COMMUNICATIONS IN APPLIED NUMERICAL METHODS, VOl. 8, 491-496 (1992)

MULTI-ORDER INITIALIZATION OF CG SOLUTION OF FE DISCRETIZATIONS

I . FRIED* Department of Mathematics, Boston University, Boston, Mass. 02215, U. S. A .

AND

N. BUDINSKY t Department of Mathematics, Southeastern Massachusetts University, North Dartmouth, MA 02747, U.S.A.

SUMMARY The conjugate gradient method holds great appeal for the solution of the stiffness equation set up with finite elements. Inexpensive, low-order finite-element discretization solutions are suggested as initial guesses for higher-order, more accurate, discretizations.

1. INTRODUCTION

There is appeal and promise in the conjugate gradient solution of finite-element static and dynamic problems. Most of the messy and uncertain aspects of sparse matrix technology are absent from the algorithm, and it stands at the threshold of universal practicality.

Early influential work' on the worst possible case, that of the one-dimensional beam equation, is possibly the reason for the slow recognition accorded to the method. Application of the CG method to higher-dimensional discretizations changed this view, and it appears that for the well conditioned two-dimensional harmonic problem CG nearly breaks even with direct methods, and that for the huge sparse, fill suffering, matrices of three-dimensional equilibrium problems the CG method is the most effective solution algorithm.

Under common circumstances convergence of the CG method is linear with rate equal to (1 - xl/*) / ( l + x1l2), where x = x(K) is the spectral condition number of the stiffness matrix K. Considerable effort is being put into preconditioning techniques whereby a truncated inverse S-I to K is set up so as to have S-'K that is spectrally near to the identity. But the construction of S- ' by the incomplete factorization of K or substructuring' again has sparseness problems.

In this paper we propose to inquire into the advantage of the possibility, which is certainly an eminently practical one, of using a low-order finite-eiement solution as an initial guess for a CG solution of a higher-order finite-element approximation. We look in particular at the CG procedure applied to the c o m p ~ t a t i o n ~ - ~ of the lowest eigenvalue of Kx = XMx via the minimization of the Rayleigh quotient p(x) = xTKx/xTMx. In this present preliminary simple analysis we consider a square membrane, discretize it with a mesh of first-order triangular finite

* Professor t Associate Profe sor

0748-8025/92/08049 1 -06$08 .OO 0 1992 by John Wiley & Sons, Ltd.

Received 26 March 1991 Revised 28 October 1991

492 I . FRIED AND N. BUDINSKY

lineor eiements cubic elemenfs Figure 1 . A square membrane discretized with linear and quadratic elements for the same number of nodes

elements, solve for eigenvalue X and eigenvector x, and then use this quickly computed eigenmode as an initial guess for a better discretization with cubic finite elements over the same nodes as in Figure 1. The advantage of this approach is that it does not require any new meshing.

2. LINEAR SEARCH

High-order finite-element eigenproblem discretizations produce the generalized Kx = XMx, where K, the stiffness matrix, is symmetric positive-semidefinite, and where M, the mass matrix, is symmetric positive-definite. We shall compute the lowest eigenpair of the eigenproblem using the CG method to minimize the general Rayleigh quotient

x *Kx x'Mx pL(x) = -

There are various ways in which the CG method can be adapted to the minimization of non- quadratic functionals but common to all the variations in a linear search for the extremum of p(x) along a search vector p.

Let xo be the current iterate vector, and PO the current search direction vector. We write x = xo + Epo and seek to minimize p(x) = p ( 4 ) with respect to scalar variable E . Some algebra results in

(2) 2E(xTKp - pxTMp)o + E2(pTKp - ~ P ' M P ) ~ (xTMx)o + 2E(xTMp)o + E2(pTMp)o b = At) - p(0) =

where ( )O refers to values at 4 = 0. In short,

where

a = (x'KP - pxTMp)o, b = (P'MP - ppTMp)o

c = (xTMx)o, d = (x'Mp)~, e = (pTMp)o

Differentiation of 6p with respect to 6 and setting the derivative equal to zero yields the quadratic equation

E2(-ae+ d b ) + t b c + ac=O (4)

MULTI-ORDER INITIALIZATION OF CG SOLUTION 493

for the [s that extremize 6 p . The two roots of the quadratic equation are:

- 2ac bc 5 ,/A' [=- A = b2c2 - 4ac(bd - ae)

and we shall show that + ,/A corresponds to the minimization of g, while -,/A corresponds to its maximization.

Indeed, with the two roots found we have the extremal

extrm(6p) = - 4,/(A 1 a (bc + ,/A - 2 ~ d ) ~ + 4a2(ce - d 2 )

By the general Cauchy-Schwarz inequality

ce - d2 = ( x ~ M x ) o ( P ~ M P ) o - (xTMp); 2 0 (7) and hence, if ,/A is positive, then 6 p is negative and vice versa.

The linear search can be given a Ritz interpretation of looking for the minimum of p(x) over the two-dimensional vector space spanned by xo and PO. In this we write x = [IXO + &PO = P[, with P = P(n x 2) holding in its columns xo and PO, and € = ([I, [ 2 ) T . Now

is a function of the vector argument I , and the extrema of p ( € ) are obtained from the 2 x 2 generalized eigenproblem

(PTKP)[ = g(PTMP)f

[iiz pdKpo] [ E J -'[pZMxo pZMpol [ E d (9) XZKPO [I - XZMXO XZMPO El

for which we have the characteristic equation

p2((XTMx)(pTMp) - (xTMp)')o

+ P( - (x'Kx) ( P ~ M P ) - ( x ~ M x ) ( P ~ K P ) + (xTKp) (p'Kp)(pTMx) + (xTMp)(pTKx))0

+ ( (x~Kx) (p T ~ p - (X 'KP i2 lo = o (10)

As originally proposed for the minimization of the quadratic functional f(x) of the vector argument x, the conjugate gradient algorithm reads:

ro = grad f(xo), po = ro

x = xo + CYOPO, f(x1) = min f(x) a 0

x = x k + l +CYk+l(rk+l +PkPk), f(Xk+2)' min f(x) W + I&k

To describe the application of the algorithm to the minimization of p ( x ) , we write

x = X I + (111 (rl + POPO) = X I + am + (YIPOPO

in the more systematic form

x = 7191 + y292 + 7 3 9 3

494 I. FRIED AND N. BUDINSKY

where q1 = XI. q2 = rt, q3 = PO. Vector x is in the three-dimensional space spanned by q1,q2, q3, and we seek the best element of this space - the one that minimizes p ( x ) with respect to yl,y2,y3. What we have, then, is a three-dimensional Ritz method similar to the one in the preceding Section, and the extremal ps are obtained from the 3 x 3 generalized symmetric eigenproblem Ay = pBy,

(14)

which is cheaply solved using Jacobi's method. If the smallest eigenvalue is the one we are interested in, then the smallest eigenvalue of

Ay = pBy is taken with its corresponding eigenvector y, and the two CG parameters 011 and Bo are computed from the corresponding eigenvector as 011 = yzfy~, PO = y3/y2.

T A , . - l, - qTKqi, B . . - i j - SiTMq,, y = ( 7 1 9 7 2 3 ~ 3 )

3 . THE SQUARE MEMBRANE

We now come to the central theme of the paper: the generation of an initial guess xo with low- order elements. We shall examine this possibility numerically by computing the finite-element approximations to the fundamental eigenvalue X = 2n2 of the square unit membrane.

Figure 2 refers to a membrane discretized with linear triangular elements. Iteration is started with the initial guess xo = (1, 1, . .., l )T, and the graph shows convergence of p to the ultimate pm and to the analytic A, as well as / / r 1 1 , r = Kx - pMx. Convergence is linear, but at a good rate,

(15) 1 pL('k) - p(".) I = 0.69k p(x-1

Note that taking the discretization accuracy into consideration there is no profit in continuing beyond ten iterations.

Figure 2. Convergence of the conjugate gradient algorithm in the case of linear elements and the initial guess x"= ( I , 1, ..., 1)T

MULTI-ORDER INITIALIZATION OF CG SOLUTION 495

-12 -

-14 -

10 20 30 40 50 60 70 0

ifer

I p - p m l / p m \ xo=(l,l,l)...) I )

J log(z) -16

Figure 3. Convergence of the conjugate gradient algorithm in the case of cubic elements and the initial guess xo = (1,1, ..., I )T

0 10 20 30 49 50 60 70 ifer

Figure 4. Convergence of the conjugate gradient algorithm in the case of cubic elements and an initial guess taken from the solution of the linear discretization (note the round-off effect)

496 1. FRIED AND N. BUDINSKY

Figure 3 shows the same for the membrane discretized with cubic elements. Convergence is once more nearly linear, with a faster rate, and a considerably better approximation to X, achieved after some 35 iterations.

Figure 4 is the essence of this paper. It shows the convergence of the CG method applied to the cubic discretization with xo taken from the solution to the linear approximation. As far as approximation to X is concerned, there is no need to iterate. The first approximation already reaches the top accuracy possible.

is acceptable for p, then in Figure 4 it is achieved in a few iterations, but if we demand a relative effort of then according to Figure 4 this is achieved in 30 steps, as compared with 65 steps in Figure 3 .

To be specific, if a relative accuracy of

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