Convergence of The Multigrid Method With A Wavelet. Abstract. This new coarse grid operator is constructed using the wavelet

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1 Convergence of The Multigrid Method With A Wavelet Coarse Grid Operator Bjorn Engquist Erding Luo y Abstract The convergence of the two-level multigrid method with a new coarse grid operator is studied. This new coarse grid operator is constructed using the wavelet transformation. For partial dierential equations with highly oscillatory coecients for which the homogenization theory is applicable, this operator is considered as being close to the corresponding homogenized operator. Under some regularity assumptions between the approximation spaces, a convergence of the method is established. The convergence rate is independent of the grid step size. Furthermore, we show that the convergence rate in general is larger than that of the two-level multigrid method with Garlerkin coarse grid operator. 1 Introduction For regular elliptic equations it is well known that the multigrid method is practically very ecient. There is a number of papers dealing with the theoretical issue of convergence with Garlerkin coarse grid operator. However, most of these papers focus on the nite element methods. Restricting to the variational form, Garlerkin form is a natural built-in construction under these methods. With the regularity assumption of the partial dierential equations, some convergence proofs are established, [, 7, 8, 9]. The nite element method induces a nested set of smooth function subspaces. The \Aubin-Nitsch" trick has been commonly used in these proofs. The multigrid method generated from nite dierence method no longer induces a nested set of smooth subspaces where similar regularity estimate Department of Mathematics, University of California at Los Angeles, LA, CA9004. Research supported by ARPA/ONR N J-1890 and NSF DMS y IMA, University of Minnesota, Mpls, MN Research supported by NSF through IMA. 1

2 can still be established. Discussions on this issue can be found in [8, 10]. For the convergence proof of the multigrid method, the regularity of the partial dierential equation is always needed. For the elliptic dierential equation with highly oscillatory coecients, standard multigrid method is not ecient. Using nite element method with smooth base functions, the convergence proof for the multigrid method in [7] does not require any regularity of the partial dierential operator. However, since the nite element method is used there, a nested set of smooth function subspaces are built in. Within such nice subspaces, one obtains easily an approximate solution, but such an approximation could be far away from the true solution of the oscillatory coecient problem, [1]. For the class of partial dierential equations with highly oscillatory coecients, for which the homogenization theory is applicable, in order to maintain a fast convergence rate for the multigrid method, a new coarse grid operator based on the corresponding homogenized operator is introduced in [3, 4, 5, 6] using nite dierence method. This operator has come to be called homogenized coarse grid operator. Since the dierence between the oscillatory solution and the homogenized solution is only of rst order of the grid step size, a large number of smoothing iteration (which depends on the grid step size) is needed to guarantee the multigrid method to converge in l-norm. The dierence of the rst derivative between both is of a constant order in general, and the use of energy norm is impossible to build a similar "approximate property". In this paper we study the convergence of the two-level multigrid method by constructing a new coarse grid operator. This operator arises from the wavelet transformation and we call it here the wavelet operator. As already discussed in [3, 4, 5, 6] for the class of partial dierential equations, this operator is assumed to approximate to the corresponding homogenized operator. It's therefore natural for us to try this operator instead of the homogenized operator directly in the multigrid method. An interesting property about this operator is that it can be explicitly written as a combination of the original operator and the operators between the grid transformation. Therefore, the use of projection theory is possible to analyze the convergence of the multigrid method with the wavelet coarse gird operator. By assuming the multigrid method with the Garlerkin variational coarse grid operator to satisfy both \approximation property" and \smoothing property" as in [8], we prove that the two-level multigrid method with the wavelet coarse grid operator is convergent independently of the grid step size. We also prove that in general the convergence rate is larger than that with Garlerkin coarse grid operator. The rest of the paper is organized as follows. In section, the wavelet operator is constructed. In section 3, the two-level multigrid method is briey introduced. The convergence of the two-level multigrid method with the wavelet operator is analyzed in section

3 4, and in section 5 the convergence rates of the two-level multigrid method with Garlerkin and wavelet coarse grid operators are compared. Wavelet Transformation Consider the following algebraic equation arising from the discretization of partial dierential equation, (.1) AU = F; where A is symmetric positive denite. To solve equation (.1) we introduce a transformation " L (.) M = ; H where and L = 1 p H = 1 p 0 0 Then M satises the following properties, (i) (ii) Multiplying M to equation (.1) becomes C A ; 1 C A : M T M = MM T = L T L+ H T H = I: LL T = HH T = I; LH T = HL T =0: MAM T MU = MF: 3

4 We have LAL T (.3) HAL T LAH T HAH T! LU HU! = LF HF Denote T = LAL T ;B = LAH T ;C = HAL T ;D = HAH T ;U L = LU; U H = HU;f L = LF, and f H = HF. From (.3),! : (.4) U L =(T BD 1 C) 1 (f L BD 1 f H ): Equation (.1) can be written as (.5) U = F: Applying the wavelet transformation (.) to (.5), we have MU = M M T Mf: That is, (.6)! UL U H L L T LA = 1 H T H L T H H T! fl f H Thus (.7) U L =(L L T )f L + L H T f H : We are now ready to establish Lemma.1! : (T BD 1 C) 1 = L L T : Proof. (T BD 1 C)(L L T ) = LAL T L L T BD 1 CL L T = LA(I H T H) L T BD 1 CL L T = I LAH T H L T BD 1 CL L T = I LAH T H L T LAH T (HAH T ) 1 HAL T L L T = I LAH T H L T LAH T (HAH T ) 1 HA(I H T H) L T = I LAH T H L T + LAH T (HAH T ) 1 HAH T H L T = I LAH T H L T + LAH T H L T = I: 4

5 3 Two-Level Multigrid Method In this section we consider only two-level multigrid method for a one dimensional problem. We discretize the partial dierential equation at a ne grid level h and obtain an algebraic equation as (.1) in a n-dimensional vector space < n.for the multigrid method we assume the simple Richardson iteration is used as the smoother G. The coarse grid operator of the Garlerkin variation form on the coarse grid level H is thus denoted by (3.8) A G = I H h AI h H ; where I H and h I h H denote the interpolation and prolongation respectively. Therefore, the two-level multigrid operator with pre- and post- smoothing iterations based on these coarse grid operators can be written as (3.9) M W = G( I h H W I H h )AG: By the analysis of the previous section, we induce the following new coarse grid operator from the wavelet transformation (3.10) where we set (3.11) A W =(I H h I h H) 1 ; I H h = 1 p L; L is a n (3.1) n matrix dened as in (.), and I h H =(IH h )T = p L: For the rest of the paper we always assume that the interpolation and prolongation are chosen to be of these forms. Furthermore, for simplicity, we only consider the one dimensional case here. However, the analysis can be extended to higher dimensional case. When the partial dierential equation belongs to the class mentioned above, the operator (3.10) is considered as an approximation to the corresponding homogenized operator. Hence, by the wavelet coarse grid operator (3.10), the operator of the two-level multigrid method with pre- and post- smoothing iterations can be written as follows, (3.13) M G = G( I h H G I H h )AG: The following lemma establishes a relationship between A G and A W. 5

6 Lemma 3.1 Proof. By Lemma.1, W G >O: A W = A G BD 1 C: Since A W ;A G and BD 1 C are symmetric and positive, A G A W >O: Therefore, W G >O: 4 Convergence Analysis Set R = L T L and consider the following two subspaces of the n-dimensional vector space < n (4.14) =Range(= R= ) and (4.15) =Range(= (I Lemma 4.1 The two subspaces (4.14) and (4.15) are A-orthogonal and form a direct sum of < n, < n = : Let S and T be the projections of < n onto and, respectively. Dene an energy norm for < n by kxk A = k= xk; 8x < n : R)): From Lemma 4.1, for any x < n,gx = SGx T Gx: Further, (4.16) kgxk A = ksgxk A + ktgxk A : Assumption 1: kgk < 1 and G and A are commute. 6

7 By Assumption 1, (4.17) kgk kgxk A A = max kxka6=0 kxk A x T GAGx = max kxka6=0 x T Ax = kgk ; we have which implies (4.18) and (4.19) Then Let operators M1 and D be dened by kgxk A kgk A kxk A <kxk A ; kgk A < 1: M1 = G(I h H W I H h A I h H G I H h A)G; D = = I h H W I H h A1= = I h H G I H h A1= : GDG = = M1= ; and by Lemma 3.1, the operator D dened in (4.19) is symmetric nonnegative denite. We now establish the following lemma. Lemma 4. Under Assumption 1, (4.0) km1k A = max kyk 6=0 y T GDGy : y T y Proof. Note rst that km1k = max km1xk A A kxka6=0 kxk A k= M1xk = max kxka6=0 k= xk x T M = max 1 T AM 1x kxka6=0 x T Ax = k= M1= k = kgdgk Since D is symmetric nonnegative denite (spd) and by Lemma 3.1, y T G(= IH h km1k A = max W I h HA1= = IH h G I h HA1= )Gy : kyk 6=0 y T y 7

8 Assumption : Set C G = I h H G I H h :C Gsatises the following approximate property, (4.1) kc G k = k I h H G I H h k C1h ; where is positive, and C1 is some constant independent of h. Remark. For regular dierential operator, this assumption is satised, [8]. Assumption 3: AG satises the smoothing property, kagk = kagk A C()h ; where is as in Assumption and C() goesto0asincreases. Lemma 4.3 Proof. k= S= k 1: Lemma 4.4 k= S= k x T = S T AS= x = max kxk 6=0 x T x y T S T ASy = max kyka6=0 y T Ay ksyk A = max kyka6=0 kyk A k= SGk C ()h : 1: Proof. By Lemma 4.3 and Assumptions 1 to 3, k= SGk k= S= k ka1= Gk k= Gk = max x T AG x = kag k C()h : kxk 6=0 x T x Lemma 4.5 (4.) km1k A C 1C(): 8

9 Proof. We have kc G k = jx T x x T IH h max G I h H xj kxk 6=0 x T x = jx T x x T = IH h max G I h H A1= xj : kxk 6=0 x T Ax Hence, by Assumption, for all x < n, (4.3) jx T x x T = I h H G IH h A1= xjc1h x T Ax: Meanwhile, for all s, we can write s = = Rx for x < n. s T Ds = x T R= (= I h H W I H h A1= = I h H G I H h A1= )= Rx = x T R(I h H W I H h I h H G I H h )Rx = x T (R R I h H G I H h )x = s T s x T = R= = I h H G I H h A1= = Rx = s T s s T = I h H G I H h A1= s: Combining with (4.3), we get s T Ds C1h s T As; 8s : Note that for y1 = SGx and y = TGx, Thus, by Lemma 4., (4.4) (Gx) T D(Gx) =(y T 1 +y T )D(y1+y)=y T 1 Dy 1: x T GDGx km1k A = max kxk 6=0 x T x C1h y max 1 T Ay 1 kxk 6=0 x T x = C1h max kxk 6=0 y = max 1 T Dy 1 kxk 6=0 x T x (SGx) T A(SGx) : x T x Together with Lemma 4.4 and Assumptions 1 to 3, (4.4) implies (4.5) km1k A C1 max kxk 6=0 k= SGxk kxk h = C1k= SGk h C1C(): 9

10 Lemma 4.6 Under the Assumptions and 3, (4.6) km G k A C1C(): Proof. Set ~ M = G(I A 1= I h H G I H h A1= )G = GQG: Then and ~ M is symmetric. Furthermore Thus That is, k M ~ jx T Mxj ~ k = max kxk 6=0 x T x jx T GQGxj = max kxk 6=0 x T x km G k A = k= M G = k = k ~ M k; jx T G= ( IH h = max kxk 6=0 k I h H G I H h k k= Gk : jx T G= ( IH h = max G I h H)A1= Gxj kxk 6=0 x T x G I H h )= Gxj x T GAGx k ~ Mk C1h C()h = C1C(): km G k A C1C(): x T GAGx Now, let's state the convergence theorem of the two-level method with wavelet coarse grid operator. Theorem 4.1 Under the Assumptions 1 to 3, x T x (4.7) km W k A C1C(): Proof. By (3.9) and (3.13) and, M W = G( I h H G I H h )AG + G(I h H G I H h I h H W I H h )AG; km W k A km G k A +km1k A : The rest of the proof is trivial. 10

11 5 Comparison of Convergence Rates In this section, we assume that the operators M W and M G of the two-level multigrid method under two dierent choices for the coarse grid operators without post-smoothing iteration G are given by M W =(I I h H W IH h A)G; M G =(I I h H G IH h A)G: Set C W = I = I h H W I H h A1= ; C G = I = I h H G I H h A1= : We are now ready to show that bychoosing the coarse grid operator to be the Garlerkin variational form, the two-level multigrid method is always convergent as long as the smoothing iteration operator kgk < 1. Note that the two-level multigrid method of M G is convergent if and only if km G k A < 1. Under the condition that kgk < 1, to show km G k A < 1it suces to show that kc G k A 1. Lemma 5.1 C G is symmetric nonnegative and (5.8) kc G k 1: Proof. Decompose the space < n into two orthogonal subspaces as (5.9) < n = S1 S; where S1 = = Range(I h H ); S = = null(i H h ): Thus for any x < n, it can be decomposed into x = x1 + x; with x1 = = IH h y 1 S1 for y1 < n and x S. Simple calculation shows C G x1 = 0; C G x = x: 11

12 Hence, Thus (x1) T (C G x1) = 0; (x) T (C G x1) = 0; (x) T (C G x) = x T x : x T C G x T = x T x 0: This shows C G is symmetric nonnegative. Furthermore x T C G x kc G k = max kxk 6=0 x T x = max x T (5.30) x kxk 6=0 x T 1 x1 + x T x 1: For the Wavelet operator, however, the above result is not true. To see this, decompose < n into two A orthogonal subspaces as where < n = S1 S; S1 = = Range(I h H); S = = null(i H h ): Then for any x < n, it can be decomposed into x = x1 + x; where x1 = = I h H y 1 S1 for y1 < n and x S. Simple calculation implies Hence, C W x1 = x1 = R= x1 =(I P)x1; C W x = x: (C W x1) T C W x1 = k(i P )x1k ; (C W x) T C W x1 = x T x 1; (C W x) T C W x = x T x : Moreover (5.31) kc W xk = k(i P )x 1k +xt x 1+x T x ; 1

13 where P = = R= is a projector which is not symmetric. The following property of Pis in general not true: k(i P )x1k kx 1k : This is because decomposing < n as above may imply that x T C W x is negative for some x < n. Hence the wavelet operator may not satisfy an analogous property to that in Lemma 5.1. We now show that the convergence rate of the two-level multigrid method with coarse grid operator of the Garlerkin variational form is always faster than that with the Wavelet operator. Theorem 5.1 (5.3) km G k A km W k A : Proof. Note rst Hence, which implies This in turn implies (5.33) = W A G= W + = W G A1= W : W A G W + G W ; W A G W G ( W G ) O: A G W A G W A G A G (A G W A G A G ) O: Decompose < n as (5.9) in the proof of Lemma 5.1. As before, for any x < n, x=x1+x; with x1 = = I h H y 1 S1 for y1 < n and x S. Wethus have Hence, C W x1 = x1 = I h H W A Gy1; C W x = x: (C W x1) T (C W x1) = x T 1 x 1 y1 T A G W A Gy1 + y1 T A G W A G W A Gy1; (C W x) T (C W x1) = 0; (C W x) T (C W x) = x T x : 13

14 This implies (5.34) (C W x) T (C W x)=x T x y T 1 A G W A Gy1+y T 1 A G W A G W A Gy10: Similarly, (5.35) (C G x) T (C G x)=x T x+y1 T A Gy1 y1 T A Gy10: (5.33) together with (5.34) and (5.35) implies (C W x) T (C W x) (C G x) T (C G x): Thus km W k A = k= M W = k = kc W Gk km G k A = k= M G = k=kc G Gk: References [1] I. Barbuska, Solution of Problems with Interfaces and Singularities, Mathematical Aspects of Finite Elements in Partial Dierential Equations, Carl de Boor, eds. 1974, pp [] R.E. Bank AND T. Dupont, An optimal order process for solving nite element equations. Math. Comp., 36(1981), pp [3] B. Engquist AND E. Luo, The Convergence of Multigrid Method For Elliptic Equation With Oscillatory Coecients, Preprint. [4] B. Engquist AND E. Luo, Multigrid Methods for Dierential Equations With Highly Oscillatory Coecients. Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, 1993, p [5] Engquist AND E. Luo, New Coarse Gird Operators For Highly Oscillatory Coecient Elliptic Problems, Preprint. [6] E. Luo, Multigrid Method for Elliptic Equation with Oscillatory Coecients. Ph. D. Thesis, Department of Mathematics, UCLA,

15 [7] J.H. Bramble, Multigrid Methods, Pitman Research Notes in Mathematics Series. Longman Scientic & Technical, [8] W. Hackbusch, Multigrid Convergence Theory, in Multi-Grid Methods, W. Hackbusch and U. Trottenberg, eds., Lecture Notes in Mathematics, Springer-Verlag, Berlin- Heidelberg-New York, 198. [9] S. McCormick AND J. Ruge, Multigrid methods for variational problems, SIAM J. Numer. Anal., Vol. 19(198), pp [10] P. Wesseling, The Rate of Convergence of A Multiple Grid Method, Lecture Note,

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