Mathematical Aspects of Finite Elements in Partial Differential Equations. Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, April 1–3, 1974

418 Pages·1974·13.59 MB·English

by  Carl De Boor

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION
Mathematical Aspects of
Finite Elements in Partial
Differential Equations
Edited by Carl de Boor
Proceedings of a Symposium
Conducted by the Mathematics Research Center
The University of Wisconsin—Madison
April 1-3,1974
Academic Press
New York • San Francisco • London 1974
A Subsidiary of Harcourt Brace Jovanovich, Publishers
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Library of Congress Cataloging in Publication Data
Symposium on Mathematical Aspects of Finite Elements
in Partial Differential Equations, University of
Wisconsin—Madison, 1974.
Mathematical aspects of finite elements in partial
differential equations.
Bibliography: p.
Includes index.
1. Differential equations, Partial—Congresses.
2. Finite element method—Congresses. I. De Boor,
Carl, ed. II. Wisconsin. University—Madison.
Mathematics Research Center. III. Title.
QA374.S934 1974 515'.353 74-23389
ISBN 0-12-208350-4
PRINTED IN THE UNITED STATES OF AMERICA
Preface
This book contains the Proceedings of the Symposium on Mathematical
Aspects of Finite Elements in Partial Differential Equations held in Madison,
Wisconsin, April 1-3, 1974, under the auspices of the Mathematics Research
Center, University of Wisconsin-Madison, and with financial support from the
United States Army under Contract No. DA-31-124-ARO-D-462. The fourteen
speakers had been invited to address mathematical questions raised by the use
of finite elements in the numerical solution of partial differential equations. Due
to the necessity of meeting the publication schedule, the text of Professor Garth
Baker's fine lecture has not been included in this volume. The sessions were
chaired by:
Professor G. Birkhoff, Harvard University
Professor H. H. Rachford, Jr., Rice University
Professor A. H. Schatz, Cornell University
Professor R. S. Varga, Kent State University
Professor B. Wendroff, Los Alamos Scientific Laboratory.
The program committee consisted of Professors J. Bramble, J. Douglas, Jr.,
J. Nitsche, and B. Noble, with the editor as chairman. The many organizational
details were handled by the experienced symposium secretary, Mrs. Gladys G.
Moran, with intelligent dedication and inexhaustible enthusiasm. The preparation
of the manuscripts for publication was in the able hands of Mrs. Dorothy M.
Bowar.
ix
Higher Order Local Accuracy by Averaging in the Finite
Element Method
J. H. BRAMBLE AND A. H. SCHA TZ
1. Introduction.
Let W be a bounded domain in Euclidean N-space
IRN with smooth boundary aw. For u a rea\valued function
defined on W we shall consider the uniformly elliptic second
order differential operator
N
~u
Lu = - L cu
ax. ( aij ax. ) +
i,~=~ ) i
where a and c are assumed smooth. The associated bi-
ij
linear form is given by
N
B(v, w) = f a ax, 2x, dx + f c 1W dx
ij
For f e IL2(W), a weak solution u of
(1.1) Lu = f in
satisfies
(1. 2) B(u,f) = f f f dx = (f,f)
W
for all functions f which are continuous and piecewise con-
tinuously differentiable in W and which vanish near aw.
We can associate with (1. 1) various kinds of boundary
1
J. H. BRAMBLE AND A. H. SCHATZ
conditions. Examples of these are
a) u = O on aw
or
on aW
b) an
or
au
u = 0 on aW ,
c) an +
where
au 3u
an aij ax. ni
i, j=1 j
Here n is the component in the direction x of the outward
i i
normal to 3 and is called the co-normal derivative.
Let S be a linear space of "finite elements" and
h
uh e Sh an approximate solution to the boundary value prob-
lem (1. 1) with a), b) or c) satisfied. For many different
finite element methods proposed for these problems, the
interior equations are as follows:
(1. 3)
B(u ,f) = (f,w)
h
for all f e Sh which vanish near 312 . In many different
specific methods which have (1. 3) in common, estimates for
norms of the error u = u in Sobolev spaces on all of W are
h
well known. (For a summary of such results cf. [2]. ) Inte-
rior estimates in Sobolev norms for u - u only satisfying
h
(1. 3) were given in [5] and in maximum-norm in [1].
Here we shall consider, instead of uh as an approxi-
mation to u , certain "local averages of u ". These, as
h
will be seen subsequently, are formed by computing K * u ,
h h
where K is a fixed function and * denotes convolution.
h
As we shall see, the function K has the following proper-
h
ties:
2
HIGHER ORDER LOCAL ACCURACY BY AVERAGING
i) K has small support;
h
ii) K is "independent" of the specific choice of
h
S or the operator L;
h
iii) K *u is easily computable from u ;
h h h
iv) Kh*uh approximates u to higher order than
does u •
h
In the remainder of this paper, we shall describe the
class of finite element subspaces which we consider, state
,x
our main result on the accuracy of K u and discuss the
h h
si nificance of the result. A local estimate in a problem
g
with certain non-smooth data is presented as a conse uence
q
of the main results. Finally, we present a numerical example.
2. Notation, subspaces and the construction of K .
h
We shall denote by Cs(W) , s = 0,1, 2, ... , the
space of functions defined on W with uniformly continuous
partial derivatives of order up to and includin s on W .
g
For v e Cs(W) we set
lv ls = sup lDav(c) l
i W
N
where a is a multi-index, jJ = a and Da = (a/ac )al
i 1
a i=1
N
.. (8/ax1) . For s real we define H5(W), the Sobolev
space with index s and for v e H5(W), liv 11 will denote
its norm (cf. [4]). For example, for s = 0, 1, 2,' . , 11 . ll
is iven by s W
g
i f lDavl2dx)a .
s,U = S
la s W
3
J. H. BRAMBLE AID A. H. SCHATZ
For s a positive non-integer, Hs(W) may be defined by
interpolating between successive integers and for s < 0 by
duality (cf. [4]).
The one-parameter family of spaces {S } , 0 < h< 1
h
which we shall consider will be assumed to have the following
properties:
1) For each h , S C H1(W) and S is finite dimen-
h h
sional.
2) For x e W CC W and U e S there are functions
1 h
which are piecewise polynomial with
fl' ' ' 'f k ,
compact support such that
k
U(x) = S S a3 f.(h-lx-a) .
7=1 aE T N
Here W1 CC W means S21 C W , a are real co-
efficients and ZN are the multi-integers. (This
property may be described as an interior transla-
tion invariance property. )
3) For some positive integer r there is a constant
C such that for y e Hs(W) , 1 < s < r ,
inf ( II v-O II + h II v-f II ) < Chs n II
feSh 0,f 1,W II s,W'
4) Let W CC W and let w be an infinitely differen-
1
tiable function with support in W1 . There is a
constant C such that for v e S
h
inf Il wn- f II ~ < Ch Il v II .
1, 1,
e Sh
1
supp f CW
It was shown in [1] that subspaces consisting of tensor prod-
ucts of one dimensional splines on a uniform mesh have all
the requisite properties. Also, in [5], it was demonstrated
that the triangular element subspaces in 1R2 defined in [3]
are examples satisfying the above four conditions provided
4
HIGHER ORDER LOCAL ACCURACY BY AVERAGING
the triangulation is uniform. We emphasize that the uniform-
ity is a condition which is only required locally. Thus we
see that many of the finite element subspaces which are dis-
cussed in the literature satisfy the above conditions.
In order to define the function K we shall need to
h
introduce the so-called smooth splines. In fact we shall
choose K to be a particular smooth spline depending on the
h
index r associated with the subspace S .
h
For t real define
and for x e IRN set
N
Y(x) = P x~x.)
j=1
For I a positive integer set
Y(~)(x) _ (Y * ... * y)(x), (I -1) times.
)
The function (1 is just the N-dimensional B-spline of
Schoenberg [6]. The space of smooth splines of order .Q on
a mesh of width h consists of all functions of the form
U(c) = S a Y ,
N a
a e 7L
for some coefficients a .
s
The proof of the following will be given in a forth-
coming paper by the authors.
Proposition.
Let I and t be two given positive integers. The
smooth spline
k y(~)(h-1c
K (x) = -a)
h
e 7LN
5
J. H. BRAMBLE AND A. H. SCHATZ
may be chosen so that
a) k = 0 when Ia, > t-1 for some j ;
b) for W0 CC W1 and v e C2t(01) there is a
constant C such that
Iv- ~ < Ch2tInI
K nI0,
0 2t W
' 1
and
2t
c) for v e H (W ) there is a constant C such
1
that
1I n - Kh*vII0 U < Ch2t IIn II
' 0 2t w1
This function K is the aforementioned function in
h
terms of which our local averages will be defined.
0
Let us denote by Sh(0 ) the subspace of Sh whose
l
elements consist of functions in Sh with support in W1 .
Let us suppose now that u e S satisfies
h h
B(u-u f) = 0
h,
a
for all f e These equations are the same as (1. 3)
Sh(Ul).
provided Lu = f. We now state our main result. The proof
of this will be given in a forthcoming paper by the authors.
Theorem 1.
Let W CC W1 CC W and p be an arbitrary but fixed
0
real number. Let I = r-2 and t = r-1. Then there is a
constant C such that for u E H2r-2(Wl)
er-2 ~~ ~~
IIu-Kh* hu II 0, w < c{ h u 2r-2, W + IIu-u h I _ p, Q }
0 — 1
and, for u e 1s(0 ) with s = 2r-2 + LN/2j + 1 ,
l
6