Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold dna .B Eckmann
953
evitaretI Solution of
Nonlinear Systems of ~noitauqE
Proceedings of a Meeting held at
Oberwolfach, ,ynamreG Jan. 13 -Feb. ,5 1982
Edited yb .R Ansorge, .hT Meis, and .W TSrnig
galreV-regnirpS
Berlin Heidelberg New kroY 1982
Editors
Rainer Ansorge
Institut f~r Angewandte Mathematik, Universit~t Hamburg
Bundesstr. 55, 2000 Hamburg ,31 Germany
Theodor Meis
Mathematisches Institut, Universit~t KSIn
Weyertal 86-90, 5000 KSIn ,14 Germany
Willi TSrnig
Fachbereich Mathematik, TH Darmstadt
SchloBgartenstr. ,7 6100 Darmstadt, Germany
AMS Subject Classifications (1980): 65B05, 65F10, 65F15, 65G10,
65H10, 65H15, 65N 05, 65N 30, 70K10, 73D30, 76-04, 76D05,
76N10, 76S05
ISBN 0-387-11602-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-11602-8 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. llA rights era reserved, whether the whole or
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DROWEROF
ehT meeting no Iterative Solution of Nonlinear smetsyS of Equations, held in the
sehcsitamehtaM Forschungsinstitut Oberwolfach, Federal Republic of ,ynamreG
during the six days of January 13 st to February 5 th 1982, saw attented yb forty
eno mathematicians dna engineers from several countries. In all ~enty four lec-
tures were given, thirteen of which are presented in these proceedings.
sisahpmE saw no three main topics: multigrid methods, enotonom dna interval arith-
metic iterations, dna applications in industrial practice.
Several contmibutors reported no the effective esu of multigrid algorithms neve
in bifurcation dna other highly nonlinear problems. ehT principle of error in-
clusion yb snaem of interval arithmetics dna enotonom iterations sah been inves-
tigated for several years. Recent secnavda in accelerating those iterations dna
emos connections with the question of global convergence were reported no at the
meeting. Finally there were stimulating contributions dna discussions no concrete
numerical problems in scimanydorea dna emos other fields of engineering.
eW want to thank the director of the Oberwolfach Institute, Prof. Barner, ohw
evag su the opportunity to organize this meeting. eW also express our thanks to
.rD Gipser dna Dipl.-Math. Kaspar, ohw coordinated the production of the -unam
script, dna last but not least to the editors of the Lecture Notes series dna the
Springer-Verlag for publishing this volume.
,grubmaH K~In, dna Darmstadt, June 2891
.R Ansorge, .hT Meis, .W T~rnig
LIST SFROOTUBIRTNOC
Alefeld, .G Prof. Dr. Kaspar, .B Dipl.-Math.
Institut f. .wegnA kitamehtaM Fachbereich kitamehtaM
Universit~t Karlsruhe HT tdatsmraD
KaiserstraBe 21 SchloBgartenstraBe 7
oo57-D Karlsruhe oo16-D tdatsmraD
Axelsson, .O Prof. .rD Meis, .hT Prof. .rD
tnemtrapeD of scitamehtaM sehcsitamehtaM Institut
University of negemjiN Universit~t nI~K
5256-LN ne~emjiN Weyertal 68 - 09
ooo5-D K~In 14
,hcsubkcaH .W Prof. Dr. Mittelmann, .H .D Prof. .rD
Abteilung f. kitamehtaM Abteilung kitamehtaM
Universit~t muhcoB U~iversit~t dnumtroD
Universit~tsstr. 15o, Geb. AN Postfach 005005
o364-D grubnereuQ-muhcoB oo64-D dnumtroD 05
,gnunroH .U .rD ,reiamueN A. .rD
Institut f. ehcsiremuN dnu Institut f. .wegnA kitamehtaM
Instrumentelle kitamehtaM Universit~t Freiburg
Universit~t retsnUM Hermann-Herder-Str. lo
EinsteinstraBe 46 oo87-D Freibur~
oo44-D retsnUM
V
Nickel, .K Prof. .rD Potra, .F .A Prof. .rD
Institut f. .wegnA kitamehtaM tnemtrapeD of scitamehtaM
Universit~t Freiburg National Institute for Scientific
.rtS-redreH-nnamreH lo dna Technical Creation
oo87-D Freibur~ .dB Pacii 022
22697 Bukarest, ainamoR
,remmahteiN .W Prof. .rD Weiland, .C .rD
Institut f. Prakt. kitamehtaM eguezgulF-BBM HbmG
Universit~t Karlsruhe Postfach o611o8
EnglerstraBe 2 ooo8-D nehcnUM 08
oo57-D Karlsruhe
.rD .W renreW
hcierebhcaF kitamehtaM
Universit~t zniaM
SaarstraBe 12
oo56-D zniaM
CONTENTS
DIRGITLUM SDOHTEM ROF RAENILNON SMELBORP
.O Axelsson: nO Global ecnegrevnoC of Iterative sdohteM 1
.W :hcsubkcaH Multi-Grid Solution of Continuation smelborP 02
.H .D Mittelmann: A Fast Solver for Nonlinear'Eigenvalue smelborP 64
ENOTONOM SNOITARETI DNA LANOITATUPMOC RORRE SDNUOB
.H Cornelius dna .G Alefeld: A Device for the Acceleration of ecnegrevnoC 86
of a Monotonously Enclosing Iteration dohteM
.B Kaspar: Overrelaxation in Monotonically Convergent 08
Iteration sdohteM
.A :reiamueN Simple sdnuoB for Zeros of smetsyS of Equations 88
.K Nickel: saD Aufl~sungsverhalten nov nichtlinearen Fix- 601
nemetsyS-negnem
.F .A Potra: nO the ecnegrevnoC of a Class of Newton-Like 521
sdohteM
SNOITACILPPA DNA LAICEPS SCIPOT
._U :~nunroH sdohteM-IDA for Nonlinear Variational Inequali- 831
ties of Evolution
.G Kolb dna .W :remmahteiN Relaxation sdohteM for the Computation of the 941
Spectral mroN
.hT Meis dna .W :eksaaB Numerical Computation of Periodic Solutions of a 951
Nonlinear evaW Equation
.C Weiland: Erfahrungen bei der gnudnewnA numerischer Ver- 271
fahren zur gnus~L nichtlinearer hyperbolischer
Differentialgleichungssysteme
.W :renreW nO the Simultaneous Determination of Polynomial 881
stooR
ON GLOBAL CONVERGENCE OF ITERATIVE r~THODS
O. Axelsson
Department of Mathematics,
University of Nijmegen
The Netherlands
We review and extend results on the local convergence of the classical Newton-
Kantorovich method. Then we discuss globally convergent damped and inexact Newton
methods and point out advantages of using a minimal error conjugate gradient method
for the linear systems arising at each Newton step.
Finally application on a nonlinear elliptic problem is considered. A combina-
tion of nested iterations, dammed inexact Newton method and two-level grid finite
element methods for the solution of the linear boundary value problems encountered
at each step are discussed.
.I Introduction
For the solution of a nonlinear problem
F(x) = 0 , F : X ~- X,
X a Banachspace, to which we assume that there exists a solution x, we consider
methods on the form
(1.1) C k(x k+1 - kx ) = -TkF(xk).
Here C k is nonsingular and in some sauce makes CkIF(x k) locally close to the solu-
tion and approximately behave like x k - "x_ There are two main types of choices of Ck:
)i( If there exists a linear operator A such that II F(x_ k) - A?I is almost inde-
k
pendent on x , then we let C k = A. In this case the problem is almost linear and we
may use iterations of Picard type. An example is given in Section 2.
(ii) If F is Fr4chet differentiable, then we may let C k = F'(Xk) and (1.1) becomes
the classical Newton-Kantorovich method (with damping parameter ~k ) . We may also
let C k be an approximation of F' (xj).
The classical Newton method suffers from two disadvantages. Firstly, it is in
general only locally convergent. This is discussed in Section 3. Secondly, at each
step we have to solve a linear system of equations "exactly", and this may not be
justified in particular when the approximations of the nonlinear system are far from
the solution.
Hence we consider inexact Newton methods where the Newton step is calculated by
an iterative method and the iterations are stopped when the residual of the linear
system is small enough.
The global convergence is achieved when we use damped steplengths. Under the
assumption of nonsingularity of F', we prove that there exists steplengths such that
convergence is achieved for all initial approximations. However we point out that it
may take many steps before we can achieve the superlinear convergence, which charac-
terizes Newton type methods when we are close to a solution. This may be improved
upon by use of a minimal error iterative algorithm instead of a minimal residual al-
gorithm for the linear systems.
Finally we discuss an application on nonlinear boundary value problems. We pro-
pose a combination of a continuation method, through finer and finer meshes, with
the use of a damped Newton method to solve the nonlinear system at each grid.
The solution of the linear systems encountered at each Newton step, may be solved by
finite elements and a multigrid method of two-level type. If we want a diseretiza-
tion error O(h p) at the final mesh, with meshparameter h; the number of continuation
steps will be O(Ilog hi). The computational complexity at each mesh is of optimal
order, i.e. only proportional to the number of meshpoints. The order of the discre-
tization error for the nonlinear nroblem follows directly from that valid for the
corresponding linearized problems.
2. Picard iteration on a mildly nonlinear singular l~ perturbed boundary value
problem.
Consider the boundary value problem
2
-eV • ?u + b • Vu + cu = f(u) , x { 9 E R
u = 0 on ~.
~f
We assume that £ > 0, c ~ 0 and that ~u is bounded on ~. If the coefficients and the
~f
boundary are smooth enough there exists a solution in C2(~) and if ~u S 0 on ~, it
is unique.
We discretize the problem by finite differences in such a way that the discre-
tization operator h satisfies
I. h is monotone (for instance, h is of Dositive type and is positive definite).
2. There exists a barrier function w 2 0 such that i h w ~ 6 > 0 V~ 6 O h.
hl elU
Definition 2.1. Let : max l ul
~e~ h
We have then the following wellknown result, often called the Barrier-Lemma.
For every u on 0 h with u = 0 on ~h'
max w
~h
(2.1) Iul0 h N min L h w ILh ul~ h "
O h
For the construction of a barrier function we assume that the first component of
the velocity vector b = (bl,b 2) satisfies
bl(~) ~ b 0 > 0 Vx ~ O .
Let
w(x,y) = Ixl~ - x 2 + 3R(Ixl~ + x)
where R is the radius of a circle with center at the origin and which covers 0. We
let L h be a central difference operator with h small enough (needed for positivity)
or we use central differences for the second order term but upwind for the first
order terms. Then we have
w_>0
and
L h w = 2e + b1(3R - 2x + Oh) + cw
2e + R b 0 V(x,y) £ 0 h ,
f 0 for central difference scheme
where 0
i for upwind difference scheme.
It follows from (2.1) that
6R 2
12.2) lul% < - 2~+~ ° IL h ul%
We shall now solve the discretized equation
(2.3) i h u h = f(uh)
by Picard iteration,
, (i-l),
(2.4) i h u h (i) = ftuh , , i = 1,2, ...
(0)
where u h may be arbitrarily chosen. From (2.3), (2.4) we get
(i)) ~f (i-i)
Lh(Uh - Uh = ~u (Uh - Uh )
and by (2.2) we then get
ul h u~i) I 6R2 I~-u(U(.))~f t
~ - - max
- ~h 2c+R b 0 u ~h uh - u~i-l) "I
Hence, if I~I is small enough
~l h
ju h - u~ )i ~ plu h - u~i-l) I9 h , i = 1,2 ....
)0(
where 0 < p < I , and we have convergence for every u h
Naturally, J~J small means that the problem is almost linear.
3. Newton-Kantorivich method, local convergence.
Simplified proofs of classical results on local convergence of Newton-Kantoro-
vich method (see Kantorovich 6, Rall i03, and Ortega, Rheinboldt 9) are pre-
sented. The results are extended.
We assume that the mapping F : X + X, where X is a Banach space, is continuous-
ly Fr~chet differentiable with derivative F' nonsingular for all $ E X, with bound
lll-)!(
B = maxll F' .
~cx
Further we assume that F' is Lipschitz continuous,
(3.~) II~'(i) -F'(~)II ~KII!-~ll V~,n~ x,
and that there exists a solution ~ of F(~) = 0.
(A solution exists, and is unique, if in addition F is norm-coercive, i.e.
limll F(~)II = +~ , II II~ ÷ =-) Let {67 } be the Newton sequence
+I: -I , k: 0,I
....
with ~O given. We have
Theorem 3. .i
Assume that d = ½~KII ! 1 - !011 < .I
Then for any F satisfying the above conditions, we have
2 ~ d(2 )i
Proof. Using the integral relation
1
(3.3) )_~(F = F(i) + F'(~) (n-~) + {f F'(~ + t(D-~)) - F'(!)dt}(~-- i) ,
0
