AD-A260 013 MARYLAND COLLEGE PARK CAMPUS A PRIORI ERROR ESTIMATES OF FINITE ELEMENT SOLUTIONS OF PARAMETRIZED NONLINEAR EQUATIONS by Takuya Tsuchiya and Ivo Babufka DTIC S ELECTE FEB05 1993D E Technical Note BN-1142 . 75;;;;d for public releosol November 1992 INSTITUTE- FOR [PVIYSICAL SCIWNCC AND TCI-INOLOCY 93-02063
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AD-A260 013
MARYLAND
COLLEGE PARK CAMPUS
A PRIORI ERROR ESTIMATES OF FINITE ELEMENT SOLUTIONS
OF PARAMETRIZED NONLINEAR EQUATIONS
by
Takuya Tsuchiya
and
Ivo Babufka DTICS ELECTE
FEB05 1993DETechnical Note BN-1142
. 75;;;;d for public releosol
November 1992
INSTITUTE- FOR [PVIYSICAL SCIWNCCAND TCI-INOLOCY
93-02063
SECURITY CLASSIFICATION OF THIS PAGE (When D04N gntEo)
REPORT DOCUMENTATION PAGE ROND OMPEUCTINoRS3l1IORll COMPLETING FrORMl
turning points, finite element solutions, a priori error estimates
2•., ABSTRACT (Coms we on r.wee alde II aoesesomp mu IdWoul r by AleS mnm•)
In this paper, a priori error estimates of finite element solutions of second order parametrized stronglynonlinear equations in divergence form on one-dimensional bounded intervals are studied. The Banachspace W" is chosen in formulation of the error analysis so that the nonlinear differential operatorsdefined by the differential equations are nonlinear Fredholm operators of index 1. Finite elementsolutions are defined in a natural way, and several a priori estimates are proved on regular branchesand on branches around turning points.
DO I FoAN 1473 E9VTI1oN OF I MOV so is ONEoT
S,'N 0102" LP 0!4- 601 SECURITY tLAII&FCATIOO OF THIS PAGE (Wfha Dao 8n,6tl
A Priori Error Estimates of Finite Element Solutions ofuf•announced []
Parametrized Nonlinear Equations Justification
Takuya Tsuchiya t Ivo Babugkat By........Distribut;•ion/ .............Availability Codes
Dist IAvail and/ori Special
Abstract. Nonlinear differential equations with parameters are called parametrized non-
linear equations. This paper studies a priori error estimates of finite element solutions of second
order parametrized strongly nonlinear equations in divergence form on one-dimensional bounded
intervals. The Banach space W0 ' is chosen in formulation of the error analysis so that the
nonlinear differential operators defined by the differential equations are nonlinear Fredholm op-
erators of index 1. Finite element solutions are defined in a natural way, and several a priori
estimates are proved on regular branches and on branches around turning points. In the proofs
the extended implicit function theorem due to Brezzi, Rappaz, and Raviart [Numer. Math., 36,
turning points, finite element solutions, a priori error estimates
AMS(MOS) subject classifications. 65L10, 65160
Abbreviated title. FE Solutions of Parametrized Nonlinear Equations
t Department of Mathematics, Faculty of Science, Ehime University, Matsuyama 790, Japan.Research was partially supported by National Science Foundation under Grant CCR-88-20279.
* IPST, University of Maryland, College Park, MD 20742-2431, U.S.A. Research waspartially supported by the U.S. Office of Naval Research under Grant N00014-90-J- 1030 andthe National Science Foundation under Grant CCR-88-20279.
I
1. Introduction.
Let X, Y be Banach spaces and A C R' a bounded interval. Let F: A x X -. Y be a smooth
operator. The nonlinear equation
(1.1) F(A,u) = 0,
with parameters A E A is called parametrized nonlinear equations.
Let (A, u) E A x X be a solution of (1.1). Intuitively, the set of the solutions of (1.1) would
form n-dimensional hypercurves in the Banach space R' x X. If D,,F(A, u) E C(X, Y), the
Frlchet derivative of F with respect to u, is an isomorphism, then, by the implicit function
theorem, the above intuition is correct, i.e. there exists a locally unique branch of solutions
around (A, u), and the branch is parametrized by A. Such branches on which D,,F(A,,u) is
isomorphism at each (A, u) are called regular branches.
However, if DUF(A, u) is not an isomorphism, the state of equilibrium defined by (1.1)
becomes unstable and the behavior of the solutions is unpredictable; the hypercurve of the
solutions might be a fold, or there might be several hypercurves of solutions intersecting at that
point. The folding points are called turning points. The points at which the hypercurves of
solutions are intersecting are called bifurcation points. (Note that the definition of bifurcation
points given by some authors includes turning points.)
In this paper we deal with the parametrized nonlinear equation F : A x H'(J) - H-'(J)
with one parameter A E A defined by
(1.2) F(A, u) = O, (A, u) E A x Hl(J),
(1.3) < F(A, u), v >:= I[a(Az,u'(z))v'+ f(A, x,u(z))vldz, Yv E H•(J),
where J := (b, c) C R is a bounded interval, and a, f : A x J x R -- R are sufficiently smooth
functions. Since F is a second order differential operator in divergence form, finite element
solutions of (1.2) are defined in a natural way.
Brezzi, Rappaz, and Raviart [BBR1-3] presented a comprehensive work on the numerical
analysis of parametrized nonlinear equations. They first proved an extended implicit function
theorem with error estimates on Banach spaces. Then, using the implicit function theorem.
they obtained several results of a priori error estimates of finite element solutions (BRRI,2]. In
[BRR3], they considered approximation of solution branches around bifurcation points, which
will not be dealt with in this paper.
2
Following [BRRI-3], Fink and Rheinboldt released several papers about numerical analysis
of parametrized nonlinear equations ([FRI,2], [R], and references therein). While the formulation
of [.BRR1-31 was rather restrictive, Fink and Rheinboldt developed their theory of a priori error
estimates of numerical solutions in a very general setting using the theory of differential geometry.
Fink and Rheinboldt employed the theory of Fredholm operators. Let X and Y be
Banach spaces and F : X -. Y a differentiable mapping. Then, F is called Fredholm on an
open set U C X if the Frichet derivative DF(z) satisfies the following conditions at any Z E U:
(1) dimKerDF is finite,
(2) ImDF is closed,
(3) dimCokerDF is finite.
We must note that, in the above prior works by Brezzi, et al. and Fink-Rheinboldt, only
mildly nonlinear equations were considered. If a(A, x,y) in (1.3) is nonlinear with respect to
y, the operator F is called strongly nonlinear (quasilinear), otherwise it is called mildly
nonlinear (semilinear).
Following the above prior works, we here develop a thorough theory of a priori and a
posteriori error estimates of finite element solutions of (1.2) on regular branches and on branches
around turning points in the case that the number of parameters is one, that is, A C R. Since
our formulation of parametrized nonlinear equations includes strongly nonliear equations, our
theory is an essential extension of the prior works.
In this paper we present the theory of a priori error estimates. In [TBI] the theory of a
posteriori error estimates and several numerical examples will be given. In the following the
outline of this paper is described.
First, we show that the exact and finite element solutions of (1.2) form one-dimensional
smooth manifolds. If F is mildly nonlinear, showing that solutions form manifolds would not
be very difficult. If F is strongly nonlinear, however, it would become very difficult, or F would
not be even differentiable in A x Hi(J).
Therefore, we redefine (1.2) and (1.3) using the Sobolev space W"'¶(J). Then, F becomes
as smooth as the functions a and f, and it is a Fredholm operator in a certain open set. From the
Fink-Rheinboldt theory, we conclude that the exact and finite element solutions form smooth
manifolds under suitable conditions.
Next, we prove several a priori estimates of finite element solution manifolds of (1.2) using the
3
extended implicit function theorem due to Brezzi, Rappaz, and Raviart [BRRI]. As mentioned
before, we need to take the Sobolev space Wl'°(J) as the stage of the error analysis of finite
element solution manifolds. However, using W",00(J) in the formulation make the finite element
analysis difficult. So we have to come up with several new tricks to overcome this difficulty. The
following is the most essential trick:
Since our operator F is defined on WI,"(J), its Fr~chet derivative D.F is a linear operator
on W"¶00(J). However, D,,F can be extended to an element of C(HO, H-') and thus the usual
theory of finite element can be applied to DuF.
Another new idea is 'rotation' or 'pivoting' of the coordinate to handle turning points. In
[BRR2], a slightly different formulation from that of [BRRI] was used to deal with turning
points. In Fink-Rheinboldt's theory, certain isomorphisms were introduced in the formulation
so that both regular branches and branches around turning points were treated simultaneously.
In this paper, we put an auxiliary equation in the original equation (1.2) so that the enlarged
operator is an isomorphism between Banach spaces around turning points or on *steep slope'.
Then we do the same thing what we do on regular branches to the extended operator.
In this paper one-dimensional case is discussed. Under certain assumptions the results
obtained here will be extended to two-dimensional case in [TB2].
This paper is a revision of a part of one of the authors' Ph.D. dissertation IT].
2. Preliminary.
In this section we prepare notation and a necessary lemma.
Let J := (b,c) C R be a bounded interval. For a positive integer m and a real p E [I, 0],
we denote by W'P,(J) the usual LP-Sobolev space of order m, that is,
W m,(J) {u E LP(J) IDku E LP(J),0 < k < m
We define the norm of W m .P(J) by
IIUllw'., := E IID'UIll,.,k=O
For p E (1, ool, we define the closed subspace Wý' (J) by
W1' (J) := { E W',P(J) ju = 0 on 8}J
As usual, we denote W- 2(J) and W0 ' (J) by Hm (J) and H'(J), respectively.
4
Note that COOO(J), the set of infinitely many times differentiable functions with compact
supports, is dense in Wl'*(J) for p, I < p < oo, but if p = oo, CO'(J) is not dense in W"'°"(J).
By the Poincar6 inequality, the norm
(2.1) II-4jW., := IIU'IILP
is equivalent to the norm 11 IIw,., in WO"P(J). We always take the norm (2.1) for WO'P(J) in
this paper.
For 1 < q < cc and p with 1 + 1, let W-",P(J) be the dual space of Wl'"(J) with the
norm
IIFIIw-,., : sup I p< F, x >q 1, F E W-'1P(J),IIXIIw.I.9=1
where p< .,. >9 is the duality paring between W-",P(J) and WOfr9(J). Then we have
Lemma 2.1. For any F E W-' 1P(J) with I < p <_ co, there exists a unique u E Wl"P(J)
so that
p< F, v >q= uV'dx, Vv E W0,(j). 3
Lemma 2.1 is a direct consequence of [B, Proposition VIII.13].
In notation of this paper, we omit '(J)' from the notation of Sobolev spaces when there is
no danger of confusion. Also, we write < .,. > instead of p <,. >q when the setting of the
duality paring is obvious.
Subscripts like a. and f,% stand for partial derivatives with respect to x and A, respectively.
3. Formulation of the Problem.
In this section we formulate our problem rigorously. To do this we define the nonlinear operator
[B] H. BREZIs, Analyse fonctionnelle: Thiorie et applications, Masson, 1983.
[C] P.G. CIARLET, The Finite Element Methods for Elliptic Problems, North-Holland,
1978.
[FRI] J.P. FINK AND W.C. RHEINBOLDT, On the Discretization Error of Parametrized
Nonlinear Equations, SIAM J. Numer. Anal., 20 (1983), pp.732-746.
[FR2] J.P. FINK AND W.C. RHEINBOLDT, Solution Manifolds and Submanifolds of
Parametrized Equations and Their Discretization Errors, Numer. Math.. 45 (1984),
pp.323-343.
[GS] M. GOLUBITSKY AND D.G. SCHAEFFER, Singularities and Groups in Bifurcation
Theory, Vol. I, Springer-Verlag, 1985.
[H] J. HUGGER, Computational Aspect of Numerical Solution of Nonlinear Parametrized
Differential Equations with Adaptive Finite Element Method, Ph.D. dissertation. Uni-
versity of Maryland, 1990.
[R] W.C. RHEINBOLDT, Numerical Analysis of Parametrized Nonlinear Equations. Wiley,
1986.
[Tj T. TSUCHIYA, A Priori and A Posteriori Error Estimates of Finite Element Solutions ofParametrized Nonlinear Equations, Ph.D. dissertation, University of Maryland, 1990.
[TBI] T. TSUCHxYA AND I. BABU9KA, A Posteriori Error Estimates of Finite Element So-
lutions of Parametrized Nonlinear Equations, submitted.
[TB2] T. TSUCHIYA AND I. BABUýXA, Error Estimates of Finite Element Solutions of
Parametrized Nonlinear Equations: Two-dimensional Case, in preparation.
32
The Laboratory for Numerical Analysis is an integral part of the Institute for PhysicalScience and Technology of the University of Maryland, under the general administration of theDirector, Institute for Physical Science and Technology. It has the following goals:
"* To conduct research in the mathematical theory and computational implementation ofnumerical analysis and related topics, with emphasis on the numerical treatment oflinear and nonlinear differential equations and problems in linear and nonlinear algebra.
"* To help bridge gaps between computational directions in engineering, physics, etc., andthose in the mathematical community.
"* To provide a limited consulting service in all areas of numerical mathematics to theUniversity as a whole, and also to government agencies and industries in the State ofMaryland and the Washington Metropolitan area.
"* To assist with the education of numerical analysts, especially at the postdoctoral level,in conjunction with the Interdisciplinary Applied Mathematics Program and theprograms of the Mathematics and Computer Science Departments. This includes activecollaboration with government agencies such as the National Institute of Standards andTechnology.
"* To be an international center of study and research for foreign students in numericalmathematics who are supported by foreign governments or exchange agencies(Fulbright, etc.).
Further information may be obtained from Professor I. Babuika,Chairman, Laboratory forNumerical Analysis, Institute for Physical Science and Technology, University of Maryland, CollegePark, Maryland 20742-2431.