respect to the energy norm is obtained if conforming elements are used. Deuflhard’s starting point for the cascadic multigrid method was the idea that it should be sufficient to start the iteration at the level i with a good approximation from the level i −1. A similar idea can already be found in Chapter 9 of Wachspress’ book [10] from 1966, i.e. from the period in which also the first theoretical investigations of multigrid methods were made. Later Shaidurov [7] established in essence a recursion relation of the form kui − vi k1 ≤ kui−1 − vi−1 k1 + c
hi mi
(1.1)
for some finite element problems with full regularity. Here ui denotes the exact solution on the level i and vi its approximation computed after mi steps. The accumulation of the error is no problem since the iteration steps on the lower levels are cheap. It is crucial for the optimality of the algorithm that the error from the previous level enters with a factor of precisely 1. Since it was not clear whether a constant factor greater than 1 is encountered in the transfer for nonconforming elements, there are no serious conjectures for the latter families. There is another difference to classical multigrid algorithms. The recursion relation (1.1) refers only to the energy norm, and it has been proved in [2] that the cascadic version is in general not optimal for the L2 -norm. This is in contrast to classical multigrid algorithms, see [6, 11], where one can more easily move between the H 1 -norm and the L2 -norm. 1
We will develop the cascadic multigrid method for saddle point problems which arise from the Stokes problem. Here we will apply the smoothing procedure proposed in [4]. Although there is the drawback with the L2 -norm mentioned above, we are able to apply a duality technique which eventually yields optimality for our saddle point problems. Since this in turn is related to a careful analysis of the transfer between the grids, the technique is also useful in the treatment of nonconforming elements. We note that the cascade algorithm offers an efficient alternative to nested iteration for obtaining a good initial guess of the finite element solution. It is not our intention to replace the standard multigrid procedure.
2 Notation and Problem Formulation Let ⊂ R d , d = 2 or 3, and let H s (), H0s () denote the usual SobolevRspaces endowed with the Sobolev norms k · ks . The space L2,0 () := {q ∈ L2 () : qdx = 0} can
be identified with L2 ()/R . The weak formulation of the Stokes problem reads: Find u ∈ X := H01 ()d and p ∈ M := L2,0 () such that a(u, v) + b(v, p) = hf, vi for all v ∈ X, b(u, q) = 0 for all q ∈ M.
(2.1)
Here, f ∈ X′ , the dual of X, is given, h·, ·i is the standard duality pairing induced by the L2 inner product and Z a(u, v) := ∇u∇vdx,
b(v, q) := −
Z
div vqdx.
(2.2)
We assume that the problem is H 2 -regular, e.g. may be a bounded convex polyhedral domain in 2-space. We are interested in approximate solutions to (2.1) obtained by finite element discretizations. To this end we assume that for each i ∈ N 0 , i ≤ J , Ti denotes a shape regular triangulation of which is generated by successively refining uniformly some initial triangulation T0 . Shape regularity means that the ratio of the diameter and the radius of the largest inscribed ball of any simplex in Ti remains bounded. Accordingly, Xi and Mi will denote the corresponding conforming finite element spaces of Taylor and Hood [3, 5]. Likewise we may use any elements with the properties listed in [9]. In particular, the finite element spaces are nested and form an ascending hierarchy of spaces X0 ⊂ X1 ⊂ · · · ⊂ XJ ⊂ X,
M0 ⊂ M1 ⊂ · · · ⊂ MJ ⊂ M.
Restricting (2.1) to the pair Xi , Mi , gives rise to the linear system of equations fi ui Ai BiT , = Bi 0 qi 2
(2.3)
where as usual the operators Ai , Bi on Xi are for ui ∈ Xi defined by (Ai ui , v) = a(ui , v),
v ∈ Xi ,
(Bi ui , q) = b(ui , q), q ∈ Mi .
Of course, as soon as one fixes bases in Xi and Mi , one obtains matrix representations of Ai , Bi which will be denoted again by Ai , Bi , respectively. For simplicity we identify the functions vi , qi in Xi , Mi with their coefficient sequences, always assuming that the bases are normalized so that kvi k0 ∼ kvi kℓ2 . (2.4) That is, both norms can be uniformly bounded by constant multiples of each other. Moreover we have the inverse inequalities kvi k1 ≤ ch−1 i kvi k0 ,
vi ∈ Xi .
(2.5)
Here and throughout the paper c will be a generic constant which is independent of the level and which may be different in different equations. Our objective is to solve (2.3) for the highest level of resolution i = J .
3 The Smoothing Operation A key ingredient of a multigrid scheme for the solution of (2.3) is a suitable smoother. In the following we will employ the smoother proposed in [4]. Since this can be described for an abstract saddle point problem, for convenience we suppress the subscripts indicating the discretization level. Thus we consider the linear sytem of equations f u A BT , (3.1) = B g q where A is a symmetric positive definite matrix. It characterizes the solution of the constrained minimum problem 1 T u Au − f T u → min! subject to Bu = g. 2 Now suppose that C is a preconditioner for A which, in particular, satisfies v T Av ≤ v T Cv,
v ∈ X,
(3.2)
d v , = e q
(3.3)
and for which the linear system
C BT B
is more easily solved. Note that the inverse is formally given by
C BT B
−1
=
C −1 (I − B T S −1 BC −1 ) C −1 B T S −1 S −1 BC −1 −S −1 3
,
(3.4)
where S := BC −1 B T is the Schur complement of (3.3). Specifically, if C = αI , then (3.2) reads v T Av ≤ α v T v, i.e., α is assumed to be not smaller than the spectral radius ρ(A) of A. In this case (3.4) becomes −1 1 T T −1 αI B T P B (BB ) , (3.5) = α T −1 B (BB ) B −α(BB T )−1 where P is the projection P := I − B T (BB T )−1 B.
(3.6)
Now, (3.1) is to be solved by an iteration of the form −1 ℓ ℓ f u u uℓ+1 αI B T A BT , − − := ℓ ℓ ℓ+1 B B g p p p
(3.7)
where superscripts will always denote iteration indices. It is important to note that uℓ+1 always satisfies the constraint, i.e., Buℓ+1 = g,
(3.8)
see [4]. Each iteration step requires solving a system of the form (3.3) with C = αI . By (3.5), this can be realized by implementing BB T q = Bd − αe,
v=
1 (d − B T q). α
Specifically, this amounts to solving an equation similar to the Poisson equation in the case of the Stokes problem. In view of the available efficient Poisson solvers this is acceptable, e.g., smoothers which incorporate Poisson solvers have been used in some efficient multigrid algorithms by Turek [8]. In particular, defining for g = 0 V := {v ∈ X : Bv = 0} the iteration remains in V . Therefore one can construct conjugate directions from the corrections in (3.7). In fact, defining the vector g ℓ := Auℓ + B T pℓ − f as the residual of the first block and computing hℓ from ℓ ℓ h g αI B T , = ℓ B p 0
4
we obtain the next conjugate direction and the next iterate from d ℓ := −hℓ + βℓ d ℓ−1
uℓ+1 := uℓ + αℓ d ℓ .
The factors αℓ and βℓ are determined as in any cg-algorithm. Note that by construction Bhℓ = 0 so that also Bd ℓ = 0, ℓ = 0, 1, . . . . (3.9) Thus one considers the cg-method confined to a subspace where A is definite. The cg-method based on (3.7) will be employed as a smoother in the cascade algorithm in accordance with the concept for scalar equations in [1, 7].
4 The Cascadic Iteration Our objective is to analyse the following CASCADE Algorithm: Compute the exact solution u0 , q0 of (2.3) on level i = 0. Set v0 := u0 . For i = 1, . . . , J : { • Compute wi as the prolongation of vi−1 . • Compute v 0 := vi0 as the projection of wi to Vi := ker Bi . • Execute m = mi steps of the cg-method. • Set vi := v m } Since the spaces Xi , Mi are nested, the prolongation qi−1 7→ q 0
vi−1 7→ wi ,
ℓ in the above scheme is simply the inclusion. However, although each vi−1 and hence vi−1 belong to Vi−1 , its prolongation wi will generally not belong to Vi . The correction can be performed by solving the sytem 0 w˜ C BT = , (4.1) B −Bwi q˜
where we again suppress the level index i in the matrices B, C. In fact, by the remarks at the beginning of Section 3, w˜ minimizes the quadratic functional (Cv, v) under the constraint B w˜ = −Bwi . One easily confirms that w˜ = −C −1 B T (BC −1 B T )−1 Bwi . 5
(4.2)
Hence, v 0 := wi + w˜ = (I − C −1 B T (BC −1 B T )−1 B) =: PC wi .
(4.3)
Thus, since for (u, v)C := (u, Cv) = (Cv, u) (PC z − z, w)C = −(C −1 B T (BC −1 B T )−1 Bz, Cw) = ((BC −1 B T )−1 Bz, Bw) = 0,
for all w ∈ Vi ,
the mapping PC is just the orthogonal projection to Vi with respect to the inner product (·, ·)C . The most convenient choice for C is αI . Noting that PαI = PI =: P (see (3.6)) for any α > 0 this gives rise to the orthogonal projector with respect to the standard L2 -inner product, i.e., kP k0 = 1. (4.4) For completeness, we note that there is also a bound with respect to the k · k1 -norm. Lemma 4.1 For P = Pi defined by (3.6) one has kPi k1 ≤ c uniformly in i ∈ N . Proof: It suffices to prove that S = Si := B T (BB T )−1 B is uniformly bounded in k · k1 . To this end, note that V = ker B is a closed subspace of H01 ()d . Therefore its orthogonal complement V ⊥ with respect to (·, ·)1 exists. Thus any v ∈ Xi can be written as v = z + w with Bz = 0 and w ∈ V ⊥ . Obviously, BSv = Bv = Bw. Hence, kSvk21 = kS(z + w)k21 = kSwk21 .
(4.5)
kwk21 ∼ (Bw, Bw) for w ∈ V ⊥ ,
(4.6)
On the other hand, since
cf. Remark III.5.5 in [3], we obtain kSwk21 ∼ (BSw, BSw) = (Bw, Bw) ∼ kwk21 ≤ kwk21 + kzk21 = kvk21 , and the assertion follows from (4.5).
5 The cg-Method and Optimal Polynomials According to (3.8) in [4] the error in the v-component for the iteration (3.7) is given by u − v ℓ+1 = P (I −
1 1 A)(u − v ℓ ) = (I − P AP )(u − v ℓ ), αℓ αℓ
(5.1)
where P is defined by (3.6). From the theory of the cg-method we know that ku − v m k1 ≤ c ||| u − v m ||| = c inf ||| u − Qm (P AP )v 0 ||| : deg Qm ≤ m, Qm (0) = 1 . 6
(5.2)
1/2
Here the energy norm ||| · ||| is defined by ||| v||| := (v, P AP v)0 , so that in the case of the Stokes problem ||| v||| := |P v|1 . (5.3)
It has been shown by Shaidurov [7] that, given m ∈ N and 3 > 0, there exists a polynomial Qm such that Qm (0) = 1√ √ 3 (5.4) for x ∈ [0, 3], | xQm (x)| ≤ 2m+1 |Qm (x)| ≤ 1 for x ∈ [0, 3].
Since P AP is selfadjoint, following Shaidurov [7] we set λmax := λmax (P AP ) and obtain from (5.4) an operator Qm with √ λmax ||| Qm v||| ≤ kvk0 , v ∈ Xi , 2m + 1 ||| Qm v||| ≤ ||| v||| , v ∈ Xi .
We emphasize that the energy norm is a mesh-dependent norm, since the projector in (5.3) depends on the grid. Therefore we reformulate the above bounds. Since Qm v belongs to Vi , we have ||| Qm v||| = |Qmv|1 , i.e. √ λmax |Qm v|1 ≤ kvk0 , v ∈ Xi , (5.5) 2m + 1 |Qm v|1 ≤ |P v|1 , v ∈ Xi . (5.6) Although during the computations Qm is only applied to functions in the kernel Vi , it is crucial for the analysis that the estimates hold for all v ∈ Xi . Moreover we note that the inverse inequality (2.5) implies λmax = λmax (P AP ) ≤ λmax (A) ≤ ch−2 .
(5.7)
6 A Recursion Relation and Final Estimates In contrast to cascadic iterations for scalar elliptic problems the prolongation of the approximate solution vi−1 on level i − 1 is followed by a correction which projects the prolongated wi to v 0 := Pi wi ∈ Vi . Since Pi is an orthogonal projector relative to the L2 -inner product and therefore generally does not have norm one in H 1 , the relation kv 0 − ui k1 = kPi (wi − ui )k1 does not allows us to directly infer the estimate kv 0 − ui k1 ≤ kwi − ui k1 which would be needed in a convergence analysis following the concepts of [1, 7]. The subsequent discussion indicates the corresponding difficulties. We will overcome them by switching to the |·|1 -projector in the analysis. The terms which arise from the compensation will be estimated by applying the following lemma. 7
Lemma 6.1 There exists a linear mapping Ri : Xi → Vi and a constant c such that k(I − Ri )zi k0 ≤ chi kzi k1 |Ri zi |1 ≤ |zi |1
for all zi ∈ Vi−1 ,
for all zi ∈ Xi .
(6.1) (6.2)
The proof of the lemma will be given in the next section. First, under the regularity assumption kuk2 + kpk1 ≤ ckf k0 , one obtains the L2 -estimate ku − ui k0 ≤ ch2i kf k0 . Thus the triangle inequality yields kui − ui−1 k0 ≤ kui − uk0 + ku − ui−1 k0 ≤ ch2i kf k0 .
(6.3)
Here we assume as usual that hi / hi−1 remains bounded. We are now prepared to analyze the error produced by the scheme described in Section 4. To this end, we recall that ui denotes the exact solution of the discrete problem in Xi , while v 0 = vi0 := Pi wi denotes the starting value for the iteration on the level i. As before wi is the prolongation of the approximate solution vi−1 = v mi−1 in Xi−1 . Abbreviating Qi := Qmi (Pi Ai Pi ), we obtain ui − vi = Qi (ui − Pi vi−1 ) = Qi (ui − Pi ui−1 ) + Qi Pi (ui−1 − vi−1 )
= Qi (ui − Pi ui−1 ) + Qi Ri (ui−1 − vi−1 ) + Qi (Pi − Ri )(ui−1 − vi−1 ). (6.4)
As for the first summand, we invoke (5.5), (5.7), and (6.3) to obtain |Qi (ui − Pi ui−1 )|1
h−1 ≤ c i kui − Pi ui−1 k0 mi h−1 ≤ c i kui − ui−1 k0 mi hi ≤ c kf k0 . mi
The second term of (6.4) is estimated by employing (5.6), Lemma 6.1, and Pi Ri = Ri : |Qi Ri (ui−1 − vi−1 )|1 ≤ |Pi Ri (ui−1 − vi−1 )|1 = |Ri (ui−1 − vi−1 )|1 ≤ |ui−1 − vi−1 |1 . The third summand on the right hand side of (6.4) can be estimated by (5.5): |Qi (Pi − Ri )(ui−1 − vi−1 )|1 ≤ c 8
h−1 i k(Pi − Ri )(ui−1 − vi−1 )k0 . mi
(6.5)
Collecting terms and using Pi − Ri = Pi (I − Ri ) we have |ui − vi |1 ≤ c
h−1 hi kf k0 + |ui−1 − vi−1 |1 + i kPi (I − Ri )(ui−1 − vi−1 )k0 . mi mi
(6.6)
The first two summands on the right hand side of (6.6) correspond to the recursion for the scalar elliptic case. In the present situation we must now deal with the third term due to the nonconformity of the prolongation. From (4.4), Lemma 6.1, and the Poincar´e–Friedrichs inequality we conclude that kPi (I − Ri )(ui−1 − vi−1 |1 )k0 ≤ k(I − Ri )(ui−1 − vi−1 |1 )k0 ≤ chi |ui−1 − vi−1 |1 .
(6.7)
By combining (6.6) and (6.7) we obtain immediately the recursion relation in the following proposition. Proposition 6.2 There exists a constant c such that |ui − vi |1 ≤ c
c hi kf k0 + (1 + ) |ui−1 − vi−1 |1 . mi mi
(6.8)
The choice of mi for the number of smoothing on level made in [1] such P steps P i was −1 −d that the term for i = J dominates in the sums i mi hi and i mi hi . (Specifically the choice in [1] corresponds to setting α := (d + 1)/2 in the next theorem.) Therefore the error of the solution uJ and the computing effort are given by the contributions of the finest grids. With the aid of Proposition 6.2 we are now in a position to establish similar properties here and show that the cascadic multigrid algorithm for the Stokes problem behaves like the cascadic algorithms investigated by Bornemann and Deuflhard [1] and by Shaidurov [7]. Theorem 6.3 Assume that 1 < α < d and that the CASCADE algorithm described in Section 4 is applied with mi cg steps on the levels 1 ≤ i ≤ J the mi being the smallest integers satisfying mi ≥ mJ 2α(J −i) . (6.9) Then the algorithm yields an approximate solution vJ on the highest level with kuJ − vJ k1 ≤ c
hJ kf0 k0 , mJ
(6.10)
where the constant c is independent of f and J . Moreover, the complexity of the algorithm is bounded by cmJ dim XJ . Proof: Since v0 = u0 , by Proposition 6.2 we obtain −1 J jY X kuJ − vJ k1 ≤ c 1+ j =0 i=0
9
c mJ −i
hJ −j kf k0 . mJ −j
From (6.9) we infer that J −1 X i=0
QJ −1 Thus the products i=0 1 +
1 mJ −i c
mJ −i
J −1 1 X 2 ≤ . 2−αi ≤ mJ i=0 mJ
are uniformly bounded by exp(2c/mJ ) and
J X hj kf k0 . kuJ − vJ k1 ≤ c mj j =0
(6.11)
The estimate (6.10) now follows from (6.11) combined with Lemma 1.3 in [1], while for the above choice of the mi the complexity estimate is a consequence of Lemma 1.4 in [1].
7 Proof of Lemma 6.1 Given zi ∈ Xi , let wi ∈ Xi be the solution of a(wi , v) + b(v, pi ) = a(zi , v) for all v ∈ Xi , b(wi , q) = 0 for all q ∈ Mi .
(7.1)
Obviously we obtain a linear projection Ri : Xi → Vi if we set Ri zi := wi . Since the finite element spaces Xi , Mi are stable, we have kwi k1 + kpi k0 ≤ ckzi k1 .
(7.2)
In order to apply Nitsche’s trick, we consider the auxiliary variational problem a(y, v) + b(v, r) = (wi − zi , v)0 for all v ∈ H01 ()d , b(y, q) = 0 for all q ∈ L2,0 ().
(7.3)
Since we have assumed H 2 -regularity, we obtain kyk2 + krk1 ≤ ckwi − zi k0 .
(7.4)
Now we insert v := wi − zi and q := pi into (7.3): (wi − zi , wi − zi )0 = a(y, wi − zi ) + b(wi − zi , r) + b(y, pi ).
(7.5)
Note that the left hand side equals kwi − zi k20 . Next we recall that (7.1) holds for v ∈ Xi−1 and q ∈ Mi−1 . Since we are only interested in estimates for zi ∈ Vi−1 , it follows that b(zi , q) = 0 for q ∈ Mi−1 . Hence, a(wi − zi , v) + b(v, pi ) = 0 for all v ∈ Xi−1 , b(wi − zi , q) = 0 for all q ∈ Mi−1 . 10
(7.6)
The general approximation results for affine families of finite elements [3] guarantee that there is a yi−1 ∈ Xi−1 such that ky − yi−1 k1 ≤ chkyk2 and ri−1 ∈ Mi−1 such that kr − ri−1 k0 ≤ chkrk1 . Combining this with (7.5) and (7.6), we obtain kwi − zi k20 = a(y − yi−1 , wi − zi ) + b(wi − zi , r − ri−1 ) + b(y − yi−1 , pi ) ≤ ckwi − zi k1 (ky − yi−1 k1 + kr − ri−1 k0 ) + cky − yi−1 k1 kpi k0 ≤ chkwi − zi k1 (kyk2 + krk1 ) + chkyk2 kpi k0 . The triangle inequality and (7.2) yields kwi − zi k1 ≤ ckzi k1 . Finally, we use (7.2) for estimating kpi k0 and (7.4) for estimating kyk2 and krk1 : kwi − zi k20 ≤ chkzi k1 kwi − zi k0 .
(7.7)
We divide (7.7) by kwi − zi k0 , set Ri zi := wi , and the proof of (6.1) is complete. In order to prove (6.2) we set v := wi , q := pi in (7.1) and obtain |wi |21 = a(wi , wi ) = a(zi , wi ) + 0 ≤ |zi |1 |wi |1 . After dividing by |wi |1 we have (6.2).
References [1] F. Bornemann and P. Deuflhard, The cascadic multigrid method for elliptic problems, Numer. Math., 75 (1996), 135–152. [2] F. Bornemann and R. Krause, Classical and Cascadic multigrid – a methodogical comparison, to appear in: Proceedings of the 9th International Conference on Domain Decomposition, P. Bjørstad, M. Espedal, D. Keyes, eds., John Wiley & Sons, New York, 1997. [3] D. Braess, Finite Elemente, Springer, Berlin–Heidelberg, 1997 or Finite Elements, Cambridge University Press, 1997. [4] D. Braess and R. Sarazin, An efficient smoother for the Stokes Problem, Applied Numer. Math., 23 (1997), 3–19. [5] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, Berlin–Heidelberg–New York, 1986. [6] W. Hackbusch, Multi-Grid Methods and Applications. Springer-Verlag, Berlin– Heidelberg–New York, 1985. [7] V.V. Shaidurov, Some estimates of the rate of convergence for the cascadic conjugate gradient method, Comp. Math. Applic., 31, No. 4/5 (1996), 161–171. [8] S. Turek, On discrete projection methods for the incompressible Navier–Stokes equations: An algorithmic approach, Comput. Methods Appl. Mech. Engrg. (1997), to appear 11
[9] R. Verf¨urth, A multilevel algorithm for mixed problems, SIAM J. Numer. Anal., 21 (1984), 264–271 [10] E.L. Wachspress, Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, N.J., 1966. [11] H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numerica 1993, 285–326. Dietrich Braess Fakult¨at f¨ur Mathematik Ruhr-Universit¨at 44780 Bochum Germany e–mail:
[email protected] Wolfgang Dahmen Institut f¨ur Geometrie und Praktische Mathematik RWTH Aachen 52056 Aachen Germany e–mail:
[email protected]
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