The convergence of iterative solution methods for. A.J. Wathen B. Fischer D.J. Silvester

Transcription

1 Report no. 97/16 The convergence of iterative solution methods for symmetric and indenite linear systems A.J. Wathen B. Fischer D.J. Silvester Iterative solution methods provide the only feasible alternative to direct methods for very large scale linear systems such as those which derive from approximation of many partial dierential equation problems. For symmetric and positive denite coecient matrices the conjugate gradient method provides an ecient and popular solver, especially when employed with an appropriate preconditioner. Part of the success of this method is attributable to the rigorous and largely descriptive convergence theory which enables very large sized problems to be tackled with condence. Here we describe some convergence results for symmetric and indenite coecient matrices which depend on an asymptotically small parameter such as the mesh size in a nite dierence or nite element discretisation. These estimates are seen to be descriptive in numerical calculations. Subject classications: AMS(MOS): to be inserted Key words and phrases: to be inserted This research was supported by NATO Grant CRG Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England OX1 3QD wathen@comlab.oxford.ac.uk October, 1997

2 2 Contents 1 Introduction 3 2 Optimal polynomials and convergence of the minimum residual method 5 3 Optimal polynomials 6 4 Ordinary Dierential Equations 7 5 Asymptotic solutions 10 6 Conclusion 14

3 3 1 Introduction Large matrix problems arise in dierent areas, the most common of which is perhaps the approximation of boundary value and initial boundary value problems for partial dierential equations. For nite dierence and nite element approximations, accuracy of the discrete solution is most often expressed in terms of an asymptotically small characteristic mesh size, h. A priori optimal error estimates will be descriptive of the observed accuracy in computations when the mesh size h is reduced. In direct correspondence with this is the recognition that the linear or linearised systems of equations which must be solved to yield the discrete solution belong to a family of matrix systems, parameterised by h, where the matrices become of larger dimension as h is reduced. Thus, for example, approximation of a 2-dimensional Dirichlet problem on [0; 1] [0; 1] for the Laplacian on a regular square mesh with mesh size h yields coecient matrices of dimension h 2 2h For direct solution methods this size (together, for example, with the bandwidth h 1 for a natural numbering) will allow an estimate of the work required to solve a particular discrete problem, however for iterative methods (for which the work at each iteration is usually xed and easy to calculate) more crucial still is the way in which the convergence will depend on h. It is usually the case that not only does the work per iteration increase as h is reduced, but also that the number of iterations required to achieve a convergence tolerance will also increase for larger problems (smaller h). For example, if the conjugate gradient method [13] is applied to the Laplacian problem mentioned above, then the iteration error will decrease at each step by a factor of 1 O(h). For positive denite problems, particularly those which are symmetric, all this is well known, see for example [5] (or for more classical iteration methods see [24]), and there has been much eort directed at preconditioning to reduce the dependence of the convergence on h or even in the best case to give a preconditioned iterative method which has a convergence rate independent of h (see for example [4] or [21]). For indenite matrices the situation is quite dierent. There is a widely held view that iterative solution of indenite systems is much less reliable and much less ecient in general, and that steps should be taken to make such systems positive denite. For example in the case of saddle-point (or KKT) systems of the form " # " # " # K B T x f = ; B 0 y 0 (which are necessarily indenite), provided that K is denite, then block elimination yields a denite Schur Complement BK 1 B T to which iteration such

4 4 as conjugate gradients, or one of the family of non-symmetric Krylov subspace methods can be applied (in the case of nonsymmetric K). Even conversion of indenite systems to normal equations by multiplying with the transpose of the coecient matrix (which can be eective for well conditioned problems) is suggested by some authors: Yousef Saad for example says \When the original matrix is strongly indenite, i.e. when it has eigenvalues spread on both sides of the imaginary axis, the usual Krylov subspace methods may fail. The conjugate gradient approach applied to the normal equations may then become a good alternative" ([21], pp 313). Research in the last ve years has provided a strong counter to this view at least in the case of symmetric and indenite systems: in general it is sensible to consider Krylov subspace iteration methods such as the method of minimal residuals, a stable implementation of which (MINRES) is described in the seminal paper by Paige and Saunders [18]. A detailed description of MINRES and alternative implementations (in particular, SYMMLQ) is given in [9]. (For a detailed roundo error analysis of MINRES and SYMMLQ see [23].) Eective preconditioners with guaranteed rapid convergence rates have been derived for a number of important problems. For problems in incompressible uid mechanics, see [20], [26], [22] or the review paper [7], for problems in elasticity see [14], [3] and [19], for problems in porous media ow, see [16] and for an example in nite element/boundary element coupling for transmission problems, see [27]. In this paper we concentrate on convergence estimates for miminum residual iteration applied to a linear system Ax = b, (so that A represents the preconditioned coecient matrix if preconditioning is employed). In particular we generalise the results of [25] to establish rigorous convergence estimates for families of matrices which depend on an asymptotically small parameter (in applications is typically a positive power of the mesh size parameter h). These results prove the superiority of the minimum residual approach over the solution of normal equations for all except one very special type of symmetric and indenite matrix. More background and an easy introduction to this problem can be found in [8], pp In section 2 we give a brief description of the minimum residual iteration and show how convergence is described by polynomial approximations on the eigenvalues of the coecient matrix. In section 3 we give some solutions of the approximation problem in terms of the classical Chebyshev polynomials and show in section 4 how an ordinary dierential equations approach can be used on these problems. In the rather more technical section 5 we establish asymptotic convergence estimates for small based upon asymptotic solution of the relevant ordinary dierential equation initial value problem. In section 6 we summarise our conclusions.

5 2 Optimal polynomials and convergence of the minimum residual method At a cost of one matrix{vector multiplication and a few vector operations per iteration, starting from x 0, the minimum residual method computes iterates x k from the shifted Krylov subspace x 0 + spanfr 0 ; Ar 0 ; : : : ; A k 1 r 0 g for which the residuals r k = b Ax k have minimal Euclidean norm. Equivalently, 5 r k = p k (A)r 0 where p k is a polynomial of degree k satisfying p k (0) = 1, and so the minimisation property yields kr k k = min p2 k ;p(0)=1 kp(a)r 0 k min p2 k ;p(0)=1 kp(a)k kr 0 k = min p2 k ;p(0)=1 kp()k kr 0 k where A = QQ T is an orthogonal diagonalisation of A. Note that it is at this point that symmetry is important: the consequent orthogonality of the eigenvector matrix Q and the invariance with respect to orthogonal transformations of the Euclidean norm have been used. For a non-symmetric problem this is not possible in general [6]. By considering the Euclidean norm of the diagonal matrix one arrives at kr k k kr 0 k min p2 k ;p(0)=1 max jp()j min p2 k ;p(0)=1 max x2e jp(x)j (2.1) where are the eigenvalues of A and E is an inclusion set for these eigenvalues. In the case of a symmetric indenite matrix, E is comprised of two real intervals on either side of the origin (which must be excluded from E because of the constraint p(0) = 1). Thus the sequence of polynomial approximation errors k = min p2 k ;p(0)=1 max x2e jp(x)j (2.2) describe the convergence of the minimum residual method. A more convenient single quantity is the asymptotic convergence factor = lim k!1 1=k k

6 6 which gives the average contraction per iteration: it is for which we will establish estimates. A comment is required at this stage: as with the conjugate gradient method, the minimum residual method would terminate with the exact solution in a number of iterations (steps) less than or equal to the dimension of A if exact arithmetic could be employed. However, the convergence estimate (2.1) and the corresponding value of describe convergence of the iteration for much smaller numbers of iterations. Indeed it is usually the case that it would be very expensive or even infeasible to iterate for as many as dim(a) iterations, so in general one seeks a rapid initial reduction of the norm of the residual. 3 Optimal polynomials Existence of a unique solution of the approximation problem (2.2), the so called optimal polynomial, is guaranteed and will not be discussed further here (see for example [17]). What is of concern to us is a characterisation of the optimal polynomial since its exact form is not known in general. The solution of such minimax approximation problems is usually associated with `equioscillation' properties of the error, and so it is here. For the polynomial of degree k there are k+1 distinct points lying in E at which the value k is attained, and at successive such points the sign of the error alternates. For the simple case where E is a single interval, say [c; d], it follows that the solution of (2.2) is explicitly given by suitably scaled and shifted Chebyshev polynomials, T k 2x d c d c p(x) = T k T k (3.1) d c d c and from this that p d p c k k 2 p p (3.2) d + c so that ( p d p c)=( p d + p c) (see for example [10], Theorem or for a derivation more in context with the present application see [8], section 7.4.5). For the case of a pair of intervals E = [ a; b] [ [c; d] (3.3) where a, b, c and d are positive, the solution is simple to express only when the intervals are of equal length: again the solution is in terms of Chebyshev polynomials 2(x + b)(x c) p(x) = T bk=2c (q(x))=t bk=2c (q(0)); q(x) = 1 + bc ad (3.4)

7 7 where bc means the integer part (see [10] or [12] though the result is originally due to Lebedev [15]). The quadratic polynomial q simply maps the two intervals onto [ 1; 1], so that p is symmetric about the central point (c b)=2 = (d a)=2. In this case p ad p bc bk=2c k 2 p p (3.5) ad + bc (see again [10]). It is unfortunate that this easily accessable estimate based on `symmetric' Chebyshev polynomials is only descriptive when the eigenvalues of a matrix are spread in intervals of essentially equal length, since it is usually quoted as the estimate for the convergence of the minimum residual algorithm regardless of any special structure that the eigenvalue spectrum may possess. In particular if the eigenvalues of a matrix are essentially symmetrically distributed about the origin so that a = d and b = c, then (3.5) implies that residual reduction after k steps is the same as would be achieved after bk=2c steps on a positive denite problem with eigenvalues in [c 2 ; d 2 ], (so that it is comparable with squaring the matrix to achieve the normal equations). Since any implementation of a Krylov subspace method for the normal equations must in some manner compute two matrix{ vector products at each iteration, the work in bk=2c steps of such a method compares with the work in k steps of the minimum residual method, and it follows that the overall work using either of these two approaches is comparable. We now embark on a dierent approach which for many problems involving an asymptotically small parameter yields tighter estimates on the convergence of the minimum residual method in all but this worst case. 4 Ordinary Dierential Equations We have already mentioned the `equioscillation' characterisation of the minimax approximation problem (2.2) when E is a single real interval which does not contain the origin: for a degree k polynomial there exist k + 1 distinct points at which the maximal error k is achieved with alternating sign. Moreover the end points of the interval E are necessarily in this set of `extreme' points. The situation is not so straightforward in the indenite case where E comprises two real intervals (except when there is an exact symmetry as in the case of two intervals of equal length as above). In particular, there is no guarantee that the four end points of the intervals will be extreme points. The issue is that though the error alternates on either side of the constraint point (the origin) we would require that there be k +2 extreme points to ensure that j k j was attained

8 8 at all four end points as well as at k 2 interior points. If there are not k + 2 extreme points then it is not possible to explicitly compute the optimal polynomial (see [9]) and further analysis is impossible. Fortunately what we require is not impossible: Proposition 4.1 If E = [ a; b] [ [c; d] then the solution of the polynomial approximation problem (2.2) has k + 2 extremal points if and only if sn q(b 1 + d)=(a + d) ; m = n k for some n 2 f1; 2; : : : ; k 1g. K(m); m = (c + b)(a + d) (b + d)(a + c) (4.1) This result, in which sn is the elliptic sine function with modulus m, and the number K is the complete elliptic integral of the rst kind, (see [28] for a comprehensive description) was proved by Achieser [2]. Though this leads to an expression for k in terms of the Jacobian theta function, for our present purposes the proposition most usefully tells us that as the number of iterations k increases, there is a more and more dense discrete net of end points for which there are k +2 extreme points including the interval end points themselves. That is, the end points of a pair of intervals are extreme points if and only if the inverse elliptic sine function for the particular given argument and modulus is an exact integer multiple of the complete elliptic integral of the modulus divided by k. Since the elliptic sine function is continuous and monotonic on the relevant domain, it follows that any given pair of intervals, E, can be embeded in a pair of generally larger intervals, ^E say, for which the optimal polynomial has k + 2 extreme points including all four interval end points and k 2 interior points. The minimax error on ^E must clearly be greater than or equal to the minimax error on E. Moreover, for larger values of k, ^E need only be larger than E by a smaller and smaller amount. In this paper we are specically concerned with indenite systems which have eigenvalues contained in inclusion intervals of the form E = [ a L ; b`] [ [c r ; d] (4.2) where a, b, c and d are positive real numbers, is an asymptotically small quantity and the exponents L, ` and r are real with L being non-negative and ` and r being positive. We clearly must have a L b` which for small enough implies that L `. Our analysis requires the further assumptions that in fact L < ` and also L < r. We note here that many other cases can be easily transformed into the form (4.2): for a simple linear change of variable, ux + v it is clear that since the

9 9 degree does not change, then the polynomial approximation problem (2.2) remains unchanged except for a shift of the constraint point if v 6= 0. The results we shall describe therefore apply to more general situations such as for example where 2 [ ; 2 ] [ [ 3 ; 2 ] when the choice u = 2, v = 0 yields (4.2) with a = b = c = d = 1, L = 3, ` = 4 and r = 5. Similarly by taking u = 1, v = 0 we cover cases when the positive upper bound in (4.2) depends asymptotically on whilst the negative lower bound is constant. The rst important consequence of the assumed asymptotic behaviour concerns Proposition 4.1. We are not guaranteed k + 2 extreme points on (4.2), however we may embed E in a pair of intervals ^E for which the result of the proposition does hold as described above. Now ^E can be selected so that for large enough k, ^E E is as small as we desire. In particular its size (measure) can be made proportional to any positive power of the asymptotic parameter,. Thus, we may embed E = [ a L ; b`] [ [c r ; d] in ^E = [ a L ; b` + ] [ [c r ; d + ] (4.3) where = O( r+`) with ^E satisfying the conditions of Proposition 4.1. The outcome is that for a set such as (4.2), we may assume that the optimal polynomial has k + 2 extreme points by ignoring higher order asymptotic quantities when k is large enough. For a more precise mathematical description in a particular case, see [25]. Figure 1: Form of the optimal polynomial Our analysis now takes an unusual turn: we derive an ordinary dierential equation initial value problem satised by the optimal polynomial. The characterisation in terms of extreme points implies that the (degree k) optimal polynomial, p, has the form as in gure 1 where a degree 7 polynomial is displayed.

10 10 Noting that p = k at k 2 interior points where the derivative p 0 vanishes as well as at the four end points we have k 2 (x t) 2 (p 2 (x) 2 k) = (x + a L )(x d)(x + b`)(x c r )(p 0 (x)) 2 (4.4) where t is the local maximum between b` and c r and the scaling factor of k 2 arises from matching the (leading) coecient of x 2k+2. Taking the appropriate square root of (4.4) and dropping dependencies, for x 2 ( b`; c r ) we have k(x t)(p 2 2 k ) 1 2 = h (d x)(x + al )(x + b`)(c r x) i 1 2 If we now rewrite this as ( dp ): (4.5) dx dp dx = k(x t)(p 2 2) 1 2 k [(d x)(x + a L )(x + b`)(c r x)] 1 2 = f(x; p); (4.6) then f(x; p) is continuous in x and Lipschitz continuous in p for x 2 ( b`; c r ) as p > k on this interval. Thus using the classical Picard existence theorem for ordinary dierential equations, (4.5) has a unique solution in this interval which satises the `initial' condition p(0) = 1 for any bounded value of t. (Note also that the solution is a polynomial if and only if the condition of proposition 4.1 is satised.) Having derived the characterising dierential equation, we now seek an asymptotic expression for (not p nor even k!) for small. Our minimum residual convergence estimate will then follow using (2.1) and (2.2). For the analogous derivation of an ordinary dierential equation in the two simpler cases of a single interval and equal length intervals as in the previous section, see [8]. We should also point out that the path followed above is a welltrodden one: Chebyshev originally computed \his polynomials" by solving the associated ordinary dierential equation in the single interval case. 5 Asymptotic solutions Our analysis now becomes more technical though we are in fact simply attempting to nd the solution to the rst order ordinary dierential equation (4.5) by using the elementary `separation of variables' technique. Directly this gives Z z y (x t) dx [(d x)(x + a L )(x + b`)(c r x)] 1 2 = 1 jp(y) + k log [p(y)2 2 k] 1 2 j e jp(z) + [p(z) 2 2 k] 1 2 j (5.1)

11 11 for any b` < y z < c r. Throughout the analysis we will only be interested in the solution between these two interval end points which are nearest the origin and so need not concern ourselves with the many singularities. We see that the left hand side of (5.1) is an elliptic integral involving the unknown point t. Our two problems now are therefore to estimate t and to simplify the elliptic integral using asymptotics. To do the second of these we note that for any x 2 [ b`; c r ] (d c r )(a L b`) (d x)(x + a L ) (d + b`)(a L + c r ) (5.2) so that taking reciprocals and square roots and using simple binomial expansions we have q 1= (d x)(x + a L ) = (ad L ) O( minf` 3L=2;r 3L=2g ) (5.3) since both the upper and lower bounds in (5.2) have this asymptotic form. Note that our assumptions L < `, L < r are necesssary here to ensure that (ad L ) 1 2 is the leading (lowest order) asymptotic term. The complexity of the higher order terms will not be important. Use of (5.3) in (5.1) will leave integrals for which there is the following explicit formula (see [1], formulae and ) Z z y (x t) dx = h [(x+b`)(c x)] 1 (y + b`)(c r y) i 1 2 h (z + b`)(c r z) i 1 2 r 2! (cr ) 1 (b`) t arcsin( cr b` 2y ) arcsin( cr b` 2z ) : (5.4) 2 c r +b` c r +b` Now since p & k both as y & b` and as z % c r, it follows from (5.1) that Z z lim y& b` and z%c r y (x t) dx [(d x)(x + a L )(x + b`)(c r x)] 1 2 = 0: (5.5) The asymptotic formula (5.3) together with (5.4) then gives t = 1 2 (cr b`) + O( r+`) (5.6) as in the limit, the rst two terms of (5.4) vanish and the trignometric terms tend to. We see that (5.6) is consistent with t = 0 in the case r = ` and b = c. That is when the intervals are symmetrically placed about the origin, then the turning point t coincides with the origin. (See the exact description (3.4) in terms of

12 12 Chebyshev polynomials when also L = 0 and a = d). In any case, provided that L < minf`; rg, t lies approximately half way between c r and b`. Our analysis is completed by integrating (5.1) from the origin to the nearer of c r or b`. Strictly speaking, we integrate from 0 to z and take the limit as z % c r or from y to 0 and take the limit as y & b`. Whichever of these two integrals we do the result is (ad L ) 1 2 (bc r+`) 1 2 (1 + O( r+`)) = log e j1 + [1 2 k] 1 2 j 1 k 1 k k : (5.7) Now j1 + [1 2 k] 1 2 j 1 k! 1 as k! 1 since 0 [1 2 k] for all k, so negating and exponentiating gives k lim k = exp q (r+` L)=2 bc=ad + O( r+` L=2 ) The outcome is k!1 1 = 1 (r+` L)=2 qbc=da + O( r+` L=2 ): Theorem 5.1 The asymptotic convergence factor when the minimum residual method is applied to a symmetric indenite linear system with eigenvalues lying in an inclusion region of the form (4.2) satises = 1 (r+` L)=2 qbc=da + O( r+` L=2 ): (5.8) In order to compare the result of Theorem 5.1 with the results of section 3 which involve Chebyshev polynomials, we must embed E in the union of (larger) equal length intervals so that (3.5) can be applied. If L 6= 0 one obvious way to do this is to take for which (3.5) gives E ^E = [ d b` + c r ; b`] [ [c r ; d] (5.9) q d 2 + O( minf`;rg ) p bc r+` bk=2c k 2 q d 2 + O( minf`;rg ) + p : (5.10) bc r+` Taking the limit as k! 1 leads to a corresponding asymptotic convergence factor s bc 1=2 1 2 (r+`)=2 + O( minfr+`=2;r=2+`g ) d s 2 bc = 1 (r+`)=2 + O( minfr+`=2;r=2+`g ) d 2

13 13 which is seen to be inferior to (5.8). If L = 0 then it matters which of a or d is bigger and the corresponding estimate is s bc 1 (r+`)=2 + O( minfr+`=2;r=2+`g ) (5.11) maxfa 2 ; d 2 g which is seen to have the same asymptotic form as (5.8) but with a smaller constant. We now consider two simple examples which illustrate the results given here. Firstly for simple illustration we use the example of an indenite matrix whose eigenvalues satisfy 2 [ ; 2 ][[ 3 ; 2 ]. Applying the simple linear mapping illustrated in section 4, the convergence of the minimum residual method for a problem of this type will be identical (excluding roundo eects) to a problem with 2 [ 3 ; 4 ] [ [ 5 ; 1]. For such a problem theorem 5.1 gives that the residual vectors satisfy kr k k kr 0 k 1 O( 3 ) k (5.12) whereas use of (3.5) gives kr k k kr 0 k 1 O( 9=2 ) k : (5.13) There is of course an advantage of (3.5) in that it is based on an estimate which holds at every iteration whereas (5.8) is only established for large k. A more practical example comes from mixed nite element approximation of the Stokes problem which describes the slow ow of an incompressible uid. See for example [11] for a general description, or [8] for a treatment more closely related to the presentation given here. For a discretisation employing a stable element on a quasi-uniform grid in a 2-dimensional ow domain, we have that the eigenvalues of the unscaled Stokes matrix satisfy 2 [ ah; bh 2 ][[ch 2 ; d] for some positive constants a, b, c and d where h is a representative mesh size. Use of theorem 5.1 then bounds the convergence of the minimum residual method by kr k k kr 0 k 1 O(h 3=2 ) k (5.14) whereas (3.5) gives kr k k kr 0 k 1 O(h 2 ) k (5.15) which is the same estimate as would be derived for iterative solution of the normal equations using the conjugate gradient method. The estimate (5.14) clearly demonstrates the superiority of the minimum residual method for such problems.

14 14 It should be mentioned in this example however that appropriate diagonal scaling of the Stokes matrix as described in [26] gives eigenvalues 2 [ a; bh][[ch 2 ; d] for which (5.8) and (3.5) yield similar results. See [25] for numerical results which demonstrate the descriptiveness of our convergence estimate for the Stokes problem. 6 Conclusion We have derived convergence estimates for the minimum residual method applied to symmetric and indenite matrix systems where the eigenvalues of the coecient matrix depend on an asymptotically small parameter, such as the mesh size in a dierential equation setting. Our estimates are an improvement on the commonly used estimate based on equal length intervals in many cases of practical interest. References [1] M. Abramowitz and I. A. Stegun, \Handbook of Mathematical functions," Dover, New York, [2] N. I. Achieser, \ Uber einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen I. Teil," Bull. Acad. Sci. URSS S.VII 9, (1932). [3] D.N. Arnold, R.S. Falk and R. Winther, \Preconditioning in H(div) and applications," Math. Comput. 66, (1997). [4] O. Axelsson, \Iterative solution methods," Cambridge University Press, [5] O. Axelsson and V.A. Barker, \Finite element solution of boundary value problems: Theory and computation," Academic Press, New York, [6] T.A. Driscoll, K-C. Toh and L.N. Trefethen, \From potential theory to matrix iterations in six steps," SIAM Review, to appear. [7] H.C. Elman, \Multigrid and Krylov subspace methods for the discrete Stokes equations," Int. J. Numer. Meth. Fluids 227, (1996). [8] H.C. Elman, D.J. Silvester and A.J. Wathen, \Iterative methods for problems in Computational Fluid Dynamics," in `Iterative Methods in Scientic

15 15 Computing', Eds. R.H. Chan, T.F. Chan & G.H. Golub, Springer-Verlag, Singapore, , [9] B. Fischer, \Polynomial based iteration methods for symmetric linear systems," Wiley-Teubner, Chichester, [10] A. Greenbaum, \Iterative methods for solving linear systems," SIAM, Philadelphia, [11] M. Gunzburger, \Finite element methods for viscous incompressible ows: a guide to theory, practice and algorithms," Academic Press, London, [12] W. Hackbusch, \Iterative solution of large sparse systems of equations," Springer-Verlag, Berlin, [13] M.R. Hestenes and E. Stiefel, \Methods of conjugate gradients for solving linear systems," J. Res. Nat. Bur. Stand. 49, (1952). [14] A. Klawonn, \An optimal preconditioner for a class of saddle point problems with a penalty term," SIAM J. Sci. Comput., to appear. [15] V.I. Lebedev, \An iteration method for the solution of operator equations with their spectrum lying on several intervals," USSR Comput. Math. and Math. Phys. 9, (1969). [16] J. Maryska, M. Rozloznk and M. Tuma, \The potential uid ow problem and the convergence rate of the minimum residual method," Numer. Linear Alg. Appl. 3, (1996). [17] G. Meinardus, \Approximation of functions: theory and numerical methods," Springer, New York, [18] C.C. Paige and M.A. Saunders, \Solution of sparse indenite systems of linear equations," SIAM J. Numer. Anal. 12, (1975). [19] L.F. Pavarino, \Preconditioned mixed spectral element methods for elasticity and Stokes problems," SIAM J. Sci. Comput., to appear. [20] T. Rusten and R. Winther, \A preconditioned iterative method for saddle point problems," SIAM J. Matrix Anal. Appl. 13, (1992). [21] Y. Saad, \Iterative methods for sparse linear systems," PWS, Boston, [22] D. J. Silvester and A. J. Wathen, \Fast iterative solution of stabilised Stokes systems Part II: using general block preconditioners," SIAM J. Numer. Anal. 31, (1994).

16 16 [23] G.L.G. Sleijpen, H.A. Van der Vorst and J. Modersitzki, \The main eects of rounding errors in Krylov solvers for symmetric linear systems," University of Utrecht, Department of Mathematics Preprint 1006 (1997). [24] R.S. Varga, \Matrix iterative analysis," Prentice-Hall, Englewood Clis, [25] A.J. Wathen, B. Fischer and D.J. Silvester, \The convergence rate of the minimum residual method for the Stokes problem," Numer. Math. 71, (1995). [26] A.J. Wathen and D.J. Silvester, \Fast iterative solution of stabilised Stokes systems Part I: using simple diagonal preconditioners," SIAM J. Numer. Anal. 30, (1993). [27] A.J. Wathen and E.P. Stephan, \Convergence of preconditioned minimum residual iteration for coupled nite element/boundary element computations," Bristol University, Mathematics Dept report AM (1994). [28] E.T. Whittaker and G.N. Watson, \A course of Modern Analysis," Cambridge University Press, A.J. Wathen Oxford University Computing Laboratory Wolfson Building Parks Road Oxford OX1 3QD United Kingdom B. Fischer Institute of Mathematics Medical University of Luebeck Wallstrasse Luebeck Germany D.J. Silvester Mathematics Department UMIST PO Box 88, Manchester M60 1QD United Kingdom