S. ZHANG (1)

ON THE CONVERGENCE OF SPECTRAL MULTIGRID METHODS FOR SOLVING PERIODIC PROBLEMS

S. ZHANG (1)

ABSTRACT - The spectral multigrid method combines the efficiencies of the spectral method and the multigrid method. In this paper, we show that various spectral multigrid methods have constant convergence rates (independent of the number of unknowns in the linear system to be solved) in their multilevel iterations for solving periodic problems.

1980 Mathematics Subject Classification (1985 Revision). Primary 65F10; Secondary 65N35.

1. I n t r o d u c t i o n

Due to its optimal computational-order, the multigrid method and its ap- plications have been studied extensively in recent years (cf. the references in [11]). Meanwhile, the spectral method has been analyzed by numerical analysts for its successful application (cs references in [20]). In [21-22], Zang, Wong and Hussaini proposed several spectral multigrid (SMG) methods to combine the efficiencies of multigrid methods and spectral methods for solving large linear systems of equations arising from collocation-spectral discretizations of partial differential equations. They are the first to apply the multigrid principle to solving pseudo(collocation)-spectral equations and to find a correct way to do the coarse-level correction (cf. (2.5) below). Numerical examples were presented in [21-22] showing the practicality and the efficiency of the method. An improved

- - Received 11 February 1991. (l) Department of Mathematical Sciences University of Delaware Newark, DE

19716 USA.

186 S. ZHANG: On the Convergence of Spectral Multigrid

spectral multigrid method was proposed later by Brandt et al. in [3]. The convergence of these spectral multigrid (SMG) methods are shown so far for constant coefficient differential equations only (in [21-22] and [3]). For such a case, the matrices for linear systems (Lk, defined in (2.4)) and for multigrid iterative operators (S~Ck, defined in (5.1)) are diagonal, and the pseudo-spectral method becomes a spectral one. The rate of convergence of SMG would simply be then the maximal diagonal numbers of skmck. But for the case of constant coefficients, the diagonal linear systems can be inverted trivially. For nonconstant coefficients problems, the collocation linear system (ofN unknowns) is full, for which O(N 3) operations are needed if the direct method is used and even when the fast Fourier transform is used, O(N31ogN) operations are needed if simple iterations are in use (for example, the one defined in (2.6) below). The purpose of this paper is to give a theoretic justification to the methods of [21-22] and [3] for solving periodic problems. We will show that the SMG method converges with constant rates in general. By virtue of its constant rate, the spectral multigrid method is of nearly optimal order, N log N, the order of fast Fourier transforms in any space dimension, for periodic problems.

To analyze the spectral multigrid (SMG) method, we follow Pasciak (cf: [12]) to reformulate the pseudo-spectral equations in their equivalent variational forms. The connection is the discrete Fourier transform, by which trigonometric functions and their collocation values are mapped back and forth. From this analysis, we naturally derive a new spectral multigrid method. In the new method, nonnested collocation points are used for the multilevel discretizations. However, the multilevel trigonometric function spaces corresponding to the sets of nonnested collocation points are naturally nested. This nonnestedness technique results in a better coupling between the discrete functions on different grids than that on the traditional coupled grids (2 n points).

By a variational formulation of the new nonnested-grid SMG method, we will prove its constant convergence-rate. Also by this variational formulation technique, we can get a clear insight into the two SMG methods ([21-22, 3]) as they have been used by engineers. The newly posed SMG method looks more efficient than the other two SMG methods. But due to the limitation of the fast Fourier transforms, the new method is not very practical for its odd numbers of points. We should remark that the finite-difference preconditioned spectral (Chebyshev collocation) multigrid method discussed in [21 ] for Dirichlet boundary value problems will not be studied in this note. A spectral element multigrid method has been studied in [16, 10], where the domain is subdivided into small pieces and the spectral method is applied to each subdomain. We refer readers to

Methods for Solving Periodic Problems 187

[15] for more related works on SMG methods. In particular, Heinrichs gives some continuation work of SMG in [8-9] and in other papers. The constant-rate convergence for a multilevel Galerkin-spectral method was studied by the author in [23]. Corresponding to the nonnested-grid SMG, nonnested-grid multigrid methods for finite element equations are proposed and studied in [23-25]. The nestedness in the multigrid method could be also caused by the nature of finite elements (cf., for example, [2, 4-7, 13-14, 17, 18, 19].

In this paper, we first propose in w a SMG method (n0rmested- discretization SMG method) and formulate it in variational forms. By the formulation, constant-rate convergence in shown for the method in w The proof is then generalized to the other two methods, the filtered one (proposed in [21 ]) and the midpoint-discretization one (proposed in [3]), in w In w some numerical tests are provided to compare the nidpoint-discretization SMG and the nonnested-discretization SMG.

2. A nonnested-discretization spectral multigrid method

In this section, a SMG method will be defined and its variational formulation will be derived. In the new method, the multiple collocation grids are not nested. This nonnestedness of grids has been used in finite element multigrid methods (cf. [24-25]).

The periodic problem to be solved by the spectral multigrid method is: Find u satisfying the boundary condition

l U(Xl,...,C i .... ,Xd) = U(Xl,...,di,...,Xd),

0U 0U U(X1 .... ,di,'",Xd), OXj (Xl'""Ci'""Xd) OXj

such that

l~<ij~d,

(2.1a) Z aid(x) - - u(x) +b(x)u(x) = f(x), x e/-/ (ci,di)c R d i=l 0X i 0Xj i-----i

for any space dimension d, where (aid) is symmetric and uniformly elliptic in the domain/-~i=l(Ci,di). For the practicality of the problem, we refer readers to [21]. If we use tensor-product collocation points and tensor-product trigonometric functions, it is enough to consider the one dimensional case since the generalization is standard (for example, a 1-D to 2-D generalization is given in [23] in detail). One specialization of the SMG method in higher space dimensions is

188 S. ZHANO: On the Convergence of Spectral Multigrid

when it is applied to highly anisotropic problems. The simple Richardson iteration (2.6) below will then deteriorate the convergence rate of SMG. Some mod- ification is needed as discussed in [3]. The weighted smoothing used there in [3] can be also covered by the theory here. But, for the simplicity of notations, we restrict ourselves to the following one-dimensional model problem.

(2.1b) - (a(x)u ' (x)) '+b(x)u(x) = f(x), x ~D=(0,1),

where a,b e H2per(g2) satisfying 0~<c~<a(x),b(x)~<a for some constants _c and 6 and f ~ L2(f2). Here Hmper(Q) is the closure of periodic C=(g?) under the usual Sobolev norm Hm($2).

(2. l b) is a simple ODE. One may use the finite difference/element method to discretize it and the resulting linear system can be solved in optimal order operations (O(N)) by a banded solver. But these methods do not have arbitrary high order of approximation if u is smooth, and also the O(N) operation counts no longer stand for direct solvers in higher space dimensions. We choose the model problem to illustrate the theory. By no means is the method applicable only to (2.1b) (cf. [21]). Another remark must be made on the differentiability assump- tion. We assume only weak derivatives up to order two for the solution in our theory. But spectral methods would be applied to smoother problems to get higher order or exponential convergence.

In collocation-spectral methods, the approximate values of the unknown function are sought at some special collocation points. In our nonnested discretizations, odd numbers of uniformly distributed collocation points are in use:

xj(k ) _ 2~r(j--l____.__~) l~<j~<nk, k= 1,2,.. " ,

nk

where

(2.2) nk = 2n'k+l and n'k = 2 k~

for some positive integer k0. By (2.2), collocation points of any two levels are completely different, and the multilevel collocation sets are nonnested. For simplicity, we may use xj for xj (k) if there is no confusion. Again we need to remark that the use of odd-points deteriorates the speed of the fast Fourier transform. In practice, 2 n points would be used.

In nonnested-discretization SMG methods, the discrete equations for (2.1 b) read: Find Uk=(Ul,U2,...unk) "r such that

Methods for Solving Periodic Problems

(2.3) LkUk = Gk, k=l,2,.. . ,

where the right hand side vector consists of the collocation values of fi

Ok := (f(Xl),f(x2),...,f(Xnk)) T,

and the coefficient matrix is defined by

(2.4) Lk := -(Fk-lDkFk)Ak(Fk-lDkFk)+Bk.

Here in (2.4)

l F k : -

- e-i(-n 'k )x~ e- i ( -n/ , )x2

e- i ( -n 'k+ l)xl e - i ( -n 'k+ 1)x2

e - i ( n~ )x~ e-~( nk )xe

. . . e - i ( -nk )x.k -

. . . e - i ( - n k + 1)xo~

... e-!( n, )• _

189

Ak := diag(a(xl),a(x2),...,a(xn~)),

Bk := diag(b(xl),b(x2),...,b(x,~)),

Dk := diag(i(-n'k),i(-n'k+ 1),...,0,i,...,i(n'k)).

We denote ~ by i. The matrix Fk in (2.4) represents a discrete Fourier transform. Therefore, --F-kkT=F~ -l, ~ T=-F~qDkFk and L--~ T=Lk. Further, the product LkUk can be evaluated by a series of fast Fourier transforms. In SMG methods, the grid functions are transformed between different grids by the intergrid transfer matrices (cf. [2t]):

(2.5) F-~_l(0 Ik-1 0) Fk =: I~ -1 : R nk--* R n~-'

Fk-l(0 Ik-1 0) T Fk-1 =: Ik-1 " R n~-' "--~ R n~

for k=2,3,..., where Ik-t is the nk_lXnk_l identity matrix. In fact Ik k-I is simply the adjoint of Ikk_l, ~1k-1 V. We now define the multilevel scheme for solving (2.3) (cf. [1] and [21]). The multigrid method is a recursively defined iteration. The itera-


tive errors and residuals are projected to lower levels. The lower level problems are again solved by the same multigrid scheme. So the multigrid scheme is defined to solve both the original problems (2.3) and the residual problems (2.7).

DEFINITION 2.1. (the kth level spectral multigrid scheme). (1) I f k= 1 (2.3) or (2.7) below is solved by any method. (2) I f k> 1, an approximation Zm+l will be generated after one cycle of the kth level iteration

from an initial guess Z0 as follows, m Richardson relaxations will be performed:

1 (2.6) Zt-Zt_l = (Gk-LkZt_l), 1 ~<l~m,

Ak

where Ak is the maximal eigenvalue Of Lk and Gk is either from (2.3) or is the 0 in (2.7) below. To define Zm+ 1, we need to construct a (k,--1)st level residual problem." Find f~ such that

(2.7) Lk-tt 7l- = Ikk-l(Gk--LkZm) =: C-k-l'

Let R be the approximation of [~ obtained by applying p(> l ) cycles of the (k-1)st level scheme to (2.7) with zero initial guess; then Zm+l is defined by

(2.8) Zm+ 1 = Zm+Ikk_lR.

To analyze the scheme, we rewrite (2.3), (2.6-8) in variational forms. This technique has been used by Pasciak [12] and many others before. The bilinear form associated with (2.1b) and the corresponding norm are defined as

a(u,v) := f~ au 'v '+buvdx Vu,v EHlper(Q)

Illulll := ~v~-~(~,~) V u ~HI~,.(D).

To relate the discrete functions (defined on collocation points) to some finite- dimensional function spaces, we define the following auxiliary, trigonometric- function spaces

/ . . . . . } (2 .9) M k : = u u = Z fije 'Jx, fij = % , k = 1 , 2 , . .

j = - n ' k

Simple computation will show that the discrete Fourier transform,


Fk : U k = (u, ..... Unk ) "-'4 V~nk(fl_n'k,...l~ln'k) ---~: ~nkl~'k,

defines a 'homomorphism between the nodal values and the Fourier coefficients for functions in the space Mk. Although the collocation sets are not nested (con- taining completely different points), the corresponding multilevel trigonometric spaces are nested naturally by (2.9), Mk-l C Mk. The latter is the reason for intro- ducing the new method. Later in w we will see that extra efforts are needed to build such links in the filtered SMG [21] and the midpoint-discretization SMG [3] methods.

Let Pk:L2(~)--->Mk be the usual L2-orthogonal projection defined by

ni Pk" ~ fije ijx --> 27 , fije ij~.

j =-r j =-nk

It is well-known (cf. [23] for example, or just by simple Fourier series calculation) that

(2.10) LIIp~ulll ~ cIIlulll and Illu-P~ulll ~ CnC'llullH--

Here and in the following, C denotes a generic constant which is independent of the functions being estimated and the level number k. We introduce two bilinear forms and a family of discrete norms to the space Mk:

ak(u,v ) := n~'lVkTLkUk, Vu,v EMk,

(2.11) (u,v)k := nk-lVkTUk Vu,v eMk,

lllulll~,,~ = nk-'UkrL~.Uk Vu ~Mk, 0~<s~<2.

LEMMA 2.2. In the space Mk, the norms III III0,k and II IIL+-<~> are the same, and the norms

IIlllll,k IllIll and II'IIH'<++I are equivalent with the equivalent constants independent of the level number k.

Illulllo,~ = llullL+<~> Vu Eak, clllulll ~ lllulll,,k ~ CllullH,<~> ~ clllulIl VU E M k.

PROOF.

(2.12)

Let u=~V~=_,, fije ux and v=-Y~L_, k ')je ux. By (2.11) we can compute that

(u,v) = v fij '-~k lei0-k)Xdx -- ~ k TOk = 1 FkVk TFkUk = (U,V)k, j,k + ' nk

- 1 n k t t ak(u,v) = nk-iVkrLkUk = nk j___X {a(xj)u (xj)v (xj)+b(xj)u(xj)v(xj)}.


Therefore, lllulllo,k--llullv<~> and

clllulll ~ cl lull~, - c ~k (u,~+u~)(xj) ~ a~(u,u) ~ +llull~, ~ cIIlulll, nk j=l

where c and ~ are defined right after (2.1b).

By the following inverse inequality (cf. [23], which can be shown easily, by the Fourier series expansion)

Ilull.,<~> ~ Cnkllullv<~,> VU~Mk,

and by the equivalence ofllll.,<~), IIl'lll,,k and Ill'Ill norms in Mk (Lemma 2.2), the maximal eigenvalue of L k (used in (2.6)) is bounded by: Ak~Cnk 2. Then, by a s tandard estimate (cf., for example, [1]), we have the following smoothing prop- erties for (2.6):

Cnk Illgk--Zmlll2'k ~ m I[[gk--Z0llll'k'

(2.13) Illuk--Zml]ll,k ~< IIIu~-zo l II ,>

LEMMA 2.3. Let z e Mk and r e Mk-1 be the corresponding (related by F k and Fk-t) trigonometric functions Of Zk and Rk-t respectively. It holds that

1 1 (z,r) = - - Zkrlk_lek_l and ak(z,r) = - - ZfLkIkk_lRk_t .

nk nk

Further, the function z + r has nodal values Zk+Ikk_tRk_t at the k-th collocation points.

PRooF.. We first note that ak(u,v) are well-defined for lower level functions due to the nestedness: Mj C Mk for j~<k. By the definitions of I~-l and Fk, we see that the vector Ikk_tRk_~ consists of the nodal values of r at the k-th level nodes, {xj(k/}. Therefore, the lemma follows the calculation (2.12). �9

(2 .14 )

In summary, we rewrite (2.3) and (2.6-8) in variational forms:

ak(Uk,V ) = (f~V)k , 1

(z l -z l -~ ,v )k = - - Ak

(gk(v)--ak(Zl-l,V)),

ak_l(~,V ) = gk(v)--ak(Zm,V ) =: gk(V),

Zm+l ---~ Zm+r"


3. Convergence of nonnested discretization spectral multigrids

We will prove a lemma showing an approximabili ty of the discrete bilinear form ak(u,v) to a(u,v). We then show the constant-speed convergence of the nonnested-discretization S M G in this section.

One can see from the first equation of (2.t4) that the discrete spectral solution Uk is not the Galerkin projection of u, due to the difference in bilinear forms ak(',') and a(', '). This is commonly seen in finite element methods whe nnonc on - forming elements or numerical quadra ture formulae are used. It is treated by the well-known first Strang lemma, where the error between the discrete bilinear forms ak(',') and a(',') need to be estimated. In our case, it is done by the following lemma.

LEMMA 3.1. I f the k0 in (2.2) is sufficiently large (depending only on a(x) and b(x) in (2. lb)), then

lak(u,v)-a(u,v)[ Cnff'llllUlll,,k-,lllvill,,k Vu EMk_I, VV E.z}//k.

PRooF. Let ~Y'~rn=-Oo ~tm eimx and Z~m=_Oo bm eimx be the Fourier expansions of the a(x) and b(x) in (2.1b) respectively, u=Xn~-'=_nL, fije ux EMk_l, and v=X~ ~- n'k vie ilx EMk. By the orthogonality of {eilx}, we can show that

nk-I nk - - a(u,v) = X X V 1 l~lj .~ (~mjl+bm)dm,l-j,

j =-n~,_, 1 =-n~, m =-oo

ak(U,V) = X~-' X~ V'~'-I]j . ~ ( a . m j l + 1 3 m ) .~ 0m,l_j+,unk. j =--n'k_ l l=--n' k m = - - ~ U=--~

Here, did is the Kroneker symbol. Noting the index ranges ofj and 1 above, we see

n' -- ' 3n'k 3nk [j-1] ~< k ' l - n k _ l = < -

2 4

where n'k and n k are defined in (2.2). Now that ak(u,v)--a(u,v) is equal to

nk-, n~ X 2:

j =-n~_, L = - n ' k

we obtain the estimate


n'~_, n~ lak(u,v)-a(u,v)[ ~< 2; Z [aj'~ll 2; Ol]~mlAt-ll~ml)

j =-n'k_ ' l = - n ' k I m l > n k / 4

Nalh +llbll.2 ~< .I7 [fij~ll(jl+l)2

m=nk/4 m 2 j,l

~< C(a,b)nk-lHUllH,HVltH, ~< Cn 'litu{ll , -,lllvt[[,, ,

where we bound the Fourier coefficients of a(x) and b(x) by I[aHH~ and [Ibllw to get ~m=O(m -2) and l~m=O(m-2), and we make use ofZ~m=n m-2~<Cn -l. Here k0 in (2.2) needs to be sufficiently large such that Iml>nk/4 is also large enough to make ~tmmO(m -2) and l~m=O(m -2) meaningful. �9

We remark that if the coarsest grid is not fine enough, i.e., k0 in (2.2) is not large enough, the spectral method does not provide approximation to the partial differential equation (2.1) on coarse grids. On the other hand, the residual solutions on such grids may not approximate the iterative errors on finer grids either. This will be tested in our numerical experiments. In the case of constant coeffi-

cients, ko can be 0. In the next theorem, we show the constant-rate convergence of the two-level

nonnested-discretization SMG method, where ~, is referred as a convergence rate. Constant-rate convergence means that y is independent of the number of unknowns in the linear system. The proof of the theorem employs the typical approach for rnultigrid methods (cf. [1]). In the proof, we compare the residual solution and the iterative error with a smooth function. A similar technique is used by Brenner in [4] to treat nonconforming finite element solutions.

THEOREM 3.2. (two-level SMG methods) Let k0 in (2.2) be sufficiently large (depending on a(x) and b(x) in (2.1b)). For any 0 < y < l , there exists an integer m~>l, independent of the level number k such that i f (2.7) is solved exactly, i.e., ~=r, then

IIlUk--Zm+ lll ,k YlIlUk--Z01111,

PROOF. Let the iterative errors be denoted as ej=uk-zj for j -0 ,1 ,2 , . . . ,m+ 1. Our goal is to show Illem+illll,k~yllIeOIIll,k. By (2.7) in definition 2.1 and its variational form in (2.14), em+l =em-r. We first introduce a smooth function w, to which both em and ~ are pseudo-spectral approximations.

Let f0 ~ Mk be the Ritz-representation of the functional ak(em,') in Mk:


(f0,v) = ak(em,V) Vv ~Mk,

and let w be the solution of the following periodic problem,

a(w,v) = (fo,v) Vv EHlper .

By definition (2.7), L e m m a 2.3 and (2.14),

ak(em,V) = (fo,v) = a(w,v) Vv e Mk, (3.1)

ak-l(r,v) = ak(tm,V) = (f0,v) = a(w,v) Vv ~Mk_ 1.

By the Fourier series expansion (cf. [23]) we can show directly the elliptic reg- ularity for problem (2.1b) to get that w e H2er(l'2) and that

IIwfrH~-< cllf0lk~.

Thus, by Schwartz 's inequali ty, we have

(3.2) CHW[IH2 • HfollL2 = ak(em,fo)/HfoJ[g 2 < fllemlf[2,kl]lfolll0,k/llf0llg~ = [][em]ll2,k.

We now est imate II]~m+,ll[,.k. By Definition 2.1, Lemma 2.2 and the triangle inequality, we have

(3.3) ][[em+llll,,k = [][em--rl[[1,k ~< ClHem-~[[[ < ciliUm-Will+el[[w-ell[

c ( Illcm-pkwlll § ] Ilr-Pk_l will § IllW-PkWlll + I Hw-Pk_ 1 will ),

where Pk is the L 2 or thogonal projection defined in (2.10). We now estimate the first term in (3.3). Let e=em--Pk-lw to shorten notations. By (3.1), L e m m a 3.1, (2.10) and (3.2), it follows that

Illellt~,k = ak(e,em)-ak(e,Pk_,W)= a(e,w)--ak(e, Pk_lW )

= a(e,w--Pk_lw) + [a(e,Pk_lw)--ak(e,Pk_zW)]

~< ]Hell][llw--Pk_tWll[ +C nC ' ]Hell] 1,klllPk-lWl[ ] l,k-1

-< CnC'llletll,.kCllwll.++lIwllH,> < Cn~'lllelll,.kllwllH2

~< Cni-t Hlelll l,k]ll~mlll2,k.

196

Thus

(3.4)

S. ZHANO: On the Convergence of Spectral Multigrid

II[~m-Pk-~wlll,,k ~ Cnk-'lll~mlll2,k.

In the same vein, since ~ is the collocation-spectral solution of w in Mk-l by the second equality in (3.1), it holds that

(3.5) IIl~-v~_,wlll,,~_, ~< Cn-L, Ill~mlll2,k.

The last two terms in (3.3) are estimated in (2.10). Combining (3.3), (3.4), (3.5), (2.10) and (2.13), we conclude that

Illem+,lll ,,~ -< Cnk "l IIl~mlll2,k ~< Cm-'/2]ll~olll ~,k < YlII~oIII ,,~,

where m is chosen larger than (C~) 2. �9

From Theorem 3.2 it is standard to derive the constant rate of convergence of W-cycle multigrid schemes. One can check [1] for details. Once the constant-rate convergence is shown for the multigrid method, it is routine (note the geometric increase of the number of unknowns in (2.2)) to demonstrate the following theorem (cf. [1] and [23]). A detailed work estimate is given in [23].

TREOREM 3.3. (Nearly-optimal order of SMG) Let Theorem 3.2 hold. With order NlogN arithmetic operations, the SMG can generate an iterative solution fik ~ Mk for a collocation-spectral linear system of N unknowns, such that

Illa~-u~lll ~ cg-~llull.~ and Illa~-ulll ~ cg-~llull~,

where u is defined in (2.1b), and uk is the exact solution of the spectral equation (2.3). �9

4. Midpoint discretization and filtered spectral multigrid methods

In this section, we consider the midpoint-discretization and the filtered SMG methods proposed in [21-22] and [3]. To avoid tedious notations and re- petitions, we simple describe the variational formulations for the two methods and the difference between the two methods and the nonnested-discretization SMG.


In both midpoint-discretization and filtered SMG methods, domains are dis- cretized by even number, uniformly distributed collocation points, i.e., the nk in (2.2) is now

(4.1) nk := 2n'k+2, n'k := 2k~

for some positive integer k0. The sets of multiple collocation points are thus nested. The discrete Fourier coefficients are indexed in the range:

nk = ~ E ule -ux', --n'k~<j~<n'k+ 1 (4.2) aj ~nk '='

where

Ul - - 1 n'~+ 1

-- ~ Z'j=_n~ fijeUX~' l~<l~<nk"

In midpoint discretizations, derivatives are evaluated at midpoints between two neighboring points,

1 n'k+l U ] + l / 2 - - gr~nk •j=_n, k fij(ijeijh/2) eijx',

where h=2~/nk, and the term (au')'(xi) in (2.1b) are approximated at the midpoints,

~ + I 1 i J - - m E ~ l a(Xm+ ~ )U'm+i/2e-iJ(•215

In this way, the Lk of the collocation equation (2.3) is now defined by

(4.3) Lk := -(Fk-IDkHkFk)Ak(Fk-IH~-IDkFk) +Bk,

where

H k :--- diag (ei(-n'k)h/2,e i(-n'k+ 1)h/2,...,ei(n'k+ l)h/2),

Ak := diag(a(xl+h/2),.. . ,a(xnk+h/2)).

The rest of the matrices in (4.3) remain the same as in (2.4) except for possible modifications in index ranges. Noting that e i(n~+l)x and e -i(n,k+l)x have the same nodal values, _1, at all k-th level collocation points {xj(k)}, we can introduce a (nested) family of auxiliary, trigonometric spaces:

198 S. ZHANO" On the Convergence of Spectral Multigrid

n~,+ 1 } (4 .4) M k :----- u =- 2 fije ijx = - ^ J=-n'k-I l~lj = U_j, U_n'k-1ER 1 k--l,2,...

The functions in Mk have 2n'k+3 Fourier coefficients, but the dimension Of Mk is only 2n'k+2=nk. Comparing the {ilk} in (4.2) and (4.4), the intergrid transfer operators should then be defined

(4.5) Ik_l := Fk -l

as

0 0 I I 0 I 0 )-rrk_,, i k-I := 77--- k_, ,

0 , / 2 [ 0 v 2 l 0

where I is the (nk_l--1)X(nk-l--1) identity. By the definitions above, the multigrid scheme and its variational forms for

midpoint discretizations remain the same as in section 2. Further, the analysis and the proof in sections 2 and 3 are almost the same here with a few minor modifications. For example, the second equality in (2.12) would be

- -

In principle, the nonnested discretization and the midpoint discretization SMG methods are the same, except that the discrete Fourier coefficients are naturally conjugate in pairs in the former, which saves an extra effort of the latter to make derivatives real. The number of midpoints between collocation points is odd too, like the case ofnonnested discretization. In particular, if the function a(x) in (2.1b) is a constant, then the Hk in (4.3) does not change Lk, i.e., the definitions

(2.4) and (4.3) are the same. We now consider the filtered SMG methods of Zang et al. When even num-

bers of collocation points are in use, the discrete Fourier coefficients are not conjugate in pairs (see (4.2)). When approximating derivatives of the solution to (2.1b), the highest Fourier term has pure imaginary values at the collocation points and the term is thus filtered. After such filtering (see (4.6) below), the coefficient matrix is singular. In order to have solutions for (4.6), the right hand side must have the highest Fourier term filtered too. The filtered collocation- spectral problem appears:

(4.6) [-(F~-IDkFk)Ak(Fs = (Fk-llkFk)Gk,

where the filtering is done by

f)k := diag(i(-n'k),i(-n'k+ 1),...,i(n'k),0) and i k := diag(1,1,..., 1,0).


We make a remark here. Although [-(Fk q I~)kFk)Ak(Fk -t I~)kFk) -b Bk] U k= G k has a unique solution, the equation is not a correct discretization of (2.1b). This is because the highest Fourier term solves another spectral equation, which is for b(x)u=f(x) rather than (2. l b). Correspondingly, the SMG method does not have a constant rate of convergence for this incorrectly filtered equation either.

In defining the coarse-level residual problem (2.7), we need to filter out the highest Fourier term of the residual (the right-hand side vector in (2.7)). In cor- recting the iterative solution, we also need to filter out the highest Fourier ~term of the coarse-level solution, the r in (2.8). Therefore, the intergrid transfer operators are defined as:

(4.7) Ik_l := Fk -l 0 I I 0 [ O)w o I 0 o I 0

t-~-" T Fk_l, Ik k-I := i k-I ,

where I is the (nk_l--1 X(nk_l-1) identity. The nested, auxiliary spaces are defined as

Mk = { u }' ^ iJ x^ : = uje uj = ~-7-J}, k= 1,2,... j =-n'k

The dimension of Mk is not nk, but nk-1. The other definitions and analysis for the nonnested discretization SMG remain almost the same for the filtered SMG with very few changes.

From the above discussion, we can see the three SMG methods are .of the same type, but the nonnested-discretization one is much simpler theoretically. Extra effort is needed in the other two SMG methods to define the discrete problems (4.3) and (4.6), and the intergrid transferring operators are complicated in the two methods too (see (2.5), (4.5) and (4.7)).

5. N u m e r i c a l tes ts

In this section, we give some numerical tests for the nonnested discretization and the midpoint-discretization SMG methods and we compare the two methods. For more numerical results of the midpoint discretization and the filtered SMG methods, we refer readers to [21] and [3]. In the latter, numerical comparisons of the two methods can be found.

We compute the contraction numbers of the SMG iterations for three test problems of (2.1b) where the a(x) in (2.1b) are different:


Tes t ( i ) b(x) = 1, a(x) = 3;

1 Test (ii) b(x) = 1, a(x) = 3+ - - sin(16n'x);

2 5

Test (iii) b(x) = 1, a(x) = 3+ - - s n ( 1 3 0 n ' x ) . 2

We note that a(x) varies rapidly in the third case. This demonstrates that the grids need to be very fine (k0 needs to be large, a condition in Theorem 3.2) for

such problems. After one cycle of two-level S M G iteration (with one fine-level smoothing),

an iterative error will be reduced from E to SkCkE, where

T Tk L-I i~-lLk. (5.1) Sk := Ik--AkHLk and Ck := *k--*k-I k-I

In columns 2-4 of Table l, we list the eigenvalues of maximal modulus of SkCk for the nonnested discretization and the midpoint discretization (listed inside pa- rentheses) S M G methods for the three cases with different discretizations (different nk). Column 5 of Table 1 shows the eigenvalues of maximal modulus of S~Ck (8 fine-level smoothings) for case (iii). People refer the absolute values of these eigenvalues as the rates of convergence of the iterations. The closer a number in Tab. 1 is to zero, the faster convergent the iteration is. A number greater than 1 means that the iteration diverges. A negative number implies that the coarse- level correction is not stable where iterative errors may be over-corrected in (2.8).

Table 1 - The contraction numbers of two-level nonnested-discretization (midpoint-discretization) SMG methods.

nk--1/~k

3

(i)&&Ck

/17 (8 /16)

17 /33 (16 /32)

.60 (.74)

Oil)& SkCk (iii)a

.61 (.74)

/5 (2, /4) .00 (.69) .10 (.68) - .31 (.77) - .02 (.15)

5 /9 (4 /8) .42 (.73) .50 (.73) .76 (.88). .39 (.40)

9

.68 (.74)

-2.70 (.92)

-4.81 (.94) ,69 (.75)

< . 2 9 -2.66 (.66)

3 3 / 6 5 (32 /64) .71 (.74) .73 (.76) - 4 . 1 2 (.95) - 2 . 8 6 (.7.2)

65/129 (64 /128) .73 (.74) .75 (.77) .90 (-2.79) .66 (.7..8)


In test case (i), both a(x) and b(x) in (2.1b) are constants. Simple calcula- tions would then show the limit of contraction numbers is 0.75 for both the SMG methods. For cases (i) and (ii), it appears that the nonnested discretization SMG method is a little better than the midpoint discretization one. This would be due to the better coupling of the intergrid transfer operator Ikk_l used in the nonnested-discretization SMG. In test three, a(x) has a large high-frequency Fourier term. The k0 in (2.2) then needs to be sufficiently large so that Lemma 3.1 and Theorem 3.2 hold. But the testing grids here are not fine enough, and thig causes the SMG methods to fail to converge sometimes. However, we should realize that the collocation equations (2.3) do not provide any approximation on such coarse grids. Since more fine-level smoothings does not help much in the nonnested discretization SMG (see the data in column 5 of Table 1), the coarse-level residual problem (2.7) must also have been perturbed in lower frequency modes too, due to ak_l(-,-)7~ak(-, -) in Mk-l. Readers can find more discussion about the violation of projection property in the coarse-level correction in [24].

Two remarks are in order. To improve the convergence rate of SMG, one can project a(x) to coarser levels to produce Ak-x, Ak-2 and so on from ak(','). This has been suggested in [3]. By this method, the rate of convergence in problems (ii) and (iii) can be improved. We also remark that, in a real computation, the spectral radius Ak in (2.6) can be replaced by an upper bound of it, or (2.6) can be modified such that local relaxation parameters are used in the smoothing (see [3]) to avoid the estimation of Ak. This technique could also improve the convergence rate for the test problems (ii) and (iii). From the analysis and the numerical experiments, we can conclude that the two SMG methods are of the same type. The midpoint discretization SMG needs 2nk extra operations (caused by the Hk in (4.3)) in each evaluation of LkUk. However the nonnested discretization SMG is not practical due to the limitation of the fast Fourier transform.

ACKNOWLEDGEMENTS. The author thanks Professor Ridgway Scott for introduc- ing the problem. The author also wishes to thank the referee who made many valuable comments.

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