method. lt uses only two grid sizes h1 and h1_ 1 (l > 0). The one-stage method as ...... sup llB(u, v)[: v E ~1⢠l!vi[Yf⢠= 1} > e llulf.tt" for all u E fft, l::?: 1 (e > 0). (4.4b).
Beiträge zur Numerischen Mathematik 9 (1981), 213-239
On the convergence of multi-grid iterations WOLFGANG HAcKBuscH
Abstract. Multi-grid methods can be applied to linear and nonlinear boun.dary value problems. These iterations are fast since usually the ,rate of convergence is bounded by a constant less than one independent of the grid size. For special problems and special discretizations there are somewhat complex proofs of convergence. Here we try to give a more transparent proof. The criteria of convergence are verified for the discretization of general boundary value problems by finite element methods. Subject Classification. AMS (MOS): 65F10, 65N20, 65N30; CR: 5.17
1.
lntroduction
Multi-grid iterations for solving boundary value problems are described by many authors (cf. FEDORENKO [9], BAOHV.A.LOV [5], AsTRAOHANOEV [1], BRANDT [7], HACKBUSCH (11, 13], NrcoLAIDES [25, 26], BANK and DUPONT [6], WESSELING {28]). These faSt methods can be applied to a very general class of problems. Numerical results of multi-grid methods are reported, e.g., in [7, 13]. Proofs of convergence are given in {1, 5, 6, 9, 25, 28]. All papers study düferent cases. Either the boundary value problems or the applied variants of the multi-grid iteration differ. In this paper we propose a more general concept of the proof of convergence. In detail we study the case of finite element equations (i.e., equations arising from a discretization by some finite element method). This proof contains the case considered in [25, 26]. lt proves convergence even in the case of less regular, non-symmetric or indefinite probleins. Thu s, the case of re-entrant boundaries is inclosed. Moreover, the behaviour of the rate of convergence as reported in [13] can be explained. The discretization matrix of a difference scheme often coincides with the stiffness matrix of a finite element method . In this case the convergence of the multi-grid iteration for finite element equations implies the convergence of the multi-grid iteration applied to difference schemes, too. . Section 2 contains the description of three multi-grid iterations. We are mainly interested in the first iteration (MGM and pre-version OSM). The second method (RN!) is a repeated application of the "nested iteration". The e:xtension of the first iteration to nonlinear problems yields the third one (NMGM).
214
WOLFGANG H ACKBUSCH
In order to obtain a transparent proof of convergence, the proof is shown b two steps. First we characterize the convergence by few conditions (Section 3). It i shown that the assumptions of the one-stage method OSM are almost sufficient for proving convergence of MGM (as already mentioned in [11]). Section 3.5 points oat that the iteration RNI requires stronger conditions than MGM. In Section 3.6 ,he convergence of the nonlinear multi-grid method is demonstrated under weak a umptions on the nonlinear equation. Finally, the amount of computational work i a· _ cussed in Section 3.7. Usually, for obtaining an approximation of the order of the discretizat ion error the wcirk is proportional to the number of unknowns. Although the conditions of Section 3 are very natural, they are not ea ily to verify. In Section 4 they ate proved in the case of finite element equations. Section 5 how that the convergence proofs of other authors can be interpreted by he condi ions of Section 3. It follows a list of symbols having a fixed meaning throughout thi paper. The sym bols are defined in the subsection mentioned in bracket . Li s t of symbols. ex (3.2); B (4.1); 0 0 (v) (3.2); .f7 1,f1-1 (2.6); ~l (2.2, 2.6, 4.8); G1 (2.2); y (2 .3); k 1 (2.1); J//?0, Jt± 1, J/?1 (4.1); Jlil8 (4.3); l, L , Lr (2 .1); .40 (4.2); m (4.8); MGM (2.3) ; M1 0SM, Jf1/SM (3.2); M, MGM, M1MGM (3.3); M1RNI (3 .5); NMGJ1 (2.6); V (2 .2); vmax(li) (3.2); Wt (4.8); Pi (4.1); P, (4.5); P1. 1- t (2.2, 4.5); Pl.1-l (2.4) · fJ 1•1_ 1 {3"3, 4.5); 1>1 (2.6); r 1- i.1 (2.2) ; r1-t.1 (3 .3, 4.5); R1 (4.4); (4.5); RNI (2.5); e 1 (2.6) · S (4.2); S1, S1 , (2.6); T (4.2); T 1 (4.4); u1_ 1 (2.6); UUi (3.6); U (4.2); U1 (4.4); V1 (2.1). The following Special norms are defined: lf·l11. rr·lf2, IHri.j (3.1 ); IHI, l·l.s (4.6).
fl,
2.
The multi-grid algorithm
2.1. The sequence of discrete problems
According to the name "multi-grid method" we consider a decrea ing equence of "step sizes" k 1 (l = 0, 1, ... ) : h0
> h1 > h2 > · · · >
h1
> ···
(l: "level number").
(2.1 )
A given abstract linear problem
Lu = / is to be discretized for all step widths k1 : L 1u 1 = / 1
(l > 0) .
(2 .3)
In the case of nonlinear problems we write L(u)
=
O.
(• .2')
The discretizations of (2.2') are denoted by
L 1(ut)
=
0.
(:..3')
On the convergence of multi-grid iterations
215
The solution u 1 of (2.3) [or (2.3')] as well as the right-hand side fi of (2.3) belong to a finite dimensional vector space V 1• In the case of (2 .3) the inverse of the mapping
L 1: Vi-+ V1 is assumed to exist. (2.3') may have more than one solution. In the sequel we assume that the domain of L 1 is restricted to a neighbourhood of only one solution u 1 of (2.3').
2.2. The one-stage methode (linear case)
The one-stage method (or two-level method) is a pre-version of the multi-grid method. lt uses only two grid sizes h1 and h1_ 1 (l > 0). The one-stage method as weil as the multi-grid algorithm are iterative processes. Each iteration is a combination of two steps - of a smoothing step and of a correction step. The latter one uses coarser grid sizes. Therefore, grid functions of V1 must be connected with grid functions of Vi-i· We need a prolongation P1,1-1:
Vi
V1-1-+
(linear, injective)
and a restriction r1- 1,l :
vl
-+
v,_l
(linear, surjective) .
Examples are given in Section 4.5 and Section 5. The smoothing step consists of v iterations of a smoothing procedure, where v is a fixed positive number. In practice relaxation (Ga uß-Seidel iteration) is often used for smoothing (cf. [7, 13}). In Section 4.8 we shall discuss a smoothing procedure similar to the Jacobi iteration. We denote the smoothing procedure by the mapping V1
~ ~1(vi. /1) =
G1v1
+ Hif1
where the matrices G1 and H 1 satisfy L 1 ~,(u,, / 1)
=
for u1
u,
=
(2.4)
(vu fi E Vi),
L 1- 1fi
=
H 1- 1 (I - G1), i.e„
(/, E Vi).
(2.5)
The name "smoothing procedure" is justified by the requirements (3.1 a, b) of Section 3.2. The one-st age iteration is started by some u, E Vi . Assume that the i tb iterate u 1 is already computed. Smoothing step. v applications of the smoothing procedure (2.4) map u 1(i) into Ul
(i+I/2). •
v/0) := u,Cil ;
v,< k+I)
: = ~,(v 1 -
u1
= L 1- 1d1,
where
d1 := L 1u 1(i+il 2> -
1),
lt,
216
\VoLFGANGHACKBUSCH
is approximated by
Hence, t he new iterate u 1Ci+l) is defined by 1
u , Ci+ l)
=
u1 -
P1,1-1Ll.!. 1r1- 1,1(L1u1 -
f i) .
(2.6 b)
2.3. The multi-grid, method (Linear case)
I n (2.6b) the exact solving of thelinearequa tion L 1_ 1v1_ 1 = g1_ 1 := r1_ 1 •1(L 1u 1 1, v1_ 1 can be approximated by a fixed n umber of iterations of the one-stage methode (applied with l - 1, l - 2 instead of l, l - 1). A further recursive application of the one-stage algorithm on the levels l - 2, l - 3, .. . yields the multi-grid method. The com.plete definition of the multi-grid iteration is given by the following procedure MGM, that performs one iteration u1 , ••. , u 1 } of solutions that can be used for estimating the discretization error and for extrapolation to the limit.
2.5. The repeated nested iteration
The following iteration can be obtained from (2. 7) by setting y = 1 and performing the smoothing step after the cotrection step. In BRANDT's paper [7] MGM is called "Oycle O" while the following iteration corresponds to "Oycle A". The procedure RN1 defines the iteration. Each iteration step of RN1 contains the nested iteration (2.8) with MGM and fk replaced by 0) as needed by NMGM. The nonlinear multi-grid iteration can also be used without the nested iteration. Let u 1 be any given approximation of u 1• Then u1_ 1 = rt-1, 1u 1, uk = rk,k+iuk+i (0 s k < l - 1) are possible choices of uk (0 < k < l) provided that h E :Fk. Hence, :Fk (O < k < l) must be a neighbourhood (cf. footnote on p. 218) of Lk(rk, 1u 1), where rk,l = ru+i ... r 1_ 1•1• During this iteration uk can be improved by setting uk := rk, 1u 1. In this case the computation of h must be repeated. The following note shows NMGM tobe the natural generalization of the procedure MGM. Note 2.1. If L 1(·) is an affine mappingand if ~o(Vo, fo) = cbo(/0 ), then the procedures NMGM and MGM are equivalent for any choice of {!k and fk (k < Z). 2.7. Multi-grid methods for sin.gular problems aruJ, eigenvalue prob"lems First we consider the case of a boundary value problem with a solution determined up to a constant. Example. Let Lu= -Llu
= fn
f be the Neumann problem
in Q,
ou/an = Ir on r,
f !rar= f !n dx.
r
g
For a natural choice of Li, p 1,1_ 1 , and r1-1,1 there is a grid function e, E V 1 with constant value one sU'eh that L 1e1 = 0, e1 = P1.1-1e1-1. and e1- 1 = r1-1,1 e1 hold. Then the assumptions of the following note are fulfilled. This note allows us to use the iteration (2.7) without any projection mapping. Note 2.2. Assume that zero is a single eigenvalue of the matrix Li. L et e1 and e1* be the eigenvectors of L 1 and the adjoint matrix, respectively: L,e, = L,*e1 = O. Then the problem L 1u 1 = fi is well-posed if e1* .L fi, i .e., the system is regular ij L 1 is consülered as a mapping from V 1' onto V/, where Vi' = {e1*} 1 • Definep/, 1_ 1 and r/_ 1•1 by p/,1_ 1 = Q1P1,t-1 and r;_1:1 = Q1_ 1 r1_ 1 , 1, where Q1 is the projection
*
Q1v1 = v1 - e1 • (e,*, v1)/(e,*, e,) . Assume that the iteration (2.7) converges if p;,1_ 1 andr~-I.l are used. Un
0)
(3.1a")
2001 (v) and Vmax(h) such that
1] ~ Oo1 (v)
for
V
~ Vmax(h,).
The desired property of the correction step is (3.2) with ex from (3.1a): p 1,1- 1L;:.\ri-i,ill1 •2 ~ Chi_ 1
llL1- 1 -
for l 2 1.
(3.2)
Here and in the following Cis a generic constant. Implicitly, (3.2) requires the existence of L 1- 1 and Lt!1• In Section 4 we shall verify (3.la, b) and (3.2) for a special choice of ll·lli> Li. 1)
holds for y;::;:::: 2.
We prepare the proof of the theorems by two lemmata.
(3.7')
On the convergence of multi-grid iterations
223
Lemma 3. 6. (3.1 a), (3.2), (3.4a), (3.5a), and (3.6) imply \[L[_\r1-1,1L1G1vl'2.2 < G
(l
> 1, 0 1, 0 < v :::;;: 'Vmax(h1)) ·
(3.8b)
Proof. The .representation P1.1-1Li.!.1r1-1,1L1G1v = G/ - [L,- 1
-
P1,1-1L/!1r1-1.d L,G,v
yields
l[L/.!.1r1-1,1L1Gt''lf2,2 < 0 · !11G1'lf2,2
+ llL,-
1
-
P1 .1-1L!.!.1r1-1.illi.2 · llL1G1'lb.1}
by virtue of (3.4a) . Hence, (3.5a), (3.2), (3.1a), and (3.6) imply (3.Sa) . Analogously, (3.8b) follows from
L1G1'P1,1-1L!.!.1r1-1,1 = L1G1·L1-1 - L1G1•[L1 - 1
P1.1-1L21r1-1,il · •
-
Lemma 3. 7. The iteration ?11atrix of the multi-grül method (2.7) is
M 1MGM
=
M 1osM
+ P1,1-J(Mf!._~M)y L/.!. 1r1-1,1L1Gt'
(l ::2: 2).
Analogously, the iteration matrix M,MGM of Theorem 3.5 takes the form
Jf!1MGM
=
M/SM + L1G1'P1,r-1L/!1(M1f!f)Y r1-1 ,1·
Proof. The multi-grid iteration on the level l - 1 for solving L 1_ 1u 1_ 1 = d 1_ 1 has the represen tation (i) MMGM (i- 1) + N d
U1 - 1 = 1-1 Ut - 1 1-1 1-1 • Since L"1.!. 1d1_ 1 is a fixed point of the iteration, N1-1
=
(I - Mlf!f) L2 1 •
(3.9)
The algorithm (2.7) starts on the level l - 1 with uf~ 1 = 0 (v := 0) and results in y-1 (y)
Ul - 1
=
'""'
L.
i=O
(MMGM)t 1-1 N 1- 1d1-1 •
Applying the multi-grid algorithm of the level l to u 1 = r 1_1,1L1G1•o1• Thus, the iteration matrix is y-1 M l MGM -- Gl • - p !,!-1 L. '""' (MMGM)t N 1-1 r :-1,! L ! Gl • • l- 1
+ o1,
we obtain dt-i
i= O
Using (3.9) we obtain M1MGM = {L1- 1
II
-
P1,1-1[l - (.Mf:. log (202 )/llog i'.21. Then the estimate
lfuk - ukj[2
is defin ed by {~e repeated nested itera.tion (2.9). Then the folloW'ing estimates hold:
1fJil1RNill1,1
0 · B- 1 • exp (lB) · Oo(v)
if ~(hk) ~
Q · l · 00(11)
if .E 'Y/(h;) < 0 f or all k, l 2 1,
B
0 für all k > 1,
~
1
j=k
0 · 0 0 (v)
if (3.14 b) holds.
If Oo(v) = O/v (cf. Section 4.7) and hk = ho/2 k, then the estimate 1llff1R·Nil[1,1 ~ requires 11 = O(h1 - ·/log 2 ), v = O(ilog hd), or v = 0(1), respectively.
Proof. Assume U1(i) = U1 + o/i) and redefine is represented by Uj o/i+l), where
+
jl
uz
Ci+i) --
' G •p
" .4.J k=l
15 Num. Ma.thema.tik 9
l
1.1-1
G'1-1 p 1-1,1-2 •• · Gk •jluuk
e
Wz) • (vt - w,)
for all v" w 1 E tJ/11
(3.15a)
lfL,- 1 • DL,(vt, w,) - 11'2, 2 s o/(l!v1 - u 11J 2
+ lfw
(3.15b)
and
1 -
u 1 J'2)
must exist, where 818) -+Ü
as
s -+ 0.
(3.15c)
Similarly, @ 1(v1, lt)
-
@ 1(w1, ft) =
DC§,(v1, w,,
/i) · (vt
- w,)
and llD@1(v1, w"
/i) - G1l'2,2 s r'.llv1 - u1l'2
are assurned. G1 is the derivative
oe§1(u1,
(v,, w, E tJ/11;
+ llwt -
/i E ff1 )
u1 1l2 + lfrJ>1(/1 )Jr2)
(3.16a)
0 ...('j)
O)/&uz.
Theore m 3.12. Assume that (3.1a, b), (3.2), (3.4a), (3.5a), (3.6), (3.15a, b, c) and (3.16a, b) are valUl for L, and G1 defined ahove. lf e1-1. llf1-1ll1, ll/11!1. and llu/0l - c/> 1(/1)11 2 are sufficiently small and if v is large enough, the results of the, nO'Yllinear ~-grü/, ueratwn (2.11a-d) satisfy J...,p~
IJu,Ci+i> - ci>1(/1)lf2 s [C · Oo(Y) where
'1/(i)
tends to zero as
+ r;,
where the argument s of DLc_ 1, DL1, and D 0).
(4.4b)
230
WOLFGANG lliCKBUSOR
i'f
Furthermore, B is assumed to be 1tcoercive (cf. [23, p. 202]): There are real ;numbers e > 0 and Ao > 0 such that Re B(v, v)
>
e [[v1J~1 - Ao lfv[[~·
for ,all v E .?/111 •
(4.5)
In case of Ji. 0 = 0, (4.5) implies (4.4a, b). In the sequel Ji. 0 satisfying (4.5) is fixed. Let T be the operator associated with
~
[B(u, v)
+ B(v, u)] + Ä.
0 •
(u, v).?ff•· By
assumption (4.5) T is selfadjoint and positive definite. Moreover, T: ;Rl--+ ytJ-1 and T- 1 : .;tf- 1 --+ .Ye1 are bounded. The domain of the operator T 112 coincides with J'f'I: D(T1f2) = .Yt'1 (cf. KATo [19, p. 331 ff.]). S is the sum S
=
T
+ U,
where U is associated with ..!.. [B(u, v) - B(v, u)] - /.0 • (u, v).?ff•· 2
4.3. Intermediate Hilbert spaces As mentioned above, D(Tlf2) = J'f'1 holds. Since D(TO) = .Yf'0 we may define the Hilbert spaces ;,1"8 := D(T8 f2);
.Yt'-s :=dual space of Jl'e&
(0
s
s
s
1)
endowed with !lu[J~, = [[T8 12u[l~o (- 1sss1). ForO s 8 s 1 wehaveJ'f'-l=:iJ'f'-" ::::> .Yf'O ::::> J/t'8 ::::::J ;,1"1 • For s = 1 the new norm is equivalent to the original norm of .Yf'I, Hence, we may use the same notation. Using the notation of LroNs and MAGENES [23] we may write .71"8 = [J'f'1 , J'f'0] 8 and .Yt'- 8 = [J'f'0 , J'f'- 1 ] 1_ 8 (0 < s s 1). Example. Consider the case of .Ye1 = H 0 1(Q) (cf. first example of Section 4.1). Then Jlt'8 = H 0 8 (Q) holds for all s E [-1, 1], s = ±1/2 (cf. [23, p. 64ff.J).
4.4. Representation of the di,scretization
Pt
Let (-, ·) be the scalarproductsof the vector spaces V1 (l > 0). Bymeansof (g, ~v1 ).*'• = (P1*g, v1) (g E ,;f'- 1, v1 E V,) the restriction R 1 := P 1*: ,;f'- 1 -r V1 (l > 0) is defined. Hence, the discretization (4.3) is equivalent to
S1u 1 =
g1 ,
where
S,
=
S,
+
UIJ
8 1 = R,SP1 , q1 = R 1g,
(4.3')
and where u1 is a perturbation (cf. Section 4.8) due to the numerical quadra;ture involved by 8 1• Although S1 and g1 coincide with L 1 and / 1 of (2.3), we shall use the notation (4.3') instead of (2.3). Note that possible errors of 'f/1 do not influence the iteration. We conclude from S = T U that S1 = T 1 U 1, where
+
T 1 = R 1TP1 ,
+
U1 = R 1TP1 •
Note that Tz is positive definite and selfadjoint, too.
On the convergence of multi-grid iterations
4.5. Oho'ices of Pt,1-1 and
231
r1-t,t
Since P 1 is a surjective mapping from V1 onto .Yf'1, the inverse matrix (R1P 1)- 1 : Vi --+ V 1 exists. For reasonable prolongations Pt the estimates
l!P11iv1--+Jf'" S 0,
l1(R1P1)- 1 llv 1--...v1
< Ot_ (l ~
(lt. t/
0)
hold uniformly with respect to l. The restriction
R,:= (R1Pt)- 1 Rt: Je-l--+ v, satisfies R1P 1 = I. Analogously, R 1P1 = I
holds for
P1:=R1* =P1(R1P1)- 1 : V1-+.Yf'1 c:.n" 1 •
We define Pl,l-I
=
r1-1,1
= R1-1P1:
R1P1-1: Vi-1--+
v,'
V, --+ V,_1,
'Pt,l-1
=
R1P1-1: V1-1 --+
Vi'
f1-1 ,1 = h,l-lpt: V1 --+ V1-1 ·
By virtue of (4.6) the mappings Ri. Ri. P1, Pi. 1- 1 , 1>1, 1_ 1, r 1- 1 ,1, and f,_ 1 ,1 (from .n"0 , Vi. Vi-t• resp. into Jf"O, V11 V1_1) are uniformly bounded for all l. Furthermore, by .n"1-1 c: .Yf'1
P1-1 = P1P1,1-1.
R1-1
=
r1-1.1R1
(l ~ 1)
hold. P}l1 = P1R 1 is the projection of ff0 onto Je1• p,, 1_ 1 and r1_ 1 , 1 satisfy the third part of the conditions (3.4a', b'). In some cases we shall need the assumptions (4.7a) or (4.7b):
llP1R11!Jf'4Jf'• so
(l ~ ü),
l!P,Rillff'-+#' s 0, 11P,Rii/.no•--....?r < 0 The inequalities (4.7b} imply (4.7a) since R 1R1 =
(4.7a) (l ~ O).
(4.7b)
P1b.1P1R1.
Note 4.1. (4.7a) is the only severe condition of this section. If we define the scalar product of V 1 by (v1, w1) = (P1v 1, P 1w1).;f7•• (4.6} holds with 0 = 1 and because of b.1 = R 1 the inequalities of (4.7a, b) are equivalent.
4.6.
Oho'ice of norms
Since T 1 is positive definite, the fractional powers T 13 12 are well-defined for all s E JR. We define the following norm on Vi :
lvila:= llT1 81 2v1 llv
1
(l > 0).
Note 4.2. By (4.6) and Ti = R 1TR1 the norms lvlls and l!P1v1 ll;f'• are equivalent for s = 0 and s = l. If (4.7a) holils, lvils is equivalent to 111'1v 1lfff• for s E [O, 1] and equivalent to l[P1v 111.;f7• for s E [-1, O]. I/inaddition (4.7b) holds, also llP1v11fff• and llP1v1lfff• are equivalent for s E [-1, 1].
232
WOLFGANG H ACKBUSCH
The proof is prepared by two lemmata. For the proof of the following interpolation theorem compare [8], [30, Lemma 4], [23, p. 27]. Lemma 4.3. LetH1 andH2 be twoHilbert spaces. Assume thatA i: D(A i) c Hi --* H; (i = 1, 2) are positive definite operators with dense domains D(Ai)· Define H {" := D(At). lf A: H 1 a --* H 2" is bounded for a = a 1 and a = a 2 , i.e. ij !fA2ciAA1-"llH._.H, < C holds for a = ix 1 and a same constant G.
=
ix 2 ,
then the estimate is valid for all oc E [ 1X 1 , a 2 ] with the
In the sequel we shall use the notation
11 ·II instead of ll · ll;f'•-v,, I[ · llv1-V1:• II· llv,7 .Jf'•, or II· ll.Yt'"-.1t'"
(l, k 2:: 0).
Lemma 4.4. (4.6) implies (4.Sa) for s E [O, 1], while (4.6) and (4.7a) ensure the estimate (4.8b) for s E [O, 1]:
llTsl2p,T1-sl2[[ < G,
llT1-sl2R1Tsl2[[ < C (l
[1T-st2p 1T/12J[ ~ C,
1JT,sf2_k1T-sf2J[
~
0),
(4.Sa)
< C (l > 0).
(4.Sb)
lf the assumptions (4.6) and (4.7b) hold, the estimates (4.8a, b) are valid for s E [-1, O].
Proof. Since R 1 = P 1* it suffices to prove only one inequality of (4.Sa). By virtue of Lemma 4.3 we have to prove (4.8a) only for s = 0 and s = ±1. For s = O use (4.6). For s = 1 the estimate (4.Sa) follows from
lfT,-1t2R,TP1T1-1/2ll
=
11111 = 1.
lfTz112R1T-1/2l12 = 11r-1t2p1T1R1T-1l21f
=
JIT1f2P1R1T- 1l2J12 < C.
[IT1/2p,p,-112112
=
For s = -1 use (4.7b):
Applying (4.7a) and (4.7b) to
11p-1/2P1Ti1'2I[ = !IT1/2ptR1T-11211
a.nd
'1r; '1 .., . 11-: "12. _„-L .:YzJ, '1 - 1; 1JT 112P,T1-112 J[ ~ lr.I'F11~R11'1)- 1--TzJ.12 tl, 0).
(4.9c)
The smoothing procedure is chosen as '9'1(v,, lt):= Vz -
w1h1 2m(S1v1 - /1),
OJ1 = 1/ (h1 2m IJ Tzli) ·
(4.10)
If the diagonal of S1 is a multiple of !, (4.10) is a d amped Jacobi iteration. Note that (4.9a) guarantees w1 2 0 > 0 (l 2 0). w1 may be replaced with w1 E [O, mi], where
0>0.
w OLF GANG HAmrnusc:e:
234
Theor em 4.8 [Oondition (4.10) and as8ume (4.6) and and l · ls„ re8pectively, where with o< 1 +[min (0, 8 1) hold8 for L 1 = S1 with a = 28m,
(3.1a, b)]. Let the 8moothing procedu re be de/ined by (4.9a, c) with o > O. Ohoo8e the norm8 I[ ·lli, 11· 112 by l · 181 2 8 1 > 8 2 and 8 1 , 8 2 E [-1, 1]. Assume either (4.9b) max (0, s2 )]/2 or (4.9b') with o < 1. Then (3.1a, b)
+
+ s)-
0 0 (v) = 0 · (v
8
,
s= 1
+ (s
s2 )/2
1 -
> 0.
Proof. Apply Note 3.1 with L 1 = S,, L/ = Tl> L/1 = U 1 + J• =
•m/w
0 ~µ~1
= (h12mmi)-s
ssv'/(v
µr
+ s)•+s ::;; h,-2sm. (8/w1)&/(v + s)s,
since the spectrum of T, belongs to [O, h1- 2 1]. From (4.9a, b, c) one obtains (3.1a" ) with (3 = (82- - 8 1+ - 2o 2 - 8 2 s1) m > 0, where 8 1+ =max (0, si) and s2 - = min (0, 82 ):
+
iiLiIIl[2 , 1
-
1sif2(Ui
+
+ at) p 1- s,/211 ::;; (hr2m/wi) 1. We make the following regularity assumptions :
11s- 1/lfffs+• ::;; 0 llfliff• for all f E :Yf'B (-1 ::;; 8 ::;; 1)' llS*- 1/lf.*'•+1 ::;; 0 11111.;f'• for all f E :YfB (-1ld fm· the prolongations P 1 defined by the choice of the coefficient space V 1 (l ~ 0). The estimates (4.9a, b) with (J < 1 are assumed for the "principle part'' T 1 and the "lower order part" Ui of S 1• The p erturbation 0) are satisfied for boundary problems with smooth coefficients and smooth boundary (cf. [22, 23]). The assumptions even hold for Dirichlet problems in nonsmooth but convex regibns with s = 0 (cf. [18]). In the case of s > 0 in (4.11a, b)
we obtain the following result. Theorem 4:12b [Oa,se of "very regular" problems]. The estimate (3.7) even hoUs with 0 0 (v) = O/vl+q (0 < q:::;; 1), if the requirements of Theorem 4.12a are changed as follows. (4.9 c) is assumed for (J = 1 + q, (4.11a, b) for s = q, (4.12) fort= 2 + q. In addition (4.7b) must hold. Proof. Choose ll·lf1
=
l· lq, 11 ·11 2 = 1·1-q; i.e.,
81
=
q,
s2 =
-q. •
In the case of non-smooth regions with a re-entrant boundary, the regularity assumptions (4.11a, b) are not valid for s = 0. E.g., NEC.AS [24] shows (4.11a, b) with s E (-3/2, -1/2) for the Dirichlet problem (.n91 = Ha1(D)) in a Lipschitz region Q (cf. [3, p. 16]). The numerical results reported in [13] show that the rate of convergence becomes worse as soon as re-entrant boundaries occur. This observation agrees with the behaviour of 0 0 (v) given in Theorem4.12c. Note thatalsoin the case of re-entrant boundaries accutate discrete soiutions u 1 ate of interest, since by the discretization of ZENGER and GIETL (29] the discretizati6n error can be made small. Note that there is a second application of Theorem 4.12c. Usually, the simplest :finite element subspaces dF1 c:: Jlf'l satisfy (4.12) for 1 < t s tmax := 1 + 1/m (2m: order of the differential operator 8). Therefore, the numbets s1 , - 82 of Theorem 4.11 must be negative (s 1/m - 1) if m > 1.
Theorem 4.12c [Oase of less regular problems]. The e8timate (3.7) holrls with 0 0 (v) = O/v1- q, 0 < q < 1, if the requirements of Theorem 4.12a are changed as follows. Either (4.9b) with 6 < 1 - q or (4.9b') with 6 < 1 must hold; moreover, (4.9c) with (J = 1 - q, (4.11a, b) for s = -q, and (4.12) fort= 2 - q are required. In addition (4.7a) must hold. Proof. Choose
5. 5.1.
11 · 1!1 = l·l-q and ll·lr2 = i · lq, i.e., s1 = -q,
82
=
q. •
Discussion of other proofs of convergence Proof of Fedorenko [9]
considers the Poisson equation Llu = f in a square with zero boundary conditions. He uses the five-point formula as discretization and applies the smoothi.ng ptocedure (4.10) with w1 E (0, h 1- 2 /4). r 1_ 1 , 1 is the trivial injection; while p 1, 1_ 1 is FEDORENKO
On the convergence of multi-grid iterations
237
defined by piece-wise linear interpolation. Vi is split into Vi' EB V/', where V/ and V/' consist of the "good" (smooth) and "bad" components, respectively. The correction step takes the form !Je = g/ + g/' r-+ vc = v/ v/', where vi' = v;, 1 vl, 2 (hc 2Lcvl, 2 is defined by the second and third terms of the equation Llvv• = · · · on page 563 of (9])- The proof of (9] corresponds to tQ.e choice of
+
+
llY1 lf1:= l 1Ji'll + q-••llgt"I!, llvd[2 := h, J[L,v;,11! + l[v;,211 + h1 2q-•• llL1v/'ll where II · II is the discrete L 2 norm. (3.1 a, b) and (3.2) hold with a = 2, 0 0 (v) = O/(v - ')10 ) (v > ')lo) 5.2.
Proqf of Bachvalov [5]
The problem (2.2) in cons1deration is a general elliptic differential equation of second order in the unit square Q with homogeneous Dirichlet boundary condition. L 1 is a certain difference scheme. r 1_ 1 ,i is the trivial injection; p 1, 1_ 1 must be an interpolation that is exact on polynomials of second degree. The smoothing procedure is (4.10). As in (9], Vi is the direct sum of A/'1• 0 containing the smooth eigenfunctions and Ai 1•1 - (3.1) and (3.2) are proved for ix = 2, l[vd[1 = Jlv/l[/00 (v) + lfvt"[[ and Jlv1Jl2 = h1 2 llLiviJ[, where v1 = vt' v,", v/ E Aoh•. 0 , v," E Aoh•.i, 0 0 (v) --;.. 0 and 0 --;.. 0 as
„ --;. 5.3.
+
00.
Proof of Astrachancev [1]
The proof of AsTRACHANCEV for a special finite element method in the case of m = 1 is similar to the considerations of Section 4. The smoothing procedure differs from (4.10) by a diagonal matrix E 1 : C§,(v1, fi) = v, - w1h1 2E1-1 (S1v1 -
The norms
5,.4.
J[vdl 1
=
l!P1v,Jl2
-111
and
![v1J[2
=
fi). IJP1vi![2
1
=
Jv1 J1 are chosen.
Proof of Nicolaides [25]
In [25] the following example is considered: m = 1, .71'1 c H 1(Q), B: symmetric bilinear form with B(v, v) > e · J[vJ[1-1cm· Furthermore the regularity assumptions (4.11a, b) must hold for s = 0. (4.10) is used for smoothing. The norms of Section 3.2 are 11·111 = ll -lb = l • lo5.5.
Proof of Wesseling (28]
The assumptions 1-3 of WESSELING (28] correspond to our conditions (3.la), (3.2), where ll -11 1 = 11 • lb = l · lo is chosen. Assumption 4 of [28] is equivalent to (3.4a). lt is proved that these assumptions are satisfied for a certain difference
238
WOLFGANG HAOKBUSOH
scheme applied to a general second order differential equation in a square with zero boundary condition. Smoothing is defined by (4.10). In opposite to [5], Lk (k < l) is defined by Lk = rk,k+1Lk+iPk+1.IP where Pk+J,k = rtk+i corresponds to piece-wise linear interpolation.
5.6.
Convergence in the case of smoothing by relaxation
All proofs mentioned above use the smoothing procedure (4.10) or a similar version. The multi-grid iteration often becomes faster if the relaxation method (Gauß-Seidel iteration) is used for smoothing. For the model problem of Llu = f in a square and Dirichlet boundary conditions, the author proved the convergence of the multigrid method with 0 0 (v) = O/v if relaxation is applied with a suitable ordering of the grid points (cf. [13] and reference [8] of [13]). It is an interesting fact that in the one-dimensional case one multi-grid iteration with smoothing by relaxation yields the exact solution, i.e„ l[M1MGMJ'2 2 = 0 (cf. [13]) . .4 .c.„; ii:,t ~ c.I°' ~ ' 'f"C.l~
