global-basis two-level method for indefinite systems. part 1: convergence studies

*Correspondence to: Jacob Fish, Scienti"c Computational Research Center, Rensselaer Polytechnic Institute, Troy, NY12180-3590, U.S.A.

sE-mail: "[email protected]

Contract/grant sponsor: O$ce of Naval Research; contract/grant number: N00014-97-1-0687

Received 28 September 1999Copyright ( 2000 John Wiley & Sons, Ltd. Revised 14 February 2000

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 49:439}460

Global-basis two-level method for inde"nitesystems. Part 1: convergence studies

Jacob Fish*,s and Yong Qu

Departments of Civil, Mechanical and Aerospace Engineering, Rensselaer Polytechnic Institute, ¹roy, N> 12180, ;.S.A.

SUMMARY

A robust two-level solver for high inde"nite system of equations arising from the "nite element discretizationis developed. It is shown that the optimal coarse model is spanned by the spectrum of the highest eigenmodesof the smoothing iteration matrix. Convergence studies conducted on a model prolongation operator revealpathological sensitivity to any deviation from the optimal coarse model. Copyright ( 2000 JohnWiley & Sons, Ltd.

KEY WORDS: multilevel; multigrid; inde"nite; convergence; global basis

1. INTRODUCTION

For symmetric positive-de"nite (SPD) systems, iterative solvers developed into mature techno-logy, which in many cases, is far more e!ective than direct methods. For such systems, multilevelpreconditioners coupled with appropriate accelerators possess an optimal rate of convergence bywhich computational work required to obtain a "xed accuracy is proportional to the number ofunknowns. For weakly inde"nite systems, for which the inde"nite term is a &compact perturba-tion' of a dominant elliptic part, multilevel solvers have been shown to possess similar character-istics provided that the coarse grid is su$ciently "ne [1]. For highly inde"nite systems, however,convergence is not guaranteed unless multilevel solvers are utilized in the context of normalsystems [6]. Often, such approach is avoided in practice because the coe$cient matrix for normalsystems is much worse conditioned than of the source system. This lack of robustness of iterativemethods causes serious obstacles to their general acceptance.

Recent research e!orts have been devoted in great part on developing robust accelerationschemes for inde"nite systems based on Krylov subspace methods, such as GMRES [4] andQMR [7]. Unfortunately, these methods may be unacceptably slow if they are used withoutpreconditioning. Robust preconditioners for highly inde"nite systems are still at the embryonic

stage of development and successes so far have been very rare. Few notable exceptions are varioustwo-level methods for Helmholtz equations [8}11] which exploit the structure of the di!erentialoperator and recently developed incomplete factorization techniques [5, 12].

In the paper, we focus on the development of a general purpose two-level preconditioner fora symmetric inde"nite system of equations

Kx"f, x3RN, f3RN (1)

where K is an N]N symmetric sparse inde"nite matrix. Examples falling into this categoryinclude the Helmholtz equation, the Galerkin or least-squares methods with constraints, and theproblems with symmetric inde"nite constitutive tensor which may arise from damage/localiza-tion in solids or shocks in #uids.

We introduce the concept of optimal prolongation operator (or equivalently the optimal coarsemodel), and show that for highly inde"nite systems such prolongator is (i) non-smooth ingeometric sense, (ii) highly sensitive to any deviation from optimality, and (iii) non-local. Thesecharacteristics preclude its e$cient construction from the local basis. To distinguish our methodfrom existing local-basis two-level methods (geometric or algebraic two-grid methods), we willrefer to it as global-basis two-level method.

The outline of this paper is as follows. In Section 2, we introduce the notion of optimality ofthe prolongation operator and show that increasing the size of the coarse model does notnecessarily improve the rate of convergence of the local-basis two-level methods. Convergencestudies on a model problem are conducted in Section 3. Computational aspects of the methodand numerical experimentation on a sequence of examples involving Helmholtz equation onbounded domains and shear banding problem with strain softening are presented in part 2 ofthis work [2].

The following notation is employed throughout the paper. We denote the prolongationoperator by Q given as

Q : RmPRN (2)

The restriction operator QT from the "ne model to coarse model is conjugated with theprolongation operator, i.e. QT : RNPRm. The coarse model matrix K

0is the restriction of K

K0"QTKQ, K

03Rm]m (3)

The smoothing preconditioner and coarse model preconditioner are denoted as M3RN]N andC"(QK~1

0QT )~13RN]N, respectively. The smoothing and coarse model correction iteration

matrices are denoted as R"IN!M~1K3RN]N and ¹"I

N!C~1K3RN]N, respectively,

where IN

is an N]N identity matrix. The error reduction, ei`1"¸ei, in a complete two-levelcycle is governed by a two-level iteration matrix.

¸"Rl¹Rl3RN]N (4)

with l post- and pre-smooth iterations. For more details we refer to References [3, 6, 9, 10, 13, 14].

440 J. FISH AND Y. QU

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 49:439}460

Figure 1. Simple bar problem.

2. MOTIVATIONS AND GOALS

Multi-level methods consist of two major elements: smoothing and coarse model correction.For SPD systems, classical iterative methods (e.g. Jacobi, SSOR, etc.) are used as smoothers,designated at eliminating the high-energy or oscillatory components of the error, while leavingthe low-energy or smooth components relatively unchanged. The e$ciency of proper multigridmethods for positive-de"nite systems stems from the fact that geometrically smooth errorcomponents are only slightly a!ected by smoothing, but can be easily approximated on a coarsergrid by solving the residual equation. For inde"nite systems, however, the low-energy modesmight be geometrically oscillatory, and thus standard smoothing methods may magnify some ofthese modes rather than reducing or eliminating them as was observed in Reference [10].

In general, the terms &smooth' and &oscillatory' may not be directly related to the geometricsmoothness or lack of it. For highly inde"nite systems it is necessary to adopt an algebraic pointof view, by which the &oscillatory' modes of errors are those which can be e!ectively reduced bythe smoother, whereas the remaining &smooth' modes are those designated to be captured by thecoarse model [14].

Let ('i, j

i) be the eigenpair of the smoothing iteration matrix, R"I

N!M~1K, which controls

the in#uence of smoothing or error reduction, i.e. ea`1j

"Reaj, where the superscript denotes the

iteration count. It is apparent that smoothing iteration matrix, R, magni"es the error correspond-ing to eigenmode '

iif Dj

iD'1. Thus an optimal prolongation of order m should span the subspace

de"ned by a linear combination of m eigenvectors corresponding to the eigenvalues of R whichare greater than one. On the other hand, if max Dj

iD(1, the convergence of a single-level method

(consisting of smoothing only) is guaranteed.As an illustration, we consider a simple bar problem as shown in Figure 1. The bar is

discretized with 10 eight-node hexahedral elements totaling 132 degrees-of-freedom. Two mater-ial models are considered: (i) a linear elastic bar, and (ii) a bar with softening zones at the centerand the two ends (with E

"!/$/E"!0.8 where E

"!/$and E are the sti!nesses inside and outside

the softening zones, respectively). The SSOR smoother is employed for all single-level andtwo-level methods considered.

For the linear elastic bar, the sti!ness matrix, K, has all positive eigenvalues, as shown inFigure 2. Figure 3 shows that the lowest-energy eigenvector of K is geometrically smooth, andmatches closely the highest-energy eigenvector of R. Since all eigenvalues of R are within the unitcircle in the complex plane, the convergence of a single-level method with only the SSORsmoother is guaranteed. On the other hand, the sti!ness matrix of the softening bar is highlyinde"nite giving rise to both negative and positive eigenvalues (Figure 2). It can be seen that the

INDEFINITE SYSTEMS. PART 1: CONVERGENCE STUDIES 441


Figure 2. Spectra for simple bar problem.

Figure 3. Algebraically &smooth' modes in a linear elastic bar.

eigenvector corresponding to the lowest eigenvalue of K is oscillatory and di!ers signi"cantlyfrom the eigenvector corresponding to the highest eigenvalue of R (Figure 4). The convergence ofthe single-level method is not guaranteed due to the existence of modes outside the unit circle(Figure 2).

For the two model problems considered, the performance of the single-level method, the twoglobal-basis two-level methods (one with prolongation operator constructed from the lowesteigenmodes of K, and the other one with prolongation operator constructed from the highesteigenmodes of R) as well as the two local-basis methods (geometric two-grid method (Figure 5)



Figure 4. Algebraically &smooth' modes in a softening bar.

Figure 5. Coarse model for two-grid method.

Table I. Convergence results of simple bar problem.

Size of the Inde"nite SPDSolution methods coarse model system system

Single-level (SSOR) * '400 47

Lowest 9 '400 18eigenvectors of K 26 '400 12

Global-basis 86 90 11

two-level Highest 9 200 17methods eigenvectors of R 26 88 11

86 11 7

Local-basis Two-grid 60 '400 36

two-level GAM 80 '400 12methods

and the generalized aggregation method (GAM) [3, 6]), are summarized in Table I. For all casesconsidered, QMR [7] method was used as an accelerator.

Several observations can be made:

(1) For the elastic bar problem the rate of convergence of the local-basis two-level methodimproves with an increase in the size of the coarse model. For the softening bar problem,however, increasing the size of the local-basis coarse model does not improve the rate ofconvergence until the size of the coarse model reaches 90 per cent of that of the sourceproblem.



(2) For the elastic bar problem the lower-energy modes of K are close to optimal (higher-energy modes of R) giving rise to identical behaviour of the two global-basis methods. Forthe softening bar problem, selection of the coarse model solely based on the spectrum of K,i.e. irrespective of the structure of smoother, fails to eliminate algebraically &smooth'components of error.

It is apparent that algebraically &smooth' error components, which might be magni"ed bya smoother, can be removed through the complementary e!ect of a properly designed coarsemodel. For SPD systems, such algebraically &smooth' error components can be adequatelyapproximated by the local-basis functions. For inde"nite systems, these modes are not necessarilysmooth in a geometric sense. Whether these modes in general can be approximated by thelocal-basis functions on the coarse model is questionable. This phenomenon was also discussed inReferences [3, 9, 10].

The primary goal of the present paper is to study the e!ect of deviation (or error) from theoptimal prolongation operator spanned by the highest eigenmodes of R. In general, the subspacespanned by the local-basis functions in the coarse model consists of both algebraically &smooth'and &oscillatory' error components; yet, for positive-de"nite systems the existence of the coarsemodel does not slow the rate of convergence. For highly inde"nite systems, on the other hand,any deviation from the optimal coarse model seems to have a detrimental e!ect on the rate ofconvergence. This suggests that a successful two-level method should possess a coarse model ableto accurately reproduce the space of algebraically &smooth'modes. The smoother, in turn, shouldbe designed to minimize the span of the algebraically &smooth' space.

In the next two sections we put this hypothesis to test by studying the rate of convergence of thetwo-level method for both SPD and symmetric inde"nite systems in an attempt to assess thein#uence of deviation (or error) from the optimal prolongation operator. The structure of theprolongation operator is simpli"ed to allow for a closed-form derivation and parametric study.

3. CONVERGENCE STUDIES

We consider a non-symmetric eigenvalue problem, R'i"j

i'

i, which is equivalent to the

generalized eigenvalue problem

K'i"c

iM'

i, K3RN]N, M3RN]N (5)

where K and M, are real symmetric matrices, ('i, c

i) is the eigenpair for the pencil (K, M) and

ji"1!c

i.

In the present paper attention is restricted to non-defective pencil (K, M). Thus, we assume thatthere are N eigenpairs for the generalized eigenvalue problem (5) such that

'TK'""N, 'TM'"I

N(6)

where '"['1'

2. . . '

i. . . '

N]3CN]N, "

N"diag (c

1c2

. . .ci. . . c

N)3CN]N.

We denote Re(x) and Im(x) as the real and the imaginary parts of x, respectively; x6 theconjugate of x; and DxD the magnitude of x with i2"!1. Since K and M are inde"nite, theeigenpairs for (K,M) might be complex. If ('

i, c

i) is the eigenpair for (K,M) such that Re('

i)O0,



Im('i)O0, its conjugate ('

i, c

i) is also the eigenpair for (K,M). This property follows from

K'i"K'

i"cM'

i"cN M'

i(7)

We "rst de"ne the set of all eigenpairs of (K, M) as

&"M('1, c

1) ('

2, c

2) . . . ('

i, c

i) . . . ('

N, c

N)N (8)

and the corresponding three sequences of eigenpairs of (K, M) as

&0"M('0

1, c0

1) ('0

2, c0

2) . . . ('0

i, c0

i) . . . ('0

m, c0

m)N (9)

&1"M('1

1, c1

1) ('1

2, c1

2) . . . ('1

i, c1

i) . . . ('1

m, c1

m)N (10)

&2"M('2

1, c2

1) ('2

2, c2

2) . . . ('2

i, c2

i) . . . ('2

l, c1

l)N (11)

which satisfy:

(i) ('0i, c0

i)3&, ('1

j, c1

j)3&, ('2

k, c2

k)3& ∀1)i, j)m and 1)k)l. For all ('

i, c

i)3&,

there exists ('kj, ck

j)"('

i, c

i) for k"0, 1, 2.

(ii) if iOj, then ('0i, c0

i)O('0

j, c0

j), ('1

i, c1

i)O('1

j, c1

j), ('2

i, c2

i)O('2

j, c2

j) and ('0

i, c0

i)O

('1j, c1

j)O('2

k, c2

k) ∀1)i, j)m, 1)k)l

(iii) D1!c0iD'D1!c1

jD ∀1)i, j)m and D1!c0

iD'D1!c2

jD ∀1)i)m 1)j)l.

(iv) if Re('0i)O0 and Im('0

i)O0, then

('0i`1

, c0i`1

)"('0i, c0

i) or ('0

i~1, c0

i~1)"('0

i, c0

i)

(v) if Re('2j)O0 and Im('2

j)O0, then

('2j`1

, c2j`1

)"('2j, c2

j) or ('2

j~1, c2

j~1)"('2

j, c2

j)

(vi) l"N!2m.

3.1. Model prolongation operator

Consider the following structure of the prolongation operator Q"Mq1

q2

. . . qi. . . q

m]3RN]m:

Q"'0Sa0#'1Sa1 (12)

where '0"['01'0

2. . . '0

i. . . '0

m]3CN]m ; '1"['1

1'1

2. . . '1

i. . . '1

m]3CN]m ; a0"diag (a0

11a022

. . . a0ii

. . . a0mm

)3Rm]m ; a1"diag (a111

a122

. . . a1ii

. . . a1mm

)3Rm]m; and S3Cm]m is a blockdiagonal matrix with either 1]1 or 2]2 blocks.

The "rst term in (12) denotes the optimal prolongation of order m, whereas the second termexpresses the error or deviation from the optimality. We further pose the following restrictions onthe error term. A more general case is considered in Section 3.2.



The blocks in S are de"ned as follows:

(i) If Re ('0i)O0 and Im ('0

i)"0 then [S

ii]"[1] (13)

(ii) If Re ('0i)"0 and Im ('0

i)O0 then [S

ii]"[i] (14)

(iii) If Re ('0i)O0 Im ('0

i)O0 and ('0

i`1, c0

i`1)"('0

i, c0

i) then

CSii

Si`1i

Sii`1

Si`1i`1D"C

0.50.5

!0.5i0.5iD (15)

The diagonal matrix a1 is de"ned as follows:

(i) If Re ('0i)O0, Im ('0

i)O0, ('0

i`1, c0

i`1)"('0

i, c0

i) and ('1

i`1, c1

i`1)O('1

i, c1

i) then

a1ii"a1

i`1i`1"0; (16)

(ii) If Re ('0i)"0 and Im('0

i)O0 and (Re('0

i)O0 or Im('1

i)"0), then a1

ii"0; (17)

(iii) If Re ('0i)O0 and Im('0

i)"0 and (Re('1

i)"0 or Im('1

i)O0), then a1

ii"0; (18)

(iv) If Re ('0i)O0, Im ('0

i)O0, ('0

i`1, c0

i`1)"('0

i, c0

i) and ('1

i`1, c1

i`1)"('1

i, c1

i) then

a0ii/a0

i`1i`1"a1

ii/a1

i`1i`1. (19)

3.1.1. The spectral radius of the iteration matrix. Prior to computing the spectral radius ofiteration matrix, ¸, we recall the following two lemmas:

¸emma 3.1. ¹Q"0. (20)

Proof. ¹Q"(I!QK~10

QTK)Q"Q!QK~10

QTKQ

"Q!QK~10

K0"0. K

¸emma 3.2. If = is K-orthogonal to Q, i.e.=TKQ"0, then ¹="=. (21)

Proof. ¹="(I!QK~10

QTK)="=!QK~10

0"=. K

For the prolongation operator Q de"ned in (12), we prove the following theorems:

Theorem 3.1. If ="'0b0S#'1b1S and a0iic0ib0ii#a1

iic1ib1ii"0 ∀1)i)m, (22)

where b0"diag (b011

b022

. . . b0ii

. . . b0mm

)3Cm]m; b1"diag (b111

b122

. . . b1ii

. . . b1mm

)3Cm]m, then

=TKQ"0 (23)

Proof. From the de"nitions of '0 and '1, we have ('0)TK'1"('1)TK'0"0 and('0)TK'0"c0, ('1)TK'1"c1 where c0"diag (c0

1c02

. . . c0i

. . . c0m)3Cm]m and c1"diag

(c11c12

. . . c1i

. . . c1m)3Cm]m.

Using (6) and (12), we obtain

P"QTK="((a0)TST('0)T#(a1)TST('1)T)K('0b0S#'1b1S)

"(a0)TSTc0b0S#(a1)TSTc1b1S (24)

Since S is a block diagonal matrix with 1]1 or (2]2) blocks and a0, a1, b0 and b1 are diagonalmatrices, P is a block diagonal with the same block structure as S. Therefore, we only need to



consider the following two cases:

Case 1: 1]1 block. From (13), (14) and (24), it follows that

Pii"a0

iiSiic0iSiib0ii#a1

iiSiic1iSiib1ii"(a0

iic0ib0ii#a1

iic1ib1ii)S

iiSii"0 (25)

Case 2: 2]2 block. If a1ii"a1

i`1i`1"0, then from (16) and (24), it follows that

CPii

Pi`1i

Pii`1

Pi`1i`1D"C

a0ii0

0a0i`1i`1D C

0.5!0.5i

0.50.5iDC

c0ib0ii

00

c0i`1

b0i`1i`1D C

0.50.5

!0.5i0.5i D"0

(26)

If a0ii/a0

i`1i`1"a1

ii/a1

i`1i`1, then using (19) and (22), we obtain

a0iic0ib0ii#a1

iic1ib1ii"a0

i`1i`1c0ib0ii#a1

i`1i`1c1ib1ii"0 (27)

a0i`1i`1

c0i`1

b0i`1i`1

#a1i`1i`1

c1i`1

b1i`1i`1

"a0iic0i`1

b0i`1i`1

#a1iic1i`1

b1i`1i`1

"0 (28)

From (19) and (24), it follows that

CPii

Pi`1i

Pii`1

Pi`1i`1D"C

a0ii0

0a0i`1i`1D C

0.5!0.5i

0.50.5iD C

c0ib0ii

00

c0i`1

b0i`1i`1D C

0.50.5

!0.5i0.5iD

#Ca1ii0

0a1i`1i`1D C

0.5!0.5i

0.50.5iD C

c1ib1ii

00

c1i`1

b1i`1i`1D C

0.50.5

!0.5i0.5i D

"0.25Ca0iic0ib0ii#a0

iic0i`1

b0i`1i`1

!i(a0i`1i`1

c0ib0ii!a0

i`1i`1c0i`1

b0i`1i`1

)!i(a0

iic0ib0ii!a0

iic0i`1

b0i`1i`1

)!(a0

i`1i`1c0ib0ii#a0

i`1i`1c0i`1

b0i`1i`1

)D#0.25C

a1iic1ib1ii#a1

iic1i`1

b1i`1i`1

!i(a1i`1i`1

c1ib1ii!a1

i`1i`1c1i`1

b1i`1i`1

)!i (a1

iic1ib1ii!a1

iic1i`1

b1i`1i`1

)!(a1

i`1i`1c1ib1ii#a1

i`1i`1c1i`1

b1i`1i`1

)D"0

Thus, P"QTK="0. K

The relation between ¹'0i, ¹'1

iand '0

i, '1

iis established by the following theorem:

¹heorem 3.2. ¹'0i"a

i'0

i!b

i'1

iand ¹'1

i"(1!a

i)'1

i!e

i'0

i∀1)i)m (29)

where ai"(a1

ii)2c1

i/((a0

ii)2c0

i#(a1

ii)2c1

i), b

i"a0

iia1iic0i/((a0

ii)2c0

i#(a1

ii)2c1

i), and e

i"a0

iia1iic1i/((a0

ii)2c0

i#

(a1ii)2c1

i) .

Proof. Substituting (12) and (22) into (20) and (21) yields

¹'0Sa0#¹'1Sa1"0 and ¹'0b0#¹'1b1"'0b0#'1b1 (30)



From (13) to (19) and (30), we have

¹'0a0#¹'1a1"0 (31)

and it follows that

¹'0"'0b0a1 [b0a1!a0b1]~1#'1b1a1[b0a1!a0b1]~1 (32)

¹'1"'0b0a0 [b1a0!a1b0]~1#'1b1a0[b1a0!a1b0]~1 (33)

Hence, using (32) and (33) together with (22), we obtain (29). K

We are now ready to evaluate the spectral radius of the iteration matrix, ¸.

Theorem 3.3. The eigenpairs of the two-level iteration matrix, ¸, are given as

(i) (i"(1!c0

i)2la

i'0

i!b

i(1!c0

i)l(1!c1

i)l'1

i;

ji"

(1!c0i)2l(a1

ii)2c1

i#(1!c1

i)2l(a0

ii)2c0

i(a0

ii)2c0

i#(a1

ii)2c1

i

∀1)i)m

(ii) (i`m

"!(1!c1i)2l(1!a

i)'0

i!b

i(1!c0

i)l(1!c1

i)l'1

i; j

i`m"0 ∀1)i)m ;

(iii) (i`2m

"'2i; j

i`2m"(1!c2

i)2l ∀1)i)N!2m ;

and the spectral radius of the two-level iteration matrix is given as

o (¸)" max1)i)m

1)j)N!2m

AK(1!c0

i)2l(a1

ii)2c1

i#(1!c1

i)2l(a0

ii)2c0

i(a0

ii)2c0

i#(a1

ii)2c1

iK , D(1!c2

j)2l DB (34)

Proof. (i) From (4), (29) and the expression of R, we get

¸(i"Rl¹Rl(

i"(1!c0

i)3la

iRl¹'0

i!b

i(1!c0

i)l(1!c1

i)2lRl¹'1

i

"

(1!c0i)2l (a1

ii)2c1

i#(1!c1

i)2l(a0

ii)2c0

i(a0

ii)2c0

i#(a1

ii)2c1

i

[(1!c0i)2la

i'0

i!b

i(1!c0

i)l(1!c1

i)l'1

i]

"ji(i

(35)

and

¸(i`m

"Rl¹Rl(i`m

"!(1!c1i)2l (1!c0

i)l(1!a

i)Rl¹'0

i!b

i(1!c0

i)l(1!c1

i)2lRl¹'1

i

"0"ji`m

(i`m

(36)



Plate 1. Variation of ρ2-level

as a function of prolongation quality parameter andoptimal spectral radius for SPD system.

Copyright © 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 49


as a function of prolongation quality parameter andoptimal spectral radius for weakly indefinite system.


as a function of prolongation quality parameter andoptimal spectral radius for highly indefinite system.




since (Ti`2m

KQ"0 and using (21), we obtain

¸(i`2m

"Rl¹Rl(i`2m

"(1!c2i)2l'2

i"j

i`2m(

i`2m(37)

Hence, from (35) to (37), we get the spectral radius of iteration matrix ¸ (34).To this end we de"ne the matrix S2 similarly to S so that '2S23RN](N!2m). We further de"ne

normalization matrix, P"['0S '1S '2S2]3RN]N, and normalize Q with respect to (PPT)~1 as

qTi(P~TP~1)q

i"1 ∀1)i)m (38)

Using (6) and (12), the normalization condition (38) reduces to

(a0ii)2#(a1

ii)2"1 ∀1)i)m (39)

3.1.2. Discussion. We "rst enumerate various factors in#uencing convergence of the two-levelmethod and de"ne corresponding measures to quantify these factors.

1. Quality of the smoother: The spectral radius of the iteration matrix of the single-levelmethod, o1-level , is given as

o1-level"o (R2l)" max1)i)m

( D (1!c0i)2l D ) (40)

o1-level is a measure of the quality of the smoother. It is apparent that a single-level method isconvergent if o1-level(1.

2. Size of the coarse model: To study the convergence characteristics of the two-level method,we "rst focus on the second term in (34), D (1!c2

j)2l D. The convergence criteria, o(¸)(1, requires

thatD(1!c2

j)2l D(1 ∀1)j)l (41)

&2

consists of algebraically &oscillatory' modes, i.e. those modes which are excluded from theprolongation operator, Q, or equivalently, modes which are K-orthogonal to Q. This implies thatthe convergence criterion is not met if the prolongation operator excludes any mode, whichcannot be smoothed out by the smoother. This also provides the necessary condition forconvergence. Therefore, the size of the coarse model should be greater or equal to the number ofmodes that cannot be smoothed out by the smoother.

If the above condition and

D (1!c2i)2l D(1 ∀1)i)m (42)

are met, and a1 vanishes, then the prolongation operator Q is optimal in the sense previouslyde"ned, and the spectral radius of ¸ satis"es.

o015

" max1)i)m, 1)j)l

( D(1!c1i)2l D, D (1!c2

j)2l D )(1 (43)



The convergence is guaranteed and the spectral radius reduces with the increase in the size of thecoarse model, i.e. Lo

015/Lm(0.

3. ¹he quality of prolongation operator: Let ei"(a1

ii)2 be a measure of quality of the prolonga-

tion operator. From (39), it follows that

0)ei)1 (44)

We further denote

o0i"(1!c0

i)2l, o1

i"(1!c1

i)2l, o2

j"(1!c2

j)2l (45)

and

ci"

eic1i

eic1i#(1!e

i)c0

i

(46)

The dependence of the convergence on the deviation from the optimal prolongation operator,'1Sa1, is governed by the "rst term in (34)

ui"Do0

ici#o1

i(1!c

i) D (47)

From (34), (40) and (43), it follows that

o (¸)"o1-level if ei"1 ∀1)i)m (48)

and

o (¸)"o015

if ei"0 ∀1)i)m (49)

In what follows, we seek for the upper bound of o(¸) for a given prolongation quality parameter,e, the optimal prolongation operator, o

015, and the quality of the smoother expressed in terms of

o1-level . The upper bound of o(¸), denoted as o2-level , is then de"ned as

o2-level" max∀c0

i, c1

i, c2

j

(o(¸)) (50)

where D1!c0iD2l)ol-level , D1!c1

iD2l)o

015, D1!c2

jD2l)o

015and e

i"e.

For convergence studies we will consider the following three cases: (1) the SPD system, (2) theweakly inde"nite system, and (3) the highly inde"nite system, which are discussed below.

1. SPD system: c0i3R, c1

i3R, c2

j3R ∀1)i)m, 1)j)l and o

1-level(1 (51)

From (51) follows that

0(c0i(2, 0(c1

i(2 and 0)c

i)1 (52)



Based on (45)}(47) it can be shown that for a given Do0iD, Do1

iD and e, the values of

c0i"1!2lJDo0

iD(1 and c1

i"1#2lJDo1

iD'1 yield the maximal value of u

i. The derivative of

uiwith respect to Do0

iD and Do1

iD satisfy

Lui

LDo0iD"( Do0

iD!Do1

iD )A

ci(1!c

i)

2lc0i(1!c0

i)2l~1B#c

i'0 (53)

Lui

LDo1iD"( Do0

iD!Do1

iD )A

ci(1!c

i)

2lc1i(c1

i!1)2l~1B#1!c

i'0 (54)

Therefore, the maximal value of uifor SPD systems is reached when

u.!9

"max∀i

(ui)"D1!c0 D2lc#D1!c1 D2l(1!c) (55)

where c0"1!2lJo1-level , c1"1#2lJo015

and c"ec1/(ec1#(1!e)c0) .Since u

.!9*o

015*Do2

jD, the upper bound of o(¸) is given as

o2-level"ol-levelc#o015

(1!c ) (56)

For a given o1-level Plate 1 depicts the variation of o2-level versus the optimal spectral radius, o

015,

and the prolongation quality parameter, e.The derivative of o

2-level with respect to e satis"es

Lo2-level

Le"

(1!2lJo1-level) (1#2lJo015

) (o1-level!o015

)

[e (1#2lJo015

)#(1!e) (1!2lJo1-level]2'0 (57)

It can be seen that the derivative of o2-level with respect to e is positive and its extreme values areo015

(minimum) and o1-level (maximum). Convergence of a single-level method is guaranteed, andthe presence of the coarse grid can only improve the performance of the iterative method. Fora given coarse model the spectral radius of the two-level method is bounded by

o015

)o(¸))o1-level (58)

The derivative of o2-level with respect to o015

satis"es

Lo2-level

Lo015

"

2lJo015

(1!2lJo1-level) (o1-level!o015

)e(1!e)

2lo015

[e(1#2lJo15)#(1!e) (1!2lJo1-level)]2

#

(1!e) (1!2lJo1-level)

e(1#2lJo015

)#(1!e) (1!2lJo1-level)'0 (59)

Since the derivative of o2-level with respect to e and o015

are both positive, the performance of thetwo-level method improves with coarse model re"nement (i.e. using a "ner coarse mesh in thegeometric multigrid method or taking more modes from each aggregate in the GAM method



[14]) or by improving the quality of the prolongation operator (i.e. smoothing the prolongationoperator [2, 7] or using larger aggregates in the GAM method).

Next, we will analyse the cases of highly and weakly inde"nite systems. We start by consideringthe weakly inde"nite system characterized by complex eigenpairs of smoothing iteration matrixand by the ability of a single-level method to converge, and then turn our attention to stronglyinde"nite systems where the smoother alone fails to converge. The upper bounds of o(¸), o2-level ,for the two cases are shown in Plates 2 and 3, respectively. For better resolution the values ofo2-level higher than 2.5 are shown to be equal to 2.5.

2. =eakly inde,nite system: c0i3C, c1

i3C, c2

j3C∀1)i)m, 1)j)l and o1-level(1.

From Plate 2 it can be seen that (d/de)o2-level could be either positive or negative depending onthe values of o

015, o1-level and e. The implication is that for a certain choice of prolongation operator

the two-level method may converge slower than a single-level method or even diverge. Thetwo-level method will converge when the upper bound of o2-level is less than one. Theorem 3.4 givesthe upper bound for o2-level .

¹heorem 3.4. If c0i3R, c1

i3R, c2

j3R ∀1)i)m, 1)j)l and o1-level(1, then o2-level(o1-level/

J1!lJo1-level).

Proof. See the appendix.The implication of Theorem 3.4 is that for the two-level method to converge the following

should hold:

D1!c0 D(0.786 for l"1 (60)

Plate 2 also demonstrates the e!ect of coarse model re"nement. As a coarse model is re"ned thevalue of o

015is reduced and the quality of the prolongation operator is improved, or equivalently

the value of e is reduced. However, unlike the case of positive-de"nite systems, o2-level may increaseespecially if the initial coarse model is very coarse and of a poor quality. For example, a smallcoarse grid can only capture geometrically smooth modes, whereas &algebraically' smooth modesare geometrically oscillatory, as has been illustrated in Figure 4. Also, a noteworthy observationis that if a coarse model is su$ciently "ne, subsequent coarse grid re"nement does improve theperformance of the two-level method. Such a critical coarse model size depends on the spectralcharacteristics of the system and the quality of prolongation.

3. Highly inde,nite system: c0i3C, c1

i3C, c2

j3C∀1)i)m, 1)j)l and o1-level'1.

As in the case of weakly inde"nite systems, coarse model re"nement may have a negative e!ecton the convergence of the two-level method unless the initial coarse model is su$ciently "ne. Butunlike weakly inde"nite systems, the tolerance to any deviation from the optimal prolongation isvery small, which is manifested by a narrow zone in Plate 3 where the two-level spectral radius isless than one. It is apparent how critical it is to include all the &algebraically' smooth modes in theprolongation operator with almost no pollution if one wants to construct a robust two-levelmethod for inde"nite systems.

Finally, Theorem 3.5 illustrates that for a certain &poor' choice of a coarse model, the spectralradius of the two-level method is unbounded.

¹heorem 3.5. If c0i3C, c1

i3C, c2

j3C∀1)i)m, 1)j)l and o1-level'1, then o2-level(o1-level

J2/(1#cos h) where h is the angle between c0 and c1.

Proof. See the appendix.



3.2. A more general prolongation operator

In this section, we conduct convergence studies for the more general structure of the prolongationoperator de"ned as

Q"

1+g/0

'gSgag (61)

where ag"diag (ag11

ag22

. . . agii

. . . agmm

)3Rm]m ; and Sg3Cm]m is a block diagonal matrix witheither 1]1 or 2]2 blocks.

The blocks in Sg are de"ned as follows:

(i) if Re ('gi)O0, and Im('g

i)"0, then [Sg

ii]"[1]; (62)

(ii) if Re ('gi)"0 and Im ('g

i)O0, then [Sg

ii]"[i]; (63)

(iii) if Re ('gi)O0 Im ('g

i)O0 and ('g

i`1, cg

i`1)"('g

i, cg

i), then

CSgii

Sgi`1i

Sgii`1

Sgi`1i`1

D"C0.5

0.5

!0.5i

0.5i D (64)

The restrictions on the error term, '1S1 a1, are given asIf Re('g

i)O0, Im ('g

i)O0 and ('g

i`1, cg

i`1)"('g

i, cg

i), then:

(i) if (Re('1~gi

)"0 or Im('1~gi

)"0) and (Re ('1~gi`1

)O0 and Im ('1~gi`1

)O0), thena1ii"a1

i`1i`1"0; (65)

(ii) if (Re ('1~gi`1

)"0 or Im('1~gi`1

)"0) and (Re ('1~gi

)O0 and Im ('1~gi

)O0), thena1ii"a1

i`1i`1"0. (66)

3.2.1. The spectral radius of the iteration matrix. From the de"nition of Q in (12), it is apparentthat '2

jKQ"0. Following Lemma 3.2, we obtain N!2m eigenpairs of the iteration matrix, ¸,

which are given as

('2j, (1!c2

j)2l) ∀1)j)N!2m (67)

We proceed to constructing the remaining 2m eigenpairs of ¸ by considering the following twocases:

Case 1: ('0i`1

, c0i`1

)"('0i, c0

i) or ('1

i`1, c1

i`1)"('1

i, c1

i). (68)

We start by de"nition of auxiliary matrices:

<"[l1

l2

l3

l4]3RN]4 (69)



where

[l1

l2]"['0

i'0

i`1] C

S0ii

S0i`1i

S0ii`1

S0i`1i`1

D and [l3

l4]"['1

i'1

i`1] C

S1ii

S1i`1i

S1ii`1

S1i`1i`1

D (70)

q"[qiqi`1

]"<a3RN]2 and w"<b3RN]2 (71)

We also de"ne vector w"<b3RN]2 such that

wTKQ"0 (72)

where

a"

a0ii

0

0 a0i`1i`1

a1ii

0

0 a1i`1i`1

3R4]2 and b"

b1

b5

b2

b6

b3

b7

b4

b8

3R4]2 (73)

The matrix b should satisfy

bTCa"0 (74)

where

C"<TK<3R4]4 (75)

From Lemmas 3.1 and 3.2, we have ¹[q w]"[0 w] and from (71), follows that ¹<[a b]"<[0 b] which is equivalent to

¹<"<D (76)

where

D"[0 b] [a b]~13R4]4 (77)

From the de"nition of <, we get

Rl<"<E (78)

where E3R4]4 depends on l and the corresponding eigenvalues of R. Let (F, ") denote theeigenpairs of 4]4 matrix EDE satisfying

EDEF"F" (79)



From (76), (78) and (79), it follows that

¸<F"Rl¹Rl<F"<EDEF"<F" (80)

From (80) it can be seen that (<F, ") contains four eigenpairs of the iteration matrix, ¸. In thefollowing, the expressions of<, C, E and b are derived for three subcases corresponding to Case 1.

Case 1a: ('0i`1

, c0i`1

)"('0i, c0

i) and ('1

i`1, c1

i`1)"('1

i, c1

i). (81)

This case corresponds to the following choice of <, C and E

<"[Re('0i) Im ('0

i) Re ('1

i) Im('1

i)] (82)

C"

0.5Re (c0i) 0.5 Im(c0

i) 0 0

0.5 Im(c0i) !0.5Re(c0

i) 0 0

0 0 0.5Re (c1i) 0.5 Im(c1

i)

0 0 0.5 Im(c1i) !0.5Re (c1

i)

(83)

and

E"

Re((1!c0i)l) Im ((1!c0

i)l) 0 0

!Im((1!c0i)l) Re ((1!c0

i)l) 0 0

0 0 Re ((1!c1i)l) Im((1!c1

i)l )

0 0 !Im((1!c1i)l) Re ((1!c1

i)l)

(84)

Since b has 8 independent variables and Equation (74) provides 4 constrains only, we exercisesome freedom in selecting the matrix b. The appropriate forms of b, which satis"es (74) and (77),are given below:

if a0iiO0, a0

i`1i`1O0, then

b"

b1

b5

b2

b6

1 0

0 1

if a0iiO0, a0

i`1i`1"0, then

b"

b1

0

b2

0

1 b7

0 b8

(85)



if a0ii"0, a0

i`1i`1O0, then

b"

1 b5

0 b6

b3

0

b4

1

if a0ii"0, a0

i`1i`1"0, then

b"

1 0

0 1

b3

b7

b4

b8

(86)

Case 1b: ('0i`1

, c0i`1

)"('0i, c0

i) and ('1

i`1, c1

i`1)O('1

i, c1

i). (87)

<"[Re('0i) Im ('0

i) S1

ii'1

iS1i`1i`1

'1i`1

] (88)

C"

0.5Re (c0i) 0.5 Im(c0

i) 0 0

0.5 Im(c0i) !0.5Re(c0

i) 0 0

0 0 (S1ii)2 (c1

i) 0

0 0 0 (S1i`1i`1

)2 (c1i`1

)

(89)

and

E"

Re((1!c0i)l) Im ((1!c0

i)l ) 0 0

!Im((1!c0i)l) Re ((1!c0

i)l) 0 0

0 0 (1!c1i)l (S1

ii)l 0

0 0 0 (1!c1i`1

)l (S1i`1i`1

)l

(90)

For this case, we use the same forms of b as in (85) and (86).

Case 1c: ('0i`1

, c0i`1

)O('0i, c0

i) and ('1

i`1, c1

i`1)"('1

i, c1

i). (91)



This case is similar to Case 1b. We use the same structure of b as in the previous two cases. Theexpressions of <, C and E are given as

<"[S0ii'0

iS0i`1i`1

'0i`1

Re('1i) Im('1

i)] (92)

C"

(S0ii)2 (c0

i) 0 0 0

0 (S0i`1i`1

)2 (c0i`1

) 0 0

0 0 0.5Re(c1i) 0.5 Im(c1

i)

0 0 0.5 Im(c1i) !0.5Re (c1

i)

(93)

and

E"

(1!c0i)l(S0

ii)l 0 0 0

0 (1!c0i`1

)l(S0i`1i`1

)l 0 0

0 0 Re((1!c1i)l) Im((1!c1

i)l)

0 0 !Im((1!c1i)l) Re((1!c1

i)l )

(94)

For a given choice of b, we can express the matrix EDE and its eigenvalues (which are also theeigenvalues of the matrix ¸) in terms of a0

ii, a0

i`1i`1, a1

iiand a1

i`1i`1. We denote the magnitude of

the maximal eigenvalue of EDE by )i.

Case 2: (Re('0i)"0 or Im('0

i)"0) and (Re ('1

i)"0 or Im('1

i)"0). (95)

For this case, the approach used to construct the eigenpairs of the matrix ¸ is similar to that inSection 3.1. We only provide the "nal results for the two eigenvalues of ¸

j1"

(1!c0i)2l(S0

ii)2l(a1

ii)2(S1

ii)2c1

i#(1!c1

i)2l(S1

ii)2l(a0

ii)2(S0

ii)2c0

i(a0

ii)2(S0

ii)2c0

i#(a1

ii)2 (S1

ii)2c1

i

, j2"0 (96)

The spectral radius of the iteration matrix ¸ is given as

o (¸)"max∀i, j

()i, D (1!c2

j)2l D ) (97)

where)

i"max ( Dj

1D , Dj

2D) (98)

As in Section 3.1, we de"ne S2 similarly to S0 and S1 so that '2S23RN](N!2m) . We further denoteP"['0S0 '1S1 '2S2]3RN]N and normalize Q with respect to P~TP~1. From the de"nition ofP and (61), we recover (a0

ii)2#(a1

ii)2"1 ∀1)i)m. The prolongation quality parameter,

0)ei)1, is de"ned as

ei"S

(a1ii)2#(a1

i`1i`1)2

2for Case 1 (99)

ei"(a1

ii)2 for case 2 (100)



3.2.2. Discussion. As in Section 3.1.2, we seek for the upper bound of o(¸), denoted as o2-level, fora given prolongation quality parameter, e, optimal prolongation operator, o

015, and the quality of

the smoother expressed in terms of o1-level. We consider the following cases:

1. SPD system: c0i3R, c1

i3R, c2

j3R ∀1)i)m, 1)j)l and o1-level(1. The model prolon-

gation operator is the same as in Section 3.1. The upper bound of o(¸) is given by (56).2. =eakly inde,nite system: c0

i3C, c1

i3C, c2

j3C ∀1)i)m, 1)j)l and o1-level(1. The

upper bound of o(¸), o2-level , is shown in Plate 4. The distribution of o1-level is plotted as a functionof the optimal spectral radius, o

015, and the prolongation quality parameter, e. To determine the

value of o2-level at each point (o015

, e) in Plate 4, we loop over all possible combinations of c0i, c1

iand

c2j

(angles and magnitudes of complex numbers) as described in Section 3.2.1, subjected toconstrains e

i"e, D1!c0

iD)o1-level , D1!c1

iD2l)o

015and D1!c2

iD2l)o

015. Note that the prolonga-

tion operator, given by (13), is a special case of the prolongation operator discussed in this section.Therefore, the value of o2-level for each point in the Plate 4 is larger than that for the correspondingpoint in Plate 2.

3. Highly inde,nite system: c0i3C, c1

i3C, c2

j3C ∀1)i)m, 1)j)l and o1-level'1. The

comparison of Plate 5 with Plate 3 is consistent with the observations made for weakly inde"nitesystems.

4. SUMMARY AND CONCLUSIONS

The paper introduced the concept of optimal prolongation operator, which spans the spectrum ofhighest eigenmodes of the smoothing iteration matrix R, termed here as algebraically &smooth'modes. We demonstrated that any deviation from the optimality has a detrimental e!ect onconvergence and in some cases makes the performance of the two-level method signi"cantlyworse than of a corresponding single-level method, especially for highly inde"nite systems.

We showed that for an iterative method to be robust it should possess a mechanism capable ofeliminating algebraically &smooth' components of error. The fact that the optimal prolongation isnon-smooth in geometric sense and su!ers from a pathological sensitivity to pollution fromoptimality suggests that it is unlikely that any single-level and local-basis multilevel methodwould be able to resolve these algebraically &smooth' components of error without pollution. Inessence, the proposed methodology serves as a bu!er zone against divergence in the form ofadditional iterative level aimed at eliminating components of error which have not been removedby the smoothing iteration matrix R. As such, the proposed method can be used to enhance theperformance of existing single- or multi-level iterative methods in particular for highly inde"nitesystems.

This notion is further investigated by numerical experimentation on the Helmholtz equationon bounded domains and shear banding problems with strain softening in the second part of thismanuscript [2].

APPENDIX

In the appendix, we provide proofs for Theorems 3.4 and 3.5. From (47), we get

ui"Do0

ici#o1

i(1!c

i) D(o1-level ( Dci D#D1!c

iD) (A1)



Figure 6. Illustration of immaginary and real part of eigenvectors.

Figure 6 is used to visualize the following quantities:

NOE"c1

i,NOF"c0

i(A2)

LEOF"h, LOAB"t (A3)

DNOA D

DNOE D"e and

NAB E

NOF (A4)

From the above de"nitions it follows that

h#t"n, DNOA D"ec1

i, D

NAB D"(1!e)c0

iand D

NOB D"ec1

i#(1!e)c0

i(A5)

Denote x"DNOA D/ D

NOB D and y"D

NAB D/ D

NOB D then Dc

iD#D1!c

iD"x#y and x2#y2#2xy

cos h!1"0.The above constraint maximinization problem is solved by "nding stationary values of the

Lagrangian

%"x#y#j (x2#y2#2xy cos h!1) (A6)

A direct solution of (A6) yields

(DciD#D1!c

iD).!9

"S2

1#cos hat x"y"S

1

2(1#cos h)(A7)



From (A1) and (A7), we obtain

ui(o1-levelS

2

1#cos h(A8)

which completes the proof of Theorem 3.5.

Furthermore, if D1!c1iD(D1!c0

iD(1, then cos h'1!2 D1!c0

iD2'1!2Jo1-level , which is

the result of Theorem 3.4.

ACKNOWLEDGEMENTS

This work was supported by the O$ce of Naval Research through grant number N00014-97-1-0687.

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global-basis two-level method for indefinite systems. part 1: convergence studies

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