computational methods for turbulent, transonic, and

COMPUTATIONAL METHODS FOR TURBULENT, TRANSONIC, AND VISCOUS FLOWS Edited by

J. A. Essers von Karman Institute for Fluid Dynamics

A von Karman Institute Book

0

HEMISPHERE PUB USHING CORPORATION 1

Washington

New York

London

DISTRIBUTION OUTSIDE NORTH AMERICA

SPRINGER-VERLAG Berlin

Heidelberg

New York

Tokyo

lntroduction to Multi-grid Methods for the Numerical Solution of Boundary Value Problems WOLFGANG HACKBUSCH

LIST OF SYMBOLS B

c C0 ( v ) , C1 f

fh,f Q, Gh , GQ.

Gh,G Q.

h 'h .2,

Hh, HQ.

Hm( n ) .Q,

Lh,L i m

r

y

r \J

bilinear form generic constants with different values at different places speci a l bounds right hand side of the boundary value prob l em right hand side of discrete problems smoother of the multi-grid method iteration matrix of Gh, Gi ' respectively grid size, discretization parameter finite element subspace Sobolev space of order m level number of multi -g rid method matrix of discrete problem 2m is order of boundary value problem iteration matri x of multi-grid method dimension of n c ~n number of unknowns prolongation, interpolation restriction unknown function of boundary value problem discrete grid function, solution of the discrete problem linear space of discrete grid functions parameter of mul ti-grid method boundary of don1ain n number of smoothing iterations 45

COMPUTATIO N AL M ETHODS FOR TURBULENT,. TRANSONIC AN D VI SCOUS F

46

Lows

upper bound o f v normal derivative domain of the bo un d ary value problem grid of size h, ht

1.

INTRODUCTION

1.1

Problems under Consideratio n

In this paper we describe mult i- grid algor i thms f o r solving discretized boundary value prob l e ms. I n the first s ub sectio n we give some exarnples of various bounda ry value problem s. The di scretization by difference schemes a nd fi nit e e l ement met hods is recal led in the following sectio n s 1 .1 . 2 a nd 1 . 1 . 3. Boundary value problems

1.1.1

Let n ctRn (n ~ 1) denote a do mai n i n th e n- dim e n s i o nal space. A boundary value problem consists of (i) a n e ll i pt i c partial d i f ferential equation for an unknown fu ncti on ( or vect or) u in Q and (ii) boundary conditions. An example for an elliptic partial diffe r e nt ia l eq u a t io n is Helmholtz 1 equation: - ~ u(x) + c 2 u(x)

g (X)

(1. 1 )

( x d:l ),

where 6 = Li~l a 2 / a x; is the Laplacian operat o r (oth er notations: v 2 , div grad). The general linear el li ptic e qu atio n of second Ot-der with variable coefficients is n

n

L

~

i=l j =l

+

a l· J· (X)

n L

i=l

a.(x) au l

+ a (x) u ( x) = g ( X)

(1 . 2 )

with n

.

l

a .. (x) r.; . ~ .

L

. 1 lJ 'J =

l

J

=I= 0

for all xc n , 0 =I= ~ = ( ~ 1 „ .. , ~ n ) eR n .

A special ell ipti c equation of fourth order is t he b ihar mo n ic

equation : t:, 2

u(x)

=

g(x)

( 1. 3)

As an example for a nonlinear equation we mention the equation minimal surface 2

2

( l+u X 2 )u X I X! -2u X 1 u X 2 u X 1 X 2 + (l+u X l )u X 2 X 2

=

fo~

0

An elliptic equation ?f order 2m must b~ ~ombi n ed with m boundary conditions. Poss1ble boundary cond1t1ons are Dirichlet values for x c: f ; ( 1. 5) u(x) = O x 1 -line -relaxa tion). That means, the points xl= (xLx1) dl h with same x1 belong to one block. The matrix Üh of the splitting (4.8) is block diagonal. Each block is tridiagonal if Lh is a nine-point formula. Again, we can choose different orderings of the blocks: lexicographic ordering: x 1-component increases; (4.lla) r e d - bl a c k o r d e r i n g : f i r s t b l o c ks wi t h " x l / h od d 11 , t he n b l o c k s with "x 1/h even". (4.llb) Ordering (4.llb) is recommended for five-or nine-point schemes. If it is not known in before, whether a k, Lik* O, j >k , Lij : = L;j-L ik lk j end incomp l ete LU-decomposition;

67

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

The results of Wesseling 1980b show that smoothing by i ncomplete LU-decomposition, namely - 1

- 1

(4.14)

Gh(uh,fh) = Sh Rh (fh+Chuh),

is well-suited not only for usual problems but also for anisotropic problems. The rate of convergence depends on the ordering of the grid points, but in any case one obtains very good results. 5.

NUMERICAL RESULTS Helmholtz' Equati on

5.1

The considerations of §2 and §3 showed that the rate of convergence of the multi-grid ite ration is bounded by some C( v ), where C(v) does not depend on the grid size h, but tends to zero for v~ 00 • The following examples shall show that, usually, good convergence can be obtained for small v (here v = 2). We consider Helmholtz' equation {1.1}: -~u

+ c 2 u = -4 +c 2 (x 2 +y 2 )

in n

(5. 1)

with Dirichlet boundary values on r .

(5. 2)

The solution is u(x,y) = x2 +y 2 . Note that the smoothness of the sulut ion does not influence the rate of convergence. The boundary value problem (5.1), (5.2) is discretized by the five-point formula with Shortley-Welier's modification at points near the boundary. The details of the used multi-grid iteration are the following: prolongation p: piece-wise linear interpo l ation* restriction r : r = p*, cf. (2.27) coarser grid matrices Lk defi ned by (3.4) smoother Gh: point-wise or line-relaxation with ordering (4.9c) or (4.llb), respectively number of smoothing iterations** v := 2 y

= 1

The FORTRAN program is contained in the report of Hackbusch 1976. modification for c 2#0: compare Hackbusch 1978. ** more precisely: v=l+3/4 (point -w ise), v=3/2 (line wise relaxation) steps of the smoother after coarse grid correction, v=l/4 ( v=l/2, resp.) before correction. Note that smoothing step and coarse-grid correction are interchanged (cf. end of §3 . 2). Compare Hackbusch 1978.

*

COMPUTATIONAL METHODS FOR TURBULENT, TRANSONIC AND VISCOUS FLOWS

68

First we give sorne results for Poisson's equation [c2=0 in (5.1)] in a circle of radius 1/2 . 1

µ

2

4

3

6

5

7

error ).l

II u1; 54- U1/ 6411 6.8 10":'1 4.010-2 3.41 0-4 1.51 0-5 2.410-7 1.01 0-8 3.510-10

ratio

10.060

lo. ooa

jo.016

10.045

10 .043

jo .034

Table 1 : Equation (5.1), c 2=0, n=circle, h=l/64, point-wise relax . Table 1 shows the errors of seven iterations of the multi-grid algorithm solving the difference equations for the grid size h=l/64 . The sequence (3 . 1) of the step sizes is 1

ho

1 h3 = -

= -

h5 = h =

16

2

32

1

64

The averaged rate of convergence is

01/ 64

=

[11 u ~ / •• -u 1 / 6 ,11/11 u : 1 64 - u 1 1 64 11]

1/ 4

=

0.032.

The computationa1 work of one multi-grid iteration (cf. §3 . 3) corresponds to nearly 5 relaxation iterations. Thus, the rate corres ponding to the wark of one relaxation iteration is about 1/2. Table 2 confirrns the fact that the rates are bounded independently of the grid size. 1/ h rate

8

. 014

12

16

24

32

48

64

96

.014

. 019

.018

.021

. 023

.023

. 025

128 . 02 7

Table 2 : rates of con vergence * (E q. (5.1), c 2 =0, n= circle, point-wise relaxation) . The results of Hackbusc h 1978 shaw that the rate of convergence depends on the shape of the domain (more precisely : on the regularity of the boundary value problem). The rate deteriorates in t he case of a re-en trant boundary. In the latter case linerelaxation is Superior to point - wise relaxation. As example we compare the results for a circle with thase for a cracked region (worst case).

* 1

These rates are obtained for the Mehrstellenverfahren (special ine-point formula of fourth order, cf. Collatz 1966).

69

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

1

1

domain

smoothing by point-wise relaxation

1

smoothing by line-wise relaxation

circle :

0

0 .030 to O.035

0.030 to 0.034

cracked ci rcle :

E)

0.113 to 0.147

0 .053 to 0.079

Table 3 : rates of convergence (Eq. (5.1), c 2 =0 ) The rates depend only very weakly on the parameter c 2 of Helmholtz' equation (5.1). Even for negative c 2 good results can be obtained. 2

c

-20 *

rate

0.26

[

1

-1 5

.038

1

-10

- .) r

0

10

.032

.031

.030

.029

100

10000 .025

.027 1

Table 4 : rates of convergence ( Eq. (5.1), h=l/64, n =circle, point-wise relaxation ) If we replace the boundary co nd ition (5.2) by other conditions we obtain similar results. Table 5 shows the rates of convergence for various boundary conditions in the case of Poisson's equation.

prescribed boundary values

u

u [~ u u

rate of convergence

0 .038

au/ an

u.

per.

au !---i ~I J~ per. n :)er. an J U anL_J an U+-L u. au/ an per. 0 .023

0.038

0.032

Table 5 : rates of convergence (Eq. (5.1), c 2=0, h=l/64, a =unit square, point - wise relaxation ) In the case of Neumann or periodic conditions the solution is determ ined up to a constant. Nev ertheless, the multi-grid itera tion** works as well as for other boundary conditions.

-*

The first eigenva l ue of (5 . 1) for this n is

c 2 ~ - 23.14.

** The equation L0 u 0 =f 0 at the lowest level i =O must be completed by an additional equation (e.g . u 0 (P)=O for some Pcn 0 ).

COMPUTATIONA L METHODS FOR TURBULENT, TRANSONIC AND VISCOUS FLOWS

70

An ALGOL and FORTRAN program for the multi-grid solution of the general linear second order equation (1.2) in a rectangle subject to the boundary conditions mentioned above is contained in the report of Hackbusch 1977 .

5.2

Indefinite Problems

The convergence rates for boundary value problems with variable coefficients are very similar to t he results of §5.1. More interesting examples are indefinite boundary value problems (i.e . , problems with positive and negative eigenvalues). The equation a 11 u

X 1 X1

+a 2 2 u

X 2 X2

+a 2 u

X2

+( A+a)u = f

in

Q=(O,l) x (O,l) ( 5 . 3)

with u = 0

on

r

( 5 . 4)

has the first eigenvalue A=O. indefinite for A (X) This technique is ca lled " full multi-grid me thod " by Brandt 1977

***

At least of order

K

72

COMPUTAT IONAL M ETHODS FOR TU RBUL EN T, T RANS ON IC A N D V I SCOUS F LOWS

i· ~ O( N 0 ) !l k

= O( i.N

Jl ma x

)

(N Jl from §3.3)

operations . One can prove that ev en f or srnall i (us ually , i= l 2) the e r ror l/ U -u II i s of order O( hK) . Th er e for e , we o bta in Jl Jl II

u1 -u Jl 11

or

K

~ 0 ( h· )

takes O(N ) o per ation s

Jl

( 6 . 4)

1

instead of (6.2). NJl is the number of gr i d poin ts of step s i z e hJl . Fora proof compare Hackbusch 198 l b . We remark that in the case of Po i sso n' s eq uati on in a square, the algorithm (6.3) is even faster t han di rect Poisso n solve r s (e . g . of Buneman 1969, Schröder & Trottenberg 1973 ) . 7. 7.1

FURTHER MODIFICATIONS Extr a polation

The nested iteration tech nique of §6 yie l ds no t onl y a n appro xi· ma t i on t o u.Q. , but also approximations to u!l -l' u _ , etc. In some 1 2 ca ses discretizations admit an expansion A

K

u1 = u

h1 e K + O(h Jl ),

+

of th e g lobal error. val ue

u:~~rap

>..

>

(7.1)

K ,

Under assumption (7. 1 ), the extrapolated

' " ""•l oH

is of accu r acy

eK independent of hJl ,

+s u 1 _ 1 , B = 1/

rl- [h~:f} , •"

1- s ,

( 7 . 2)

O( h ~ ) .

Applyi ng extrapola t ion ( 7.2) to the res ul ts ~ 1 and u ! - 1 of the nested iteration, we ca n obtai n an O{h ~ )-appr ox imat io n to the co ntinuous solutio n u, provided that UJl and u t -1 are compute d up to an iteration er r or O(h ~ ) . The extrapolati on {7.2) is equivalent to the solution of a discrete proble m L 1 _ 1 u 1 _ 1 = f : ~~rap with an extrapo l ated right-hand s id e fext1 rap The co nst r uct io n of fext rap and the mu l ti-grid i Jl - 1 solution of the corres pond ing equation is called 11 T-extrapolation" by Brandt 1979 (p . 67 ). 7. 2

Multi-Grid Algo ri th m with De fect Correction Eve

if an expansion (7. 2 ) of t~e globäl err?r does not exist,

we can impr ove the accuracy of the d1screte solut1on u ~ by using a defect correct ion techn1que (cf . Brandt 1979 {~. 97) • Hackbusch 1979c ) .

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

73

Let h=h 1maxdenote the finest step size and assum e that we have two discretizations of the boundary value problem at the level 1max : ( 7. 3 a)

is a discretization of consistency order K· (7.3a) is used for the coarser grids, too. In addition, there must be a more accurate discretization 1

L

t max

u

(7 . 3b)

t max

of consistency order K'> K. For instance, in the case of Helmholtz' equation the first discretization (7.3a) may be the five-point scheme, while (7.3b) is the Mehrstellenformel * of Collatz 1966. In· this case we have K=2, K'=4. The formula (7.3b) may also contai n complicated discretizations at points near the boundary. Here, it is important that the scheme L' may be ünstable or even not int vert~ble. Li should be a very sparse matrix, whereas multiplication by Lt may require more computational work. Now we modify the multi-grid iteration as follows smoothing step : use a l ways Li (0 h1 > . .. > hJI.> ... we also write. (8 . 3)

.C Jl. ( uJI. ) = O

f or (8 . 2 ) with h

=

hJI. .

Th e re ma y be more than one solution of the co ntinuous and dis cre t e pr obl ems ( 8 . 1 ), ( 8 .3). In the following we fix one solution u* of ( 8 .1) and the re l ated solutio n u ~ of ( 8 .3) .

In order to de fine the nonlinear mu l ti-grid iteration, the system (8.3 ) mu s t be generalized to ( P small).

11 f Jl. 11.,.;; p

(8.4)

I f u ~ is an isolat e d solution of (8.3), the inverse mapping t he orem ens ur es th e e xistence of a unique solution uJI. of (8.4) in the ne i ghbourhood of u; , prov i ded that p> O is smal 1 enough . Formal l y, we wr ite u

8.2

JI.

u*

s olution of (8 . 4),

=

=

JI.

k is the scalar product of level k . at l e vel R. reads as follows :

Then the iteration

i

i-th iterate approximating the eigenfunction e

Ul

!.

u R.

.=

; i i < L1 u 1 ;u 1 > RJ11u II

2

2m

i

= (1- wi h i

\1

LAl )

(Rayleigh quotient);

ut

1

;

(11.Sa) (11.Sb)

{result of v iterations of smoothing by Jacobi 's i te ration);

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

d i := L;u u i d 9., - 1 := Q9, - 1 rd i

(defect ) ;

(ll. 5c)

{re stricted and projected defect);

(11.5d)

v t -l : approximate solution of the (singular)equation

LA

0

i -1' " i+l u „n :=

-lv t -l = d i -1;

uR, - pQ

.Q. -

85

lv 2 - 1

(cor rection step).

(ll.5e) (ll.5f) (11. 5g)

Note that the solution Q9., - 1v 9., - of (11.5f) is well-defined if l 1 d 2-1 €{ e i -1 } . (11 . 5) must be completed by an algorithm performing step (ll . 5e). Here we shall apply a multi-grid iteration for the almest s i ngular problem (11 .Sf} tha t is very similar to the linear multi-grid method of §3 . In order to understand the iteration (11.5), split the iterat e ul into vl+ ae i , where vii e 2 . If v ~ is sufficiently small compared with a , A from (11.5a) is a good approximation to the eigenvalue Ai · The part ae g, remains almest unchanged by the smoothing step (11 . SbJ and the coarse-grid correction in (11.5c-g) . But the term v ~ is reduced by the usual convergence rate of the linear multigrid iteration . A precise analysis is given by Hackbusch 1979a. The full process of solution is described by the following three programs: the nested iteration (11.6), the eigenvalue multi~rid ~ethod emgm performing (11 .5), and the ~inguTar ~ulti-grid ~ethod smgm performing step (ll.5e). Multi-grid method for the eigenvalue prob lern Li u i =A. u i approximate eigenfunction eo .. -k:=l 1 until i

at level 9.. = 0; for do ~ 2 begin ek - P*ek-1; l k - 1 : =< Lk - 1ek - 1 'e k - 1> k- 1 /II e k- 111 , (11.6) for j:=l ~ 1 unt il i do emgm (k,ek) end nested iteration; erocedure emgm( i ,u); beqin real ,}. · inteaer j ; arra ]'._ d, V; )., . - 1 (e 1 ) =h 1 ~ 1 ( e i ) = h 1 X EQ.2, X El"l 1 The eigenfunction e 2 can be normalized by the condition

*

*

(11 . 9) Combining this equation wit h (11.4) we obtain the equivalent nonlinear equation (11.lO)

Any eigenfunction e ~ norma l ized by (11.9) is a solution of (11.lO) Any solution u 1 of tll.10 ) i s an eigenfunction corresponding to th~ eigenva l ue ;\ = ~ i( u1). If A.t is a single eigenvalue, then the Solution u 1 of (11.10) i s isolated. The nonlinear problem (11 . 10) can be solved by the nonlinear mu l ti-~rid ite~ation of §8 . Minor but useful _modifications and an analys1s are g1ven by Hackbusch 1980b . Numer1cal examples, in Particular, the solution of the Stek l o v eigenvalue problem (11 . 3) is contained in the paper mentioned above. 12 . 12.l

MULTI-GRID METHOD OF THE SECOND KIND Equa t ion s of the second kind

The multi-grid method of the foregoing paragraphs (now named "multi - grid method of the first kind") has bee n applied to diff e r e nti a l eq uation s . In this sect i on we consider abstract equat ion s of th e sec ond kind. *

I n a ddition to (2.32) we need Gg, e 1 = e 1 (e .1

eigenfunction)

87

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

u

Ku + f

(linear case)

(12.1)

K ( u)

(nonlinear case)

(12. 2)

=

u

where K ( K ) is some linear (nonlinear) operator. A model problem for equation (12.1) is Fredholm 1 s integral equation of the second kind u(x)

=

f

(x En ).

k(x,y) u(y) dy + f(x)

(12.3)

Q

This integral equation is of the form (12.1) where K is the integral operator defined by (Ku)(x)

f

=

(1 2.4 )

k(x,y) u(y) dy.

Q

If k is continuously differentiable, the operator K maps continuous functions u into differentiable functions Ku. This is the essential condition we need in the following. A precise formulation of this smoothing effect of K is given by Hackbusch 198la, 198lc. In the nonlinear case we must assume that the derivative satisfies the condition mentioned above.

v . - K1

Let ut

=

Kt ut + f i

or

ui

=

Kt (u i ), respectively,

(12.5)

be a discretization of (12 .1 /2) for step size ht , where hi is a In case of problems (12.3), member of the sequence (3.1). equation (12.5) may be obtained e.g. by some quadrature formu la. In many other applications Ku is defined implicitly as the solution of a boundary value problem or of a parabolic differential equation. Then, Kt ut may be defined to be the discrete solution corresponding to the parameter hi . An examp l e is given in §12. 3. 12.2

Algorithm

We obtain the multi-grid algorithm of the second kind from the multi-grid algorithm of §3 (linear case) and § 6 (nonlinear c ase) by the following forma l replacements : replace L

i

by I-K

i

replace c i (u i ,f i ) by Ki u i +f i set v := l, y := 2.

by I- Ki ) ( by Ki (u i ) + f .Q. )

(.C i

The resulting algorithm is simpler than the multi-grid method of the firs t kind, since one need s not l oo k for a suitable choice of v and Y· Nevertheless, the convergence is usually much faster than for the multi-grid method of the first kind. The rate of convergence tends to zero for h t 0, whereas the rate of the first algorithm is uniformly bounded from above. More precisely,

88

COMPUTATIONAL METHODS FOR TURBULENT, TRANSONIC AND VISCOUS FLOWS

one can prove convergence

rate~

a

C h.e. ,

et

> 0'

where et depends on the increase of differentiability the consistency order of K.e. · The constant ·c de pends (i .e., on K). Rates of the size Ch ~,.;; 0.001 or even practical choice of h.e. are possible (compare examp l e section). 12.3

by K and on on the problem better for a of the next

Application to Differential Equations

The multi-grid algorithm of the second kind can be appl ied not only to integral equations but also to boundary value problems. As a simple example consider the nonlinear boundary value problem - 6u = f(u)

i n n,

u = O on r

with some nonlinear right-hand side f(u) and discretize the problem by - 6.Q. UR. = f(u .Q. ).

(12.6)

6.e. may be the five-point scheme. Assume that linear problem - 6i W.e.= 9.e. is easily solvable, for instance by a direct Poisson solver or by a linear multi-grid program. Define the discrete nonlinear mapping KR. by K.e. (u .e. )

= - 6~

1

f(u .e. ).

Then obviously, problem (12 . 6) is equivalent to (12.2) . The following numerical results are cited from Hackbusch 1979b . The boundary value problem is - 6 u = eu in n

=

(O,l) x (O,l), u

=

0 on r .

(12.7)

We choose the sequence ho = 1/2, h 1 = 1/4, . .. , hs = 1/64 and apply the nested iteration (8.17) with R. max = 5 and i = 1. Hence, we perform only one iteration of the nonlinear multi-grid method per level. At the mid point (1/2, 1/2) we obtain the fol lowing results. Level i

0 1 2 3

4

5

Grid Si ze 1/ 2 1/4 1/8 1/16 1/32 1/ 64

Tabl e 8

Resu lt of the Nested Iteration 0.066 0 .074 0.077 0.077 0 .078 0.078

819 715 200 872 043 086

05 481 649 724

685

Exact Discrete Solution ut 0.066 0 .074 0.077 0.077 0.078 0.078

819 716 206 874 044 086

77

541 047 063 769

Iteration Error

Di screti zat; on Error

-

1.131 0- 2 3.391 0-3 9.011 0-4 2.2810-4 5.731 0-5 1.431 0-5

1. 710 -6 6 . 110 - 6 1.410 -6 3.410 - 7 8 .410 - 8

Results of problem (12 . 7) at the midpoint (1/2,1/2)

89

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

The amount of computational work is mainly determined by perfor- 1 mance of the mapp i ng u + K (u ) = - ß exp(u ) . For computing -1 i i i i i ßi we used the direct Poisson solver of Buneman 1969. The follow ing table shows how often the Poisson solver must be performed for ob ta inin q all results of table 8 . Step Size

1/4

1/8

l/16

1/32

1/64

Number of Calls

48

24

12

8

2

Tab le 9 : number of calls needed for computing u 1164 by (B.17) In further papers of t he author one can find more applications of the multi-grid method of the second kind : applications to elliptic and integral eigenvalue problems, to optimal control problems for elliptic and parabolic equations, and to time-periodic par abolic diffe rential equations. Re ferences ASTRACHANCEV, G.P. ( 1971): An iterative method of solving elliptic net ' problems. Z. vycisl. Mat. Mat. Fiz., Vol. 11, pp 439 448 (= U.S.S.R. Comp. Math. Math. Phys . Vol. 11, No . 2, pp 171 -1 82). '

ASTRACHANCEV, G.P. (1976) : The it 0 rative imorovem ent of e i ge nvalues . Z. vycis l . Mat. Mat . Fiz . Vol. lS, No. l, pp 131 - 139 (= U.S . S.R . Comp . Math . Math. Phys ., Vol. 16, No. l, 123 -1 32). AZIZ, A.K . (1972) (ed . ): The mathema tical foundations of the finite element method with applications to partial d i fferential equat ions. Academic Press, New York. BACHVALOV, N.S. (1966): On th e convergence of a relaxation metho d with natural constraints on the elliptic operator. Z. vycisl. Mat. Mat. Fiz., Vol. 6, No. 5, pp 861 - 883 (= U.S.S.R Comp . Ma th. Math. Phys, Vol. 6, No . 5 , 10 1-135 ) . BANK, R.E. & DUPONT, T. ( 19 81): An optimal order process for sol ving elliptic finite element equations. Math. Comp. Vo l. 36, PP 35-5 1. BANK, R.E. & SHERMAN, A.H. (1981): An adapt iv e, multi-level meth od for elliptic boundary value problems. Computing, Vol . 26 , PP 91 -1 05. BERCOVIER, M. & PIRONNEAU, 0. (1979) : Error estimates for fin ite element method solution of the Stokes probl em in the pr imiti ve variables. Numer . Math . Vol. 33, pp 211-224 . BRAESS, D. (1981): The contraction number of a multi-grid meth od for solving the Poisson equation. Numer . Math. Vol. 37, pp 38 7-404. BRAMBLE, J.H. & HUBBARD, B. E. (1965) : Approximation of solution s of mixed boundary val ue problems for Poisson's equation by fin ite di fferences. J. of the ACM, Vol . 12 , pp 114-123.

90

COMPUTATIONAL METHODS FOR TURBULENT, TRANSONIC AND VISCOUS F LOWS

BRANDT, A. (1977): Multi-level adaptive solutions to boundaryvalue problems. Math . Comp. Vol. 31, pp 333-39 0 . BRANDT, A. & DINAR, N. (1979) : Multigrid so lu tions to el l ipti c flow problems. In: Numerical Methods for PDEs ( S. Parter, ed.). Academic Press, New York. BUNEMAN, 0. (1969): A compact non-iterative Poisson solver. SUIPR Report No. 294, Institute for Plasma Research, Stanford University, Cal iforni a . CIARLET, P.G . (1978): The finite element rnethod for elliptic problems. North-Holland, Amsterdam . COLLATZ, L. (1966): The numerical treatment of differential equations. 3rd ed., Springer, Berlin-Heidelberg-New York. FEDORENKO, R. P. (1961): A relaxation method for solving elliptic difference equations. Z vycisl. Mat. Mat . Fiz. Val. 1, No. 5, pp 92 2- 927 (= U.S.S.R . Comp. Math. Math. Phys. Vol 1, pp 1092-1096). FEDORENKO, R.P. (1964): The speed of convergence of dne iterative process. Z. vycisl. Mat. Mat. Fiz., Vol. 4, No. 3 pp 559 - 564 (= U. S.S.R . Comp. Math. Math. Ph~ 1 s. Vol. 4, No. 3, pp 227-235). FOERSTER, H.; standard multigrid mediate grids. In: Academic Press New

STUBEN, K.; TROTTENBERG, U (1980) Nontechniques using checkered relaxation and interElliptic Problem Solvers (M. Schultz, ed . ) York. ,

FREDERICKSON, P.O . (1975): Fast approximate inversion of large sparse linear systerns. Report 7- 75, Lakehead University. GIRAULT, V. and RAVIART, P.-A.(1979): Finite element approximation of the Navier-Stokes equations. Lecture Notes in Math. 7 49 Springer, Berlin-Heidelberg -New York. GUSTAFSSON, l. (1978): A class of first order factorization methods. BIT, Vol . 18, PP 142-156. HACKBUSCH, W. (1976) : Ein iteratives Verfahren zur schnellen Auflösung elliptischer Randwertprobleme. Report 76-12, Mathematische Institut, Universität zu Köln . HACKBUSCH, W. (1977): A multi-grid method applied to a boundary value problem with variable coefficients in a rectangle Report 77-17, Mathem. Institut, Universität zu Köln . · HACKBUSCH, W. (1978): On the rnulti-grid method applied to difference equations . Computing, Vol. 20, pp 291-306. HACKBUSCH, W. (1979a): On the computation of approximate e igenv alues and eigenfunctions of elliptic operators by means Of a multi-grid method. SIAM J. Numer. Anal., Vol. 16, pp 201-215.

NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS

91

HACKBUSCH, W. (1979b): On the fast solution of nonlinear el liptic equations. Numer . Math., Vol . 32, pp 83-95 . HACKBUSCH, W. (1980a): Convergence of multi-grid iterations applied to difference equations . Math . Comp. Vol. 34 , pp 425 - 440. HACKBUSCH, W. (1980b) : Multi-grid solutions to linear and nonlinear eigenvalue problems for integral and differential equations. Report 80 - 3 , Math. Inst., Un iversität zu Köln. HACKBUSCH, W. (1930c): Analysis and mult i-grid solutions of mix ed finite element and mi xed difference equations. Preprint Oct. 80, Math. Inst., Ruhr-Universität Bochum. HACKBUSCH, W. (1980d): Survey of convergence proofs for multi-grid iterations. In: Special Topics of Applied Mathematics (J. Frehse et al ., eds. ) . North -Holland, Amsterdam . HACKBUSCH, W. (198la) : Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art. Beiträge zur Numerischen Mathematik, Vol . 9, pp 47 -62 . HACKBUSCH, W. (198lb): On the convergence of multi-grid iterations . Beiträge zur Numer ischen Mathematik, Vol. 9, pp 213 - 239 . HACKBUSCH, W. (198l c ) : Error analysis of the nonlinear multigrid method of the sec ond kind. Aplikace Matematiky, Vol. 26, pp 18-29 . HACKBUSCH, W. ( 198l d) : Bemerkungen zur iterierten Defektkorrektur and zu ihrer Kombina tion mi t Mehrgitterverfahren. Rev. Roumaine Mat h. Pures Appl., Val. 2 6, pp 1319-1329. HACKBUSCH W. and HOFMA NN, G. (1980) : Results of the eigenvalue problem for the plate equation . ZAMP, Vol. 31, pp 730 -739. HACK BUSCH , W. an d T R0T TE MB ERG , U. ( e d s . ) ( 19 81 ) : Mul t i g r i d Methods 1 Proceedings, Köln Nov . 1981 . Lecture Notes in Math. 960, Springer, Berlin-Heidelber g- ew York (tobe published). HEMKER, P. W. (1980a): The incomplete LU-d ecomposition as a relaxation method in multi-grid algorithms. In: Boundary and Interior Layers-Computational and Asymptotic Method s (J.J . H. Miller, ed.). Boole Press, Dublin .

HEMKER, P . W. (1980b): On the structure of an adaptive multilevel algorithm. BIT, Val . 20 , pp 289 -301. HEMKER, P. W. an.:! SCHIPPERS, H. (1981) : Multigrid methods for the solution of Fredhol m equations of the second kin d . Math. Comp., Vol. 36, pp 215-232 . KRONSJÖ, L. (1 975): A note on th e "nested iteration 11 method . BIT, Vol. 15, PP 107-110. MALLINSON, G.D. a nd de VAHL DAVIS, G. (1 977 ): Three dimensional nat ural convectior. in a box: a numerical study . J. Fluid Mech., Vol. 82, pp 1-31.

92

COMPUTATIONAL METHODS FOR TURBULENT, TRANSONIC AND VISCOUS FLOWS

MEI s , T h . a nd MA RCO\H T Z , U. ( 1 9 7 8 ) : Nume r i s c ne Be h a n d l u n g partieller Different1algleichungen. Springer, Berlin-HeidelbergNew York. English translation: Numerical solution of partial differential equations. Springer, New York-Heidelberg-Berlin (1981).

MOL, W.J.A . (1980): Numerical solution of the Navier-Stokes equations by means of a multigrid method and Newton-iteration. Preprint NW 92/80, Mathematisch Centrum, Amsterdam. NICOLAIDES, R.A. (1977) : On the i 2 convergence of an algorithm for solving finite element equations. Math. Comp., Vol. 31, pp 892-906. NICOLAIDES, R.A. (1979): aspects of multigrid methods .

On some theoretical and practical Math . Comp. Vol. 33, PP 933 - 952.

OGANESJAN, L.A. and RUCHOVEC, L. .A„(1979): Variacionnoraznostnye metody resenija ellipticeskich uravnenij (Variation'aldifference methods of solving elliptic equations) . Isdatel-stvo Adadem i i Nauk Armjanckoj CCP, Erevan. SCHRODER, J . and TROTTENBERG, U. (1973): Reduktionsverfahren für Differentialgleichungen bei Randwertaufgaben I. Num. Math . , Vol. 22, PP 37-68 . SHORTLEY, G.H. and WELLER, R. (1938): Numerical solution of Laplace• s equation. J. Appl. Phys. Vol. 9, pp 334 - 348 . STRANG, G. and FIX, G.. l. (1972): An analysis of the finite element method. Prentice Hall, New York. THOMASSET, F. (1980): Fin i te element methods for Navier-Stokes equations. In: Lecture Series 1980-5, Computational Fluid Dynamics. von Karman Institute, Rhoue Saint Genese . VARGA, R.S. (1962): Matrix iterative analysis. Prentice Hall, New Jersey. WESSELIN G, P. (1977): Numerical solution of the stationary Navier-Stokes equations by means of a multip l e grid method and Newton iteration. Report NA-18, Delft University. WESS ELING, P. (198~a): The r~te of convergence of a multiple grid method. In: Numer1cal Analysis (G.A. Watson, ed . ) . Lect . Notes in Math. 773, Springer, Berlin-Heidelberg - New York . WES SE LING, P. (1980b): Theoret i cal and practical aspects of a mu ltigrid method. Report NA-37, Delft University. WESSELING, P . and SONNEVELD, ~· p980): Numerical experiments with a multiple g rid and a precond~t1oned Lanczos type method . In: Approxim ation methods for Nav1er-Stokes problems (R . Rautmann ed.), Lecture Notes in Mathematics. 771, Springer, Berlin Heidelberg-New York. ZIENCKIEWICZ O.C. (1977): The finite element method in engineering scie nce. MacGraw-Hill, New York.