Multigrid methods for 3-D definite and indefinite problems

ELSEVIER Applied Numerical Mathematics 26 (1998) 377-398 MATHEMATICS

Multigrid methods for 3-D definite and indefinite problems

Yair Shapira

Computer Science Department, Technion, Haifa 32000, Israel

Abstract

A multigrid method for the solution of certain finite difference and finite volume schemes for elliptic PDEs is introduced. A parallelizable version of it, suitable for two-level and multi-level analysis, is also defined and serves as a theoretical tool for deriving suitable implementations for the main version. For indefinite equations, this analysis provides a prediction of a suitable mesh size for the coarsest grid used, Numerical experiments show the efficiency of the method for 3-D diffusion problems with discontinuous coefficients and highly indefinite Helmholtz equations. © 1998 Elsevier Science B.V.

I. Introduction

The multigrid method is a powerful tool for the solution of linear systems arising from the discretization of elliptic PDEs [4,5], The basic multigrid method works well for the Poisson equation in a square or a cube, but difficulties arise with nonsymmetric and indefinite problems and problems with variable coefficients, complicated domains or non-uniform grids. In the following we give a brief explanation for this phenomenon.

In order to supply proper coarse grid correction terms for the fine grid approximate solution, the equation should be well approximated on the coarse grids. The most difficult to approximate are error components consisting of eigenfunctions with nearly zero eigenvalues with respect to the coefficient matrix. For such grid functions, denoted hereafter by "nearly singular functions", the coarse grid operator might have considerably different eigenvalues from those of the fine grid operator. When the coarse grid nearly zero eigenvalues differ from those of the fine grid by a scalar multiplication, as is the situation for the Poisson equation (with the scalar 4 when the problems are taken in the undivided form), this can be fixed easily. For other classes of problems involving two or more types of nearly singular functions, however, each type requires a special treatment. For non-elliptic problems, for example, the error components which are smooth in the characteristic direction are scaled by factor 2 when approximated on the coarse grid, whereas those which are smooth also in the orthogonal directions are

1 Present address: Los Alamos National Laboratory, MS B-256, Los Alamos, NM 87545, USA. E-mail: yairs@c3serve. c3.1anl.gov.

0168-9274/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0168-9274(97)00070-6

378 Y. Shapira /Applied Numerical Mathematics 26 (1998) 377-398

scaled by factor 4 as before. (This is handled in [7] by using residual over-weighting and multigrid W-cycle rather than the usual V-cycle.) For diffusion problems with discontinuous coefficients, the nearly singular functions are those with smooth flux. Multigrid algorithms for handling these functions properly are presented in [1,8-10,21]. For indefinite equations, it is common to distinguish between two classes of problems: (a) slightly indefinite problems, for which almost all of the spectrum of the coefficient matrix is positive and very few negative eigenvalues (say two or three) exist, and (b) highly indefinite problems, for which many more negative eigenvalues exist. For class (a), the method of [6], which is based on "filtering out" nearly singular modes, yields convergence rates close to those for the Poisson equation. The Cyclic Reduction Multigrid (CR-MG) method of [2] is also superior to standard multigrid. For class (b), a projection method (suitable for finite element schemes) is presented in [3]. The AutoMUG method of [20,21] and the method of [17] also yield satisfactory convergence rates when supplemented with an outer acceleration scheme. It should be kept in mind, though, that, unlike the methods of [8-10,17], which apply to problems with (3d)-coefficient stencils (for d-dimensional PDEs), AutoMUG and CR-MG use (2d + 1)-coefficient stencils at all levels.

The aim of this work is to provide a proper implementation for AutoMUG for highly indefinite (3-dimensional) problems. To this end, we introduce a parallelizable version of AutoMUG, named Par- allelizable AutoMUG (PAMUG). This method may be considered a generalization of the Parallelizab!e Superconvergent Multigrid (PSMG) of [11] to nonsymmetric and indefinite problems. PAMUG uses the fine grid at all levels, hence is suitable for parallel architectures with large number of processors. However, we do not use it as a solver but merely as a theoretical tool for deriving a proper implementation for AutoMUG. Due to its simple algebraic formulation, PAMUG is suitable for two-level analysis for a certain class of problems. For another class of model problems, including indefinite Helmholtz equations, the whole spectrum of the multi-level iteration matrix is available. This enables one to choose in advance a suitable mesh-size for the coarsest grid used and a suitable acceleration scheme (if necessary). Due to the similarity of AutoMUG and PAMUG, this implementation is suitable also for AutoMUG, as is illustrated numerically. It is also shown numerically that AutoMUG is applicable for problems with discontinuous coefficients.

The contents of the paper are as follows. In Section 2 AutoMUG and PAMUG are defined. In Section 3 variants of PAMUG and AutoMUG are analyzed. In Section 4 the multi-level analysis technique is used for computing the spectrum of the PAMUG iteration matrix for several model problems. In Section 5 numerical experiments, using AutoMUG with the suitable implementation, are reported. In Section 6 concluding remarks are made.

2. The multigrid methods

2.1. The abstract multi-level method

We start with an abstract definition of a multi-level (ML) method for the solution of the linear system of equations

A x = b.

The notation of the present definition is useful in the sequel. Let "+-" stand for substitution. Let : z ---+ Sz be a smoothing procedure for the linear system with the initial guess x and s, r, t and o

Y. Shapira /Applied Numerical Mathematics 26 (1998) 377-398 379

nonnegative integers denoting, respectively, the cycle index, the number of presmoothings, the number of postsmoothings and the minimal order of A for which ML is called recursively. (In (1) we use the term "order" loosely, considering A to be of order o if, for some ordering of the variables, A is block- diagonal with block submatrices of order at most o.) The operators R (restriction), P (prolongation) and Q (coarse grid coefficient matrix) will be defined later.

ML(xin, A, b, Xout):

if A is of order < o

Xout ~-- A - 1 b

otherwise:

Xin ~ Sxin (repeat r times),

e *-- 0, (1)

ML(e, Q, R(Axin - b), eout) / repeat c times,

e +-- eou t J Xout +-- Xin - - P e ~

Xout +- Sxout (repeat t times).

An iterative application of ML is given by

xo = 0, k = 0,

while IIAxk -- bil2 >1 threshold. IlAxo - bll2

ML(xk, A, b, xk+l), (2)

k+---k+l

endwhile.

Below we define the operators R, P and Q used in (1) for AutoMUG, its variant AutoMUG(q) and the parallelizable versions PAMUG and PAMUG(q).

2.2. Some matrix functions

For any square matrix M, let spect(M) denote its spectrum and p(M) denote its spectral radius. Let K be a positive integer. Let I be the identity matrix of order K. For any matrix M, M =

(?Tti,j)l<~i,j<~K , define the matrix functions

0 j = l l<~i~K

D(M) = diag(M),

R(M) = 2 I - M ( D ( M ) ) - ' ,

Q(M) = R(M)M,

P ( M ) = 2 I - ( D ( M ) ) - I M ,

S(M) = r o w s u m ( P ( f ) ) .

380 Y. Shapira / Applied Numerical Mathematics 26 (1998) 377-398

These definitions apply to the AutoMUG and PAMUG methods defined below. For the variants AutoMUG(q) and PAMUG(q), replace the above definition of S(M) by S ( M ) =_ (2 + q)I. (The role of the parameter q will be discussed later.) Let VK be the space of the K x K x K-grid functions. (It is assumed hereafter that the first point in a 3-dimensional grid is numbered (1, 1, 1).) Define the injection operator O: Vh- ~ VLK/2 j by ( O v ) i , j , k ~- V2i,Zj,Zk. Define the permutation U by

( U Y ) i , j , k : Y j , k , i , V E V K .

For any matrix B, we say that t3 is a K-block matrix if B is block-diagonal with tridiagonal blocks of order K, that is,

/3 = blockdiag (B (j)) l<.j<~t(2,

with

B (j) = tridiag (b} j), c} j) , dl j)) 1 <<. j <~ K 2. l<~i<~K ~

By the notation "tridiag" we mean a periodically extended tridiagonal matrix, that is, bl j) = B (j) 1,K and d~ ) r~(J) We assume that either bl j) = d~ ) = O, 1 <~ j <~ K 2, or K 2 k for some --- L,K,1. ---~ positive integer k. This guarantees that A and the coarse grid coefficient matrices defined below are of property-A. As a matter of fact, the block submatrices B (j) need not be of the same order, which enables handling problems on non-rectangular domains. Here, however, we assume for simplicity that these blocks are of the same order.

2.3. Transfer and coarse grid operators

Here we define the operators R, P and Q used in (1) for linear systems which arise, for example, from finite difference discretization of elliptic PDEs in 3-D.

Let N be a positive integer. Let n ~< [log2 N] be the integer denoting the number of levels minus one. Assume that the coefficient matrix A is of the form

A = X + Y + Z, (3)

where X, U Y U y and U2ZU 2v are N-block matrices. For example, if

X = u Y U T = U 2 Z U 2 T = ( 2 - 1 3 h 2 ) -I ( ( /3h2 1 ) ) (4) 3 blockdiag tridiag - 1,2 ~ - , ,

where/3 is a parameter and h is the cell size, then A represents a seven-point second order scheme for the Helmholtz equation

- - U x x -- Uyy -- Uzz - - /3U = f (5)

in a cube. (The default in this paper is the unit cube.) We now define the operators R, P and Q used in (1). Define Ao = A, Xo = X, Yo = Y and

Zo = Z. For i = 1 ,2 , . . . ,n, define the matrices Ri, Pi and Ai by

X i = S(Y/-1 ) S ( Z i - 1 ) (~ (X i -1 ), =

Z i = S ( X i _ I ) S ( Y i _ I ) Q ( Z i _ I ) ,

Y Shapira /Applied Numerical Mathematics 26 (1998) 377-398 381

= OR( 1 ) ) R( ),

Pi = P(X i - I )P(Y i - I )P(Z i -1 ) OT,

Ai = O ( X i -+- Yi 4- Zi) OT.

These definitions apply to AutoMUG and AutoMUG(q). For the parallelizable versions PAMUG and PAMUG(q), they are modified as follows: omit the operators O and O T in the above definitions and replace the definition of Pi by Pi ----- I. (Note that, with these definitions, Ri and Pi are square matrices, whereas for the AutoMUG versions they are rectangular.) The parameter q in AutoMUG(q) and PAMUG(q) is chosen by the user such that S(Xi-1), S(Yi-1) and S(Zi-1) are approximated by (2 + q)I. For example, if t3 in (5) varies with the spatial location, then a reasonable q is the average of 2/(2 - ~3h2/3) - 1 in the domain. PAMUG(q) and AutoMUG(q) are suitable for two-level analysis. For simplicity, q = 0 is used in most of this analysis.

The ML procedure, that is, ML(xin, A, b, Xout) defined in (1), is called n + 1 times per iteration. In the (n + 1)st time (with a suitable choice of o in (1)), it is an exact solver. The ith call to the ML procedure, 1 ~< i ~< n, uses the operators

Q +--- Ai, R ~--- Ri and P +- Pi.

The purpose of defining all these versions is learning about the convergence properties of AutoMUG (which is actually used in our applications) from the analysis of the other versions. Indeed, it is possible to draw connections among the various methods. Note that, for the PAMUG versions, A1 includes eight independent subsystems, each of which corresponds to odd (even) numbered variables in the x, y and z spatial directions. The coarse grid equation in a PAMUG version which uses even numbered variables in every spatial direction is identical to that of the corresponding AutoMUG version. Therefore, AutoMUG and PAMUG have similar effects on low-frequency error components and, hence, it is expected that they have similar convergence rates, as is indeed verified in Corollary 1 below. For certain model problems, e.g., convection~iiffusion equations with periodic boundary conditions, AutoMUG and PAMUG are equivalent to AutoMUG(0) and PAMUG(0), respectively (since all the row-sums used in AutoMUG and PAMUG are equal to the constant number 2). (As a matter of fact, AutoMUG is equivalent to AutoMUG(0) for other types of boundary conditions as well, provided that N is a power of 2 minus 1.) Furthermore, such equivalences hold also for the example in (4), provided that an appropriate q ¢ 0 is used. Hence, one can learn about the behavior of the actually used method AutoMUG from those of the theoretical methods PAMUG, PAMUG(q) and AutoMUG(q).

3. Analysis of the multigrid methods

3.1. Two-level analysis for PAMUG(O)

In this subsection, we derive an upper bound for the asymptotic convergence factor of PAMUG(0) for a class of problems. This class is characterized by the coefficient matrix having a positive spectrum and the problem being separable and isotropic. (For anisotropic problems, a similar analysis method can be employed for analyzing line relaxation in PAMUG or semi-coarsening PAMUG-Iike algorithms.) In particular, the present theory applies to Symmetric Positive Definite (SPD) Helmholtz equations (e.g.,/3h2/3 < 4 sinZ(rch/2) in (4)). The upper bound for the convergence rate presented in Theorem 1

382 E Shapira /Applied Numerical Matheraaffcs 26 (1998) 377-398

is independent of the size of the problem and the clustering of the eigenvalues near zero. This may be interpretecl to inclicate that AutoMUG can hanclle nonnegative eigenvalues of highly indefinite plobiems, Dovided that the negative ones ale handled efficiently by some acceleration technique (see Secuon 4.2).

Since PAMI3G is clesigued for palailel implementations, we assttme that the perfectly palallelizable damped Jacohi smoother is used. (For some architectures, two damped Jacobi relaxations are less expensive than one ted-black Gauss-Seidel sweep.) This simplifies the analysis considerably.

The olcler in which smoothing al~d comse grid conecting ale performed is immaterial, due to the commutalivlty of the smoothing and coalse-gfid correcting operators. For consistency, however, we consider damped Jacobi itelations for ptesmoothing and other methods (such as Jacobi) for postsmoothing.

l heo , e,Jl i . Assume that , X , Y and Z commme with each other; ~, D ( X ) -- D ( Y ) -- D ( Z ) -- I (isotropy assumption); ~, the spectra of X , Y and Z lie in the interval [0, 2] (e.g., X , Y and Z are symmetric M-matrices

oY symmetric and diagonally dominant matrices); u at least one of the matrices X , Y or Z has its spectrum in the open interval (0, 2) (e.g., it is an

M-matrix oc block-diagonal with irreducibly diagonally dominant blocks, see [25]). 7hen the as_vmpiotic convergence factor of a two-level implementation of PAMUG(O) with r damped 3acobi pcesmoothings (with damping faclor ¼) and no postsmoothing is bounded from above by

max 2 -'~, 1 2 + r 2 + ~ J " (6)

For the pmoL see Appendix A. Note that the aasamptions in Theorem 1 do not necessalily imply that A is symmetric or even

normal. For example, assume that A arises from the second-ruder central diffelencing of the PDE

- A u + ~. Vu = f

in me unit cube with Dilichlet boundm'y conditions, where • = (el, c2, c3) is the convection coefficient vector. Assmne that max i~<i~<3 I cil < 2 /h (where h = 1 / (N + 1) is the cell-size) so that A is irreducibly diagonally domi,ant. A is then similar ro the symmetric matrix E - l AE, where E is a diagonal matrix of the folm E = E(I)E(z)E(3), with

( ((, ) Ui - 'E ( i ) (UT) i - ' = blockdiag diag 1 +cih/2 J ]I<~j<~N

(The permutation matrix U is defined in Section 2.2.) Consequently, A has a real spectrum and 3heo,em 1 applies. For further de~ails see [17, Table 2].

3.2. Two-level analysis for Au~oI14UG(O)

Hc;e we use q heoiem 1 for deriving an upper bomld for the asymptotic convergence factor of AutoMUG(0). (A similar approach may he used for analyzing AutoMUG(q) with q ¢ 0.)


Lemma 1. Assume that ® A is diagonalizable; ® X, Y and Z commute with each other; • D ( X ) = c[, D ( Y ) = 7 I and D ( Z ) = r l I for some constants c, ~/and r I.

Assume that two levels and damped Jacobi presmoothing are used and that PAMUG(q) is implemented with no postsmoothing and AutoMUG(q) is implemented with a single postsmoothing of the form x +-- POx. Then the asymptotic convergence factor of AutoMUG(q) is bounded from above by

max ((1 +11 - x/el2) (1 +11 - y/~'12) (1 + 1 1 - z/rile)) '/2 (x,y,z) Espect(X) x spect (Y) x spect(Z)

times that of PAMUG(q).

For the proof, see Appendix B. We combine Theorem 1 and Lemma 1 to have:

Corollary 1. Assume that AutoMUG(O) is implemented with two levels and r damped Jacobi pre~- moothings as in Theorem 1 and a single postsmoothing of the form :c ,-- POx . Assume that A is diagonalizable and that the assumptions in Theorem 1 haM. Then the asymptotic convergence factor of AutoMUG(O) is bounded from above by

~/8max 2 -~, 1 2 + r 2 + r "

3.3. Multi-level analysis for PAMUG

Theorem 1 yields an upper bound for the asymptotic convergence factor for the two-level implementation of PAMUG(0) for a class of problems of positive spectrum. As discussed in the begim~Mg of Section 3.1, this may be interpreted to indicate that indefinite problems may also be solved, provided that the negative eigenva!ues are handled efficiently by an acceleration method. In tbis subsection. we give a quantitative support for this heuristic argument. In particular, we introduce a method for computing the spectrum of the iteration matrix of tbe mlflti-level implementation of P/~MIJG in model cases. This method will be especially helpful in Section 4.2, where highly ipdefinite oroblems are treated.

In the following, the number of damped Jacobi presmootbJngs is denoted by s. Tbe re~son for this is that r is already used for denoting a parameter in the function relgreseoting tbe eigenv3hm, s of the iteration matrix.

Theorem 2. Assume that the block~ in X , U Y U T and U2 Z U 2T are circulant Toeplitz matrices, that

is,

X = blockdiag(tridiag(bo, co, do)),

U Y U T = blockdiag(tridiag(/3o, "7o, 8o)),

U2ZU 2T = blockdJag(tridiag(Co, rio, 0o)),


f o r some constants bo, co, do,

bo + co + do Po = 2

co

For i = O, 1 , . . . , n - 2, define

/30, 7o, 60, fro, r/o and 00. Let

/3o +~o + 6o qo = 2 , ro = 2

% fro + r/o + 00

bi+ l = -qir ib2 / ci,

di+l = -q i r id2 /ci ,

/3i+1 -- -piri/32i/'/i, =

~'/+1 = -Piqi~2/rli ,

0i+1 = -piqiO2 /rli,

Ci+l = qiri(ci - 2bidi/ci) ,

pi+l = 2 - (bi+l + ci+l + di+l) /c i+l ,

qi+l = 2 - (/3i+1 + 7i+1 + ~i+I) /7i+1,

r l~+~ = p ~ q ~ ( r l ~ - 2~0i/~),

r i+l = 2 - ((~+1 + ~/i+1 + 0i+1)/~i+1.

Define

(2 - x / c ) ( 2 - y / ' y ) (2 - z / r l ) ( x + y + z) g(c, 7, ~I;P, q, r; x, y, z) = 1 - q rx (2 - x / c ) + p r y ( 2 - Y / 7 ) + pqz (2 - z/~?)'

c + , , / + r l g ( c ' % r l ; p ' q ' r ; x , Y , Z ) 1 a ( c + ~ + r / ) '

fs(n-1), ,t Vx, Y, z) = f s , t ( C n - l , ~ / n - l , r l n - l ; P n - l , q n - l , r n - 1 ; x , Y , Z ) "

For i = n - 2, n -- 3 , . . . , O, define

,t ~ y, z) = fs,t(ei, "/i, r/i;pi, qi, r i ; x, y, z)

+ ~(i+1)~/ . , fs,t ~qirix(z -- x / c i ) , p~r i y (2 - Y /T i ) ,P iq i z (2 - z /r l i ))

Then there exists an orthogonal matrix T such that the iteration matrix o f PAMUG (implemented with n + 1 levels, cycle index z, s damped Jacobi presmoothings with damping factor a -1 and t Jacobi postsmoothings) is given by

T* diag( f~?) (x, y, z ) ) T (x,y,z)Gspect(X)×spect(Y)×spect(Z) "

For the proof, see Appendix C.

4. Multi- level analysis-- -applicat ions

4. l. Computation o f the spectral radius o f the PAMUG iteration matrix

Theorem 2 enables the computat ion of the spectral radius of the iteration matrix of P A M U G by scanning spec t (X) × spect(Y) × spec t (Z) and comput ing the m a x i m u m of If(°)l there. Here we do this for several PDEs in the unit cube with periodic boundary conditions discretized by the usual second order 7-coefficient central difference scheme.


First we consider the Poisson equation with periodic boundary conditions, for which

spect(X) = spect(Y) = spect(Z) = {1 -cos(ZTrj/K)}o~<j<~.

Since our aim is to simulate Dirichlet problems, we exclude the zero eigenvalue of the coefficient matrix. This is equivalent to assuming that the right hand side does not include a constant Fourier component and the equation is projected onto the orthogonal complement of the constant vector. Poles of the functions f(i) of Theorem 2 are excluded if the arguments of such a function x, y and z and the numbers 2 - x/ci , 2 - Y/'Yi and 2 - z/~i are positive or zero since the proof of Theorem 1 implies that these poles are then of the first kind (removable). K = 64, c = 1 and c~ = 4 are used. We

have found that [f(°0) l ~< 0.4988 for both the 2-level (n = 1) and 4-level (n = 3) implementations of PAMUG. This implies that the upper bound of Theorem 1 is tight and that the rate of convergence is practically independent of the number of levels.

An approximation of the spectrum of the iteration matrix is also available when the ILU(0) smoother of [24] is used. This smoother is based on the incomplete decomposition of the coefficient matrices

Ai as

Ai - (LiHi 1 + I ) H i ( I + Hi 1Ui),

where Li and Ui are the strictly triangular lower and upper parts (respectively) of Ai and Hi is the diagonal matrix consisting of the pivots in the incomplete decomposition. Define

ci + "Yi + rh + V/(Ci + "/i + rli) 2 - 4(bidi +/3i5i + ~iOi) # i = 2

Following [22], one may approximate Hi by #iI, provided that the above square root is real and ci + "Yi + ~/i > 0. (This approach is used in [15] for estimating the smoothing factor.)

Similarly, an approximation of the spectrum of the PAMUG iteration matrix is also available when the Parallelizable Truncated ILU (PTILU) smoother of [22] is used. This method is an approximation of standard ILU in which the triangular systems of ILU are solved approximately by two consecutive Jacobi iterations (with zero initial guesses). (As a matter of fact, in [22] a block-Jacobi iteration is used, with blocks corresponding to subdomains. Here, however, we take the extreme case in which each subdomain contains one point only.)

Note that -- b,e-2rcC-212~i/K + ~ie-2rrx/-12ia~2/K _L T.,::.-27rx/Z12iaJ3/K

and

Ti,col,a~2,w3 ~ die2~rx/~12~aJl/K + ~ie27rx/-z12iaJ2/K ~_ Oie27rxfz~ 2iw3/K

approximate the eigenvalues of the Fourier vector

{ ezTr C~-I (~l k+aJ2J+a331)/K }0~<k,j,/<K

with respect to Li and Ui, respectively. In order to obtain an approximation for the eigenvalue of this vector with respect to the PAMUG iteration matrix with p ILU(0) or PTILU(0) smoothings, one has to multiply the right-hand side in the definition of the funct ion f(i) in Theorem 2 respectively by

( _ x + y + z --1 )P 1 (/%i,Wl,CO2,CO3/oLi -1 + 1)#i(1 + #i 7"i,,~,,~2,~3)

386 Y. Shapira / A p p l i e d Numer ica l Mathemat ics 26 (1998) 3 7 7 - 3 9 8

Table 1 Spectral radius of the PAMUG iteration matrix with various smoothers for 3-D problems with periodic boundary conditions

Equation Levels Damped Jacobi ILU(0) (approx.) PTILU(0) (approx.)

(a) 2 0.092 0.019 0.094

(a) 4 0.128 0.027 0.094

(b) 2 0.960 0.673 0.839

(b) 4 0.960 0.673 0.839

(c) 2 0.992 0.876 0.944

(c) 4 0.992 0.876 0.944

(d) 2 0.204 0.125 0.195

(d) 4 0.640 0.240 0.500

o r

( -, )p. 1 - ( x + y + - -

Pi

In Table 1 we display the spectral radii of the PAMUG iteration matrices for the equations

- u ~ - V,y~ - u ~ = f , ( a )

- u x x - 10 " u y v - u ~ z = f , (b)

- U x x - 10-4uyv - U z z = f , (c)

- u x z - v,y.~ - U z z + 100ux + 50u v + 25Uz = f (d)

in the unit cube with periodic boundary conditions. K = 64, z = 1 and four smoothings per level are used. The Jacobi smoothing (third column in Table 1) is implemented as follows: two relaxations with damping factor ½, one with damping factor 3 and one with damping factor 3 are used at each level of the V-cycle. These damping factors are chosen in order to cancel possible poles of the function 9 of Theorem 2 (and the proof of Theorem 1) and guarantee that the functions f ( i ) there are bounded. This choice seems optimal: using, e.g., damping factor 1 for all the four relaxations yields somewhat poorer results. It should be kept in mind. though, that in practice (Section 5) we use AutoMUG with the more efficient red-black Gauss-Seidel relaxation.

The results in the first row of Table 1 in conjunction with Lemma 1 imply that the asymptotic convergence factor of AutoMUG (which is equivalent to AutoMUG(0) for the present periodic examples) implemented with two levels and the above damped Jacobi presmoothing is bounded by x/ '8 .0 .092. The results for examples (b)-(c) indicate that the isotropy assumption in Theorem 1 is essential. Nev- ertheless, the method of proof of Theorem 1 and Lemma 1 may be used for algorithms which are more suitable for anisotropic problems such as multigrid algorithms involving l ine- re l~a t ion and/or semi-coarsening. It can be seen from Table 1 that the ILU smoother for 3-D anisotropic equations is not as efficient as it is in the 2-D case. (We have found that this is the case also for anisotropic

Y Shapira I Applied Numerical Mathematics 26 (19081 377-398 387

equations in which both Uyy and ' t~ are multiplied by small parameters.) Alto, tbe PT!LU smoother is not much better than the damped Jacobi one.

4.2. The indefinite Helmho!tz equation

The main application of the multi-level analysis method of Theorem 2 is for ip_defi~ite problems. As discussed in [6] and in Section 1. the most prob!ernatic eigenvalues in_ this case are tho~e which are close to zero. Theorem ! a_nd Corollary 1 show that P&MUG(0) and AutoMUG(0) can handle arbitrarily small positive eigenva!ues, yielding convergence factors which are independent of the size of the problem a~d the clustering of the eigenva!ues. Although this applies to the solution of definite prob!ems using two levels, it in_dleates that the algorithm might work efficiently nko wi_th rnulti-!eve! implementatlon_s for i~definite problems. In this case. however, the cell-size o f the coarsest grid cnnnot be arbitrarily large, as is discussed p e×t.

When the coarsest grid is not too coarse, the PAMUG iteration matrix has o,nly few eigenvn,!ues of magnitude larger than one. These eigenval,es are a_nnihilated (the error components corresponding to them are significantly red~!ced) by an appropriate IZrylov sp~ce acceleration method applied to the multlgrid iteration. The remaining eigenvalues are sigt~ificantly smM!er (i~n rnagnit-de) thon o~e; good convergence rates are thus achievable, provided that the dlmension of the 14rvlov sauce is large enough, say twice as large as the number of eigenwb~es of magnitude gre~ter tbnn one. W/hen the number of levels is large, so tbat very co~rse grids are used, the spectrum of the P~MIJG iterntlon matrix deteriorates significantly: the magnitude of many eigenvalues then anproaches one ~nd exceeds

it. Thus. Theorem 2 rn~y help in choosing in advance an appropriate dimension for the 14rvlov space

in the (possibly restarted) acceleration method. For highly ir~definite problems, this dimension mu~t be rather large. In this case, a conventlonM acceleration method, such as GMI~ES or [! 61, will not dO. since the required amoun_t of storage (respectively arithmetical operations) increases linearly (respectively quadratically) with the dimension of the I<rylov sl0ace used. Tbe Ouasi Minimal Residual (OMIt) acceleration method of [121, which uses an arbitrarily laree ICrylov SO~ce with a fixed reaHiremeot of work and storage, is thus oreferab!e.

Consider the Helmholtz eou~tion (3)-(A) in the unit cube with periodic bmmdarv conditions. O!lr aim is to compute the sr~ectrum of the PAMIJG iteration matrix for this rwob!em. In this case,

{ ,. )} spect(X) = spect(Y) = sheet(Z) = 1 -- 2cos 2 3~:2 14~<U

The coustant eigenvector of tbe matrices )(. Y and Z is excluded. This is eonivalent to ~ssumiof that the right hand side includes no Fourier cornoonems which ~re constant in one of the spatial directions and the eouation is projected prop the orthogonal complement of the set of these components. This situation sirnJflates problems with Dirichlet bourMary conditions, since the soectn~m of Y, Y and Z is not enlarged under the transformation

periodic boundary col~ditions -~ Dirichlet boundary conditions

K ~ K / 2 - 1

~ , ! ~ I~ .


-'1 b t 2 • • • g e e • • o o o •

-'1 6 t 2

-'1 ~) I 2 • • • o ~ o * • o • •

-'1 O 1 2

K = 64, 3 levels:

K = 64, 4 levels:

K 128, 4 levels:

K = 128, 5 levels:

Fig. 1. Eigenvalues of the iteration matrix of PAMUG of magnitude 1> 0.5 for the 3-D indefinite Helmholtz equation with /3 = 800 and periodic boundary conditions.

It is verified that no pole of the functions f(i) of Theorem 2 is met during the scanning. As in Section 4.1, we use two damped Jacobi relaxations with damping factor ½, one with damping factor 43 and one with damping factor 3 at each level of the V-cycle.

First we consider a slightly indefinite equation, for which/3 = 120 and K --- 64. For this problem, the spectral radius of the PAMUG iteration matrix is 0.094 for two-level and 0.160 for four-level implementation. Lemma 1 implies that the asymptotic convergence factor of AutoMUG (implemented with two levels and the above damped Jacobi presmoothing) is bounded by

0 . 0 9 4 ( 1 + ( 2 2 2 3 / 2

A numerical support for this is given in Table 2, example (2). Next, we turn to a highly indefinite equation, for which /3 = 800. In Fig. 1, eigenvalues of the

iteration matrix of PAMUG with absolute value exceeding 1 are displayed. When the coarsest grid is

of size 16 x 16 x 16, only few isolated eigenvalues exceed ½ in magnitude, which indicates that the method is suitable for acceleration. On the other hand, when 8 x 8 x 8 coarsest grid is used, many eigenvalues approach one (in magnitude) and exceed it, which implies that no acceleration will do. It may be inferred from Fig. 1 that, for a fixed coarsest grid, the spectrum is nearly independent of the number of levels. This may be considered as a generalization to the 3-D case of known theoretical [3] and computational [19] results.

In light of the above discussion, it is expected that, for Dirichlet boundary conditions and/3 = 200, the choices K = 31 and K = 63 (respectively) yield pictures which are much the same as those of the two former and two latter rows (respectively) of Fig. 1. Consequently, a 7 x 7 x 7 coarsest grid is suitable in this case.

5. Numerical experiments

Here we apply AutoMUG and two versions of Black Box Multigrid to problems of the form

- V ( D V u ) - cru = f in f2 - (0, ~2) x (0, (2) × (0, ~2),

with

Y. Shapira /Applied Numerical Mathematics 26 (1998) 377398 389

f 0, j(t) =

t 1,

D(x,y,z) = {

a(x,y,z) = {

f(x,y,z) = {

0 < t < ~ l ,

E1 < t %~2,

d~, (x,y,z) E f2, j (x)+j(y)+j(z) = O m o d 2,

db, (x,y,z) E ~2, j (x)+j(y)+j(z) = 1 mod 2,

do, (x,y,z) ¢[2,

ar, (x,y,z) E ~, j (x)+j(y)+j(z) = O m o d 2,

ab, (x,y,z) e f2, j (x)+j(y)+j(z) = 1 mod 2,

(z,y,z)

O, (x,y,z) E Y2, j (x)+j(y)+j(z) = O mod 2,

1, (x,y,z) C~2, j ( x )+ j ( y )+ j ( z )= l mod 2,

O, (x,y,z) ¢£2,

and mixed boundary conditions of the form

Dun +'you = O, x = O , y = O or z = O ,

Du~ + 71u = O, x : ~2, Y = ~2 or z = ~2

(where g is the outer normal vector and El, ~2, '~0, 71, dr, db, do, at, O'b and O-o are parameters). The finite volume discretization of [9] is used. However, in light of [1], it cannot be applied directly to the original PDE, since this would result in strong coupling between subdomains which are only weakly connected in the PDE and, therefore, yield an inadequate scheme. Hence, it is applied to the modified PDE

- V ( / g V u ) - ~ u = f ,

where

dr+db 2

-~ f i r + f i b

2

b(x, y, z) = {

man {d~/db, db/dr} ,

min {d~/db, db/dr},

C~

D(x, y, z),

min{[x-~ll+ly-~ll, lX-~l i+lz-~l l ,

otherwise,

390 Y, 5hapi,'a /Applied NumecicaI Mathematics 26 (1998) 377-398

I s, r a i n { m a x (1~' - ~, I, I~ - ~, l ) , m a x {i~" - ~, I, i~ - ~, I } ,

G(x. g. :). otherwise.

A unil-o,m 31 × 31 × 3i firte g, ia is USed. When Dificniet bOundary conditions aJe imposed, this is denoted by "?0 -- +)l -- ~c. hi ufis case, no grid point lie on ~).(2 (tna~ is, no trivial equations ate used). We also consJaei examples wim comptex oouilamy conoiaons of the third kind, ior which % and "/j might nave no.vanisnmg ,magmaiy pa, ts. The initial guess is always zero.

Foi nigniy lnoemnte prooiems, mat is, when maxlc~,., ob} > 100 in our exampies, the basic multi- level i~eration (2) ,s accelerated Dy tnc Transpose Free Quasi Minimal Resiciuai (TFQMR) method [13, Algorithm 5.2], which avoids the computaaon of the transpose of the coefficient rnatrix arid preconditioner (the maer is onty implicitly g~ven la (i), so its transpose is not avaiiaNe). As a matter of fact, the TFQiviR memo(] may ue considered as a modification of the Conjugate Gradient Sqamed (CGS) method oi- [23,24]. The costs of these acceie, ation techniques ale compmable to that of the Conjugate Gradients memod, that is, appmxnnately one work unit per iteration. We found tha~ CGS and "I-i~QMR nave similar coavergenee lares; we p~efe, red the laaer beca,se of its smooth convergence curve.

AutoMUG is implemented with tile rea-biack Gaass-Seiaei (RB) smoother. The Black Box Multi- gim versions aie implemented with an 8-color Gauss-Seidel smoomer [ 18, Method A]. A V( 1, 1)-cycle is u~ed (v = t = ~ = i m (ij). in light of the discussion in Section 4.2, when/3 = 200 tile third-level equation stiould be solved exactly. This is done by appiying to ~t the RB iteration (accelerated by TFQMR) until me comsest gild ,esiclaai is ,educed by 6 o~ciers of magnitude. The maximal number of iterations lequired t0r mis is clispiayeo in TaDie 2 under the heading "coarse". In practice, of coarse, the coarsest grid proNem can be solved di~ecdy; tide parameter "coarse" gives information regarding the condition number of the co~,sest grid coefficient matrix.

The convergence factor is defined by

c f = L 4 ~ : j ~ , - blI2

and the averaged conve, gence factor is defined by

a v e r = (":4~""~'-~ bl 2 ) '/''~~t 11~4:,:0 - b i Ix

where last is ti~c smallest positive integer for which

II-a.,,~,.,, - - h i !2 ~< thleslaold 11.4~:o-- bil~_

and ti,reshold is approxlmateiy 10 -°. When acceleration is u~ed, cf often oscillates and does not give valuaDle information: therefore, for the highly indefinite examples only avcf is reported.

in Table 2 we display ,esults for the following methods: (A) AutoMUG, (B) Black Box Multigrid (the first method in [9]), and (C) the second method in [9]. All thiee multigrid methods are implemented with coarse grids consisting of even numbered variables of the next finer grid. (Similar results were obtained when odd numbered variables were used for this purpose.) The oft-diagonal row-sums used

E Shapira / Applied Numerical Mathematics 26 (1998) 377-398 391

Table 2 q-hree multigrid methods: (A) AutoMUG, (B) Black Box Multigrid, and (C) the second method of Dendy [9], applied to definite anti indefinite problems with discontinuous coefficients. Uniform 31 x 31 x 31 fine grid is used

Description of examples

Example ~ ~2 "~0 2q dr db do ¢x~. ab Cro Acceleration

1 o c 1 (1) oc

(2) 1 :x3 ~<3 1

(3) 1 oc ~c 1

(4) 1 10i 10i 1

(5) 14/30 1 10i 10i 1

(6) 15/30 1 10i 10i 1

(7) 14/30 1 10i 10i 1

(8) 15/30 1 10i 10i 1

(9) 30 0 0.5 1

(101 30 0 0.5 1

(111 14 30 0 0.5 1000

(12) 15 30 0 0.5 1000

1000

1000

1

1

1

1

l 0 0 0 n o

I 30 30 30 no

l 200 200 200 yes

0 200 200 0 yes

0 200 0 0 yes

0 200 0 0 yes

0 200 0 0 yes

0 200 0 0 yes

0 0 0 0 no

0 0 0 0 no

0 0 0 0 no

0 0 0 0 no

Numerical results

Example Levels A B C

coarse cf avcf coarse cf avcf coarse cf avcf

(1) 5 l 0.224 0.234 1 0.199 0.159 1 0.208 0.224

(2) 4 8 0.237 0.245 13 > 1 > 1 10 > 1 > I

(3) 3 54 0.382 I 18 0.678 66 0.97

(4) 3 54 0.493 85 0.463 80 0.910

(51 3 48 0.488 51 0.412 57 0.795

(6) 3 38 0.424 52 0.344 47 0.789

(7) 3 90 0.474 124 0.327 100 0.715

(8) 3 39 0,483 87 0.509 57 1/.761

(9) 4 10 0.425 0,425 12 0.199 0.170 11 (/.331 0.350

(10) 3 28 0.270 0,230 25 0.200 0.169 24 0.301 0.320

(11) 3 51 0.376 0.397 49 0,207 0.201 60 0.234 0.247

(12) 3 51 0.376 0.397 41 0.987 44 0.989

392 Y Shapira /Applied Numerical Mathematics 26 (1998) 377-398

in [9] are not used here, since coarse grids do not include boundary points of the next finer grid (see [17]). Also, unlike in [8], the prolongation does not use the right hand side, since, as reported in [17], this does not improve the convergence for indefinite problems.

Examples (1) and (2) in Table 2 show that, as indicated by Lemma 1, AutoMUG (with no acceleration) performs for nearly singular Helmholtz equations almost the same as for the Poisson equation. For more highly indefinite problems, however, acceleration must be used.

Although for most of the indefinite examples Black Box Multigrid performs well, it should be kept in mind that it is more expensive than AutoMUG due to the 27-coefficient stencils used on the coarse grids. The last four rows of Table 2 deal with examples which are equivalent, apart from a slight change in the size of the domain, to [9, Problems 4.1 and 4.2]. It is evident from example (12) that Black Box Multigrid stagnates when the breaking point E1 lies on all the coarse grids. The reason for this is that the 27-coefficient stencils of the coarse grid operators of the third level involve strong coupling between subdomains which are only weakly coupled in the PDE (see [17]). In this case, the 7-coefficient stencils of AutoMUG are preferable.

It is interesting to note that when D, rather than /~, is used in the finite volume discretization of example (11), Black Box Multigrid converges rapidly whereas AutoMUG diverges. However, in light of the remarks made in [1], it is not clear whether the resulting scheme is adequate.

6. Conclusions

In this work we apply AutoMUG to 3-D definite and indefinite problems with discontinuous coefficients. We introduce a theoretical (not practical) method which is closely related to AutoMUG, named Parallelizable AutoMUG (PAMUG), which is suitable for two-level and multi-level analysis. For a certain class of problems, including nearly singular SPD problems, we derive upper bounds for the asymptotic convergence factors of certain two-level implementations of PAMUG and AutoMUG, independent of the size of the problem and the clustering of the eigenvalues of the coefficient matrix. This can be interpreted to indicate that good rates of convergence can be achieved also in the indefinite case, provided that a suitable acceleration scheme is used for handling the negative eigenvalues. This assertion is confirmed by the multi-level analysis for PAMUG, provided that the coarsest grid is not too coarse. The optimal implementation derived for PAMUG applies also for the practical method Au- toMUG, as is illustrated numerically for several indefinite examples. It is also shown numerically that AutoMUG is applicable for problems with discontinuous coefficients.

Acknowledgements

The author wishes to thank Moshe Israeli and Avram Sidi for their valuable advice.

Appendix A. Proof of Theorem 1

Although the theorem applies to PAMUG(0), we consider PAMUG(q) with a general q so long as this is possible, Let ~ be a common eigenvector of X, Y, Z and A with the corresponding eigenvalues


x, y, z and x + y + z , respectively. Then g is also an eigenvector of the iteration matrix of the two-level implementation of PAMUG(q) with the corresponding eigenvalue fr,o(x, y, z), where

( 2 - x ) (2 - y ) ( 2 - z ) (x + y + z) g(x ,y ,z ) = 1 - (2 + q)2(x(2 - x) + y ( 2 - y) + z(2 - z ) ) '

3 g(x ,y ,z) 1 x + Y + Z 4

(r is the number of damped Jacobi presmoothings and t is the number of Jacobi postsmoothings). In order to prove the theorem, it is sufficient to bound Ifr,01 in the region 0 ~< x, y, z ~< 2, 0 < x + y + z < 6. It is easily verified that, for q ) O, 0 <, Ifr,ol < 9 ~< 1. Since g is symmetric, it is reasonable to rewrite it as a function of the symmetric variables

c = x + y + z , d = x 2 4- y2 4- z 2

e = ( 2 - x ) ( 2 - y ) ( 2 - z).

Note that c2/3 ~< d. In the region under consideration, we have also d ~< c 2 and 0 < c < 6. Consider first the case 0 < c < 2. For fixed c and d, consider the problem

minimize e = (2 - x)(2 - y)(2 - z)

subject to x 4- y 4- z = c,

X 2 4- y2 4- Z2 ~_ d.

By introducing Lagrange multipliers A and #, one obtains

r e = 2 x ' 2 - - - y ' £ : z

For fixed ~ and #, the equation

- 1 -- ~ 4- # x

2 - - x

has at most two distinct solutions. Hence, at least two of the variables x, y and z are equal to each other. Suppose that x = y. Then we have

2x + z = c, 2 x 2 4- Z 2 ~ d,

the solutions of which are

x = ~ + 6d - 2c 2,

Since c < 2 and d ~< c 2,

6 - c 3 2c2 > 0.

Consequently,

z = ~ z~ 6 d - 2c 2.

394 E Shapira /Applied Numerical Mathematics 26 (1998) 377-398

( 6 @ f + ~ V/-~ _ 1 2c 2-)2( 6 - c 3 31 v / 6 d - 2c2)

Q 6 - c 1 ) 2 Q 6 _ ~ l ) ~< e ~< ~ g V / ~ - 2c 2- + ~ V/-~ - - 2c 2 .

With the above symmetric variables, we have ce 9(c,a,e) = 1-

(2 + q ) 2 ( 2 c - d)"

Note that 2c - d > 0 in the region under consideration. The derivative of 9 with respect to e is

09 - c - - = < 0 . 0e (2 + q)2(2c - d)

Hence, for a fixed 0 < c < 2, the maximum of 9 is obtained somewhere along the curve )2( ) 1 v / 6 d _ 2 c 2 6 - c 1 e = + g -~ ~ V / ~ - 2c 2

(or, possibly, at a boundary of the feasible region of the form x = 0, y = 0 or z = 0; but them g and fr,0 are actually functions of two variables and can be analyzed as in [19]). In order to find this maximum, we compute the partial derivative of 1 - 9 ( c , d, e(c, d)) with respect to d along this crave:

3 ( 1 - 9 ( c , d , e ( c , d ) ) ) ( ( 6 d - 2 c 2 ) -1/2 ( 6 d - 2 c 2 ) - ' /2 l

0d = \ _¢~ + ~ v/6d _ 2c 2 - 6-__~3 -- I v/6d _ 2e2 + 2c-----d / (1 - 9)

_ ( - 1 / 2 1 ) - ' d - - - - _ - - ( l - g ) . + - 2c2)( 2c2) +

In order to show that 9 is monotonically decreasing with increasing d along the above cmwe, it is sufficient to show that the last quantity is positive or, equivalently, that

c-d/2 < (~@ f- +lV/6d- 2c2) ( 6-c3 3 1 v / 6 d - 2 c 2 )

4c c 2 1 6 - c 1 ( 6 d - 2 c 2) = 4 5 + - 9 - - 6 5 v / ~ - 2 c 2 18

or, equivalently, that

0 < 7 2 - 4 2 c + 5 c 2 - ( 6 - c ) a + 0 . 5 a 2, w h e r e a = v / 6 d - 2 c 2.

A sufficient condition for this is

(6 - c) 2 - 2(72 - 42c + 5c 2) = - 9 c 2 + 72c - 108 < 0,

which is valid for 0 < c < 2. Hence, the maximum of 9 for a fixed 0 < c < 2 is obtained at d = c2/3 and x = y = z = c/3. At this point,

9 = 1 - ( 2 - x ) 3 3 x = 1 ( 2 - - x ) 2 (2 + q)23x(2 - x) (2 + q)2

Y. Shapira /Applied Numerical Mathematics 26 (1998) 377-398 3 9 5

and, with q = 0,

~ c(12 c) X - C C -

g - - - - x - - - - 4 3 36 36

The maximum of Ifr,01 is also obtained there:

f~,0 ' 3 ' 36 ~<

We find the maxima of h:

h ' ( c ) = ( lc 1 2 - c l 6--r ) h(C) O c = or, equivalently,

( 1 2 - c ) ( 6 - c ) - c ( 6 - c ) - r c ( 1 2 - c ) = 0

or, equivalently,

(2 + r)c 2 - (2 + r ) 1 2 c + 72 = 0

or, equivalently,

72 - - - - O .

r (12 - - _

36

,.)

c- - 12e+ 2 + r

The solution is

c = 6 - 3 6 - 2 + r

and the value of h at this point is

1 2 + r 2 + r "

The theorem follows from

If,-,0] ~< = 2-"

in the region 2 ~< c < 6. [] Let ct -1 denote the damping factor in the above damped lacobi relaxation. It can be seen from

the above proof that the choice c,-i = 43_ comes from the property that 3c~ is the root of the shifted Chebyshev polynomial of degree one on the interval [2, 6]. This, however, is not the optimal choice if varioas damping factors for various relaxation sweeps are considered. Bound (6) may be improved by using in the kth damped Jacobi relaxation, 1 ~< k ~< r, damping factor c,~ -l for which 3c~k is the kth root of the shifted Chebysilev polynomial of degree r on the interval [2, 6]. With these choices, the first argument in the maximum function in (6) is replaced by the Loc norm of this polynomial on [2, 6] and the second argument also improves. We leave these details to the reader, since our main objective is to show that the convergence is independent of the size of the problem and the clustering of the eigenvalues.

396 Y Shapira /Applied Numerical Mathematics 26 (1998) 377-398

Appendix B. Proof of Lemma 1

For i E {0, 1 }, define the injection operators O~,i, Oy# and Oz# from VN ---, VN by

l = i m o d 2 , J'Vz,m,n, m = i m o d 2 , I ¢ i mod 2, (Oy,iV)Z,m,n = ~ O, m ¢ i rood 2,

and

I Vl,m,n~ n = i mod 2, (Oz#V)l,m,n = O, n ~ i mod 2,

v E VN (Ox,i injects onto every other yz-plane, Oy,i injects onto every other xz-plane and Oz,i injects onto every other xy-plane).

Let v be a common eigenvector of X, Y and Z with the corresponding eigenvalues Xv, Yv and z~, respectively. Since X, Y and Z are of property A, it follows from [26, Section 7.1] that the following is a set of common eigenvectors of X, Y and Z:

W=~ { E (--1)°d+~J+~kOx,iOy,jOz,kV~ i,j,ke{O,1} ) ~,/3,'yE{0,1 }

Denote by xw (respectively yw, z~) the eigenvalue of an element w E W with respect to X (respectively Y, Z). Define the set of vectors

3/2 V ~- {2 Ox,iOy,jOz,kV}i,j,kE{O,l}.

Define the symmetric orthogonal discrete Haar transform

3 5 . H = (h%5)%6e{o,1}3, h-y,5 = 2-3/2(-1)}--~i="~i ~

Note that W = H V and V = H W . Let MR and MA denote the iteration matrices of PAMUG(q) and AutoMUG(q), respectively (implemented as in Theorem 1 and Lemma 1, respectively). Note that o T o = Ox,oOy,oOz,o and OMA = OMp. The assumption that a postsmoothing of the form x ~-- P O x is performed in AutoMUG(q) is equivalent to replacing the substitution Xout ,--- Xin -- Pe in (1) by Xout + - P ( O x i n - e). From these observations it follows that, for any w E W,

MAW = fr,o(x~,,y~,z~) E (i -- Xv/C)i(1 -- y~/"/)J(1 -- z~,/rj)kOx,iOy,jOz,kV i,j,kc{O,1}

(with fr,0 as defined in the beginning of Appendix A). Consequently, span(W) is an invariant subspace A A

of MA. Let MA denote the restriction of MA to span(W). The representation of MA in the basis W

is of the form ~rA = 2-3/2Hpu T, where p and u are the following 8-dimensional vectors:

p {(1 x~/c)i(1 y~/~/)J(1 k T T . . . . Zv/?]) }i,j,kc{O,1} and u = {fr,o(Xw,yw,Zw)}wc W.

Consequently, we have

p( rA) < II AII -< 2--3/elIHIIP(p T 'pT)'n ---- 2--3nl lPl l I1 11 IlPll max Ifr,o(x ,y ,zw)l wEW

= ( ( 1 +11 - - Xv/Cl2) (1 +11--yv/~12)(1 + ll -- zv/~712 )1/2 max ]f~,o(xw, Yw, Zw)l. wEW


Consequently,

p(MA) <~ max P MA {v 13A, IAv-Av[=O} ( )

×p(Me). []

The result in Lemma 1 can be slightly improved by finding the maximum of the function ]]p[[2 ilul12 over the set of eigenvectors v of A. (Note that xw is equal to either xv or 2 - xv and similarly for yw and Zw.) The present result, however, is sufficient for our purpose, that is, showing that the convergence is independent of the size of the problem and the clustering of the eigenvalues of the coefficient matrix.

Appendix C. Proof of Theorem 2

Define

A 0 = A and Di--=diag(Ai), 0 ~ < i ~ < n - 1 .

Consider the ith call to the PAMUG procedure in the PAMUG method (1), 1 ~< i <~ n. This call is designated for solving the equation A~-I~' = ~'. For this equation, denote the two-level PAMUG iteration matrix by N~_ l and the multi-level PAMUG iteration matrix (with n - i + 2 levels) by Mi_ 1. For a PAMUG cycle with index e, we have (see [14]) Mn-t = Nn-1, and, for i -- n - 2, n - 3~. . . , 0,

M~ = (I - D~ 1Ai) t (I - (I - M~+I)A~IR~+IAi ) (I - a -1D~ l Ai) ~

= Ni + (I - D~-' A~)tM~+IA~-~,R~+,A~(I - a - ' D~ 1A~) ~.

It is easily verified by induction that all the operators Xi, R(Xi) , UY~U T, UR(Yi)U T, U2ZiU 2T and U2R(Zi)U 2T, for every i, are block-diagonal with circulant Toeplitz blocks. Hence, all the operators A~, Di and R/, for every i, are diagonalizable by the 3-dimensional discrete Fourier transform. Hence, so are also the operators Ni and, by induction, also the operators Mi. [3

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Multigrid methods for 3-D definite and indefinite problems

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