Superconvergence in Galerkin Finite Element Methods (Lecture Notes in Mathematics, 1605) 9783540600114, 3540600116

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

1605

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

1605

Springer

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Lars B. Wahlbin

Superconvergence in Galerkin Finite Element Methods

Springer

Author Lars B. Wahlbin Department of Mathematics White Hall Cornell University Ithaca, NY 14853,USA E-mail: [email protected]

Library of Congress Cataloging-in-Publication Data. Wahlbin, Lars B., 1945Superconvergence in Galerkin finite element methods/Lars B. Wahlbin. p.cm. (Lecture notes in mathematics; 1605) Includes bibliographical references (p. -) and index. ISBN 3-540-60011-6 (acid-free) 1. Differential equation, Elliptic - Numerical solutions. 2. Convergence. 3. Galerkin methods. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1605. QA3. L28 no. 1605 [QA377] 510s-dc20 [515' .353]

Mathematics Subject Classification (1991): 65N30, 65N 15

ISBN 3-540-60011-6 Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany SPIN: 10130328

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Preface. These notes are from a graduate seminar at Cornell in Spring 1994. They are devoted mainly to basic concepts of superconvergence in second-order timeindependent elliptic problems. A brief chapter-by-chapter description is as follows: Chapter 1 considers onedimensional problems and is intended to get us moving quickly into the subject matter of superconvergence. (The results in Sections 1.8 and 1.10 are new.) Some standard results and techniques used there are then expounded on in Chapters 2 and 3. Chapter 4 gives a few selected results about superconvergence in L2-projections in any number of space dimensions. In Chapter 5 we elucidate local maximumnorm error estimates in second order elliptic partial differential equations and the techniques used in proving them, without aiming for complete detail. Theorems 5.5.1 and 5.5.2 are basic technical results; they will be used over and over again in the rest of the notes. In Chapters 6 through 12 we treat a variety of topics in superconvergence for second order elliptic problems. Some are old and established, some are very recent and not yet published. Some of the earlier contributions have benefitted from later sharpening of tools, in particular with respect to local maximum-norm estimates. In Chapter 6 we consider tensor-product elements. Using ideas of [Douglas, Dupont and Wheeler 1974b] we show that in some situations one-dimensional superconvergence results automatically translate to several dimensions. Chapter 7, "Superconvergence by local symmetry", presents recent fundamental results from [Schatz, Sloan and Wahlbin 1994]. Chapter 8 treats difference quotients for approximating derivatives of any order on translation invariant meshes. Here we follow the basic ideas of [Nitsche and Schatz 1974, Section 6]. In Chapter 9 we briefly comment on how, in many cases, results about superconvergence in linear problems automatically carryover to nonlinear problems. The essential idea is from [Douglas and Dupont 1975], which in turn is essentially the quadratic convergence of Newton's method. Chapter 10 is concerned with superconvergence on curved meshes which come about via isoparametric mappings of straight-lined meshes; it is based on [Cayco, Schatz and Wahlbin 1994]. Chapter 11 reverts back to the seventies. It is mainly concerned with the K -operator of [Bramble and Schatz 1974]; the presentation follows [Thomee 1977]. We give an application to boundary integral equations, [Tran 1993]. Also, we briefly mention a method for obtaining higher order accuracy in outflow derivatives, [Douglas, Dupont and Wheeler 1974a], and an averaging method of [Louis 1979]. Finally, in Chapter 12, we review the computational investigation of [Babuska, Strouboulis, Upadhyay and Gangaraj 1993] and comment on it in light of the theories of Chapters 6 and 7. Previous treatises of superconvergence are [Chen 1982a] and [Zhu and Lin 1989]. Both are in Chinese and, although there is a description of Zhu and Lin's book in Mathematical Reviews, it is hard for me to judge how they compare with the present account. The use of local maximum-norm estimates seems common to all three. For what appears to be a major difference of approach, see Remark 7·4·i. Surveys of superconvergence with a more limited scope have appeared in [Krizek and Neittaanmiiki 1987a] and [Wahlbin 1991, Chapter VII]. Let me next list some topics that are not included in these notes. The first is superconvergence in collocation finite element methods for differential equations. For

vi

this topic I have not even included references; the reader is referred to [Krizek and Neittaanmaki 1987a]. The second omitted topic is superconvergence in boundary integral methods or in integral equations (except for Section 11.5). Third, extrapolation methods, involving computation on two or more meshes, and fourth, the use of superconvergence in construction of smooth stress fields and the related use of superconvergence for a posteriori error estimation and adaptive refinement. For the second through fourth topics I have included a fair number of references so that the interested reader may, by glancing through the list of references, easily gain an inroad to the literature on the subjects. Finally, I have not considered superconvergence in Galerkin finite element methods for time­dependent problems. Here, though, I have included all references that I know of. The references are likewise "complete" with respect to the mathematical literature for the main topics treated. Of course, many of these references touch only briefly on superconvergence. In preparing the references I have used Mathsci Mathematical Reviews since 1972 on line) which does not systematically cover the vast engineering literature. The number of references given is more than it makes sense to actually refer to in the text, unless one resorts to plain listing, which I have not done. I hope that nevertheless some readers will find the list of references useful. (As an example, if a reader interested in early history wants to find references to papers on superconvergence in finite element methods before 1970, our list gives: [Stricklin 1966], [Filho 1968]' [Stricklin 1968]' [Oganesyan and Rukhovetz 1969J and [Tong 1969].) Basic discoveries continue to be made at present. Furthermore, the theory of superconvergence is very immature in carrying results up close to boundaries (or, internal lines of discontinuity). Today, such investigations are carried out on a caseby­case basis, and, of course, not all results hold all the way up to boundaries (see Section 1.7). Most results that are proven up to boundaries in several dimension pertain to axes­parallel parallelepipedes, or, locally, to straight boundaries. For these and other reasons I have decided to offer these notes essentially as they were written week­by­week during the seminar, rather than rework them into "text­book" form; such a textbook would most likely be out­of­date when it appears. The reader may be warned that, reflecting my lecturing style, proofs often appear before theorems; indeed, the "theorem" may be only an informal statement. Also, true to the principle that repetition is the mother of studies, there is a fair amount of such. E.g., symmetry considerations are first met with in two­point boundary value problems, then in L 2-projections and finally in multidimensional elliptic problems; difference quotients first ocur in two­point boundary value problems and later in many dimensions; and, tensor product elements are considered as well for L2-projections as for elliptic problems. Let me penultimately remark that there are three concepts which may be confused with superconvergence. "Supraconvergence" is a concept in finite­difference theory for irregular meshes, cf. [Kreiss, Manteuffel, Swartz, Wendroff and White 1986], [Manteuffel and White 1986], [Heinrich 1987, p. 107], and cf. also [Bramble 1970J. "Superapproximation" will be explained in these notes and "superlinear convergence" occurs in the theory of iterative methods, cf. e.g. [Ortega and Rheinboldt 1979, pp. 285 and 291].

vii

Finally, I thank Arletta Havlik for her superior typing, Al Schatz for many stimulating discussions, Stig Larsson for timely help and the National Science Foundation for financial support. Ithaca, January 1995 Lars B. Wahlbin

Table of Contents. Chapter 1. Some one-dimensional superconvergence results. 1.1. Introduction 1.2. Nodal superconvergence for function values in continuous elements (IJ- = 0). 1.3. Reduction to a model problem. 1.4. Existence of superconvergence points in general. 1.5. Superconvergence for interior points of mesh-intervals for continuous elements (IJ- = 0). 1.6. Superconvergence in derivatives at points about which the meshs are locally symmetric (r even). 1.7. Necessity of staying C 1hln1/h away from the boundary for superconvergence in the uniform mesh case (IJ- = 1). 1.8. Finding all superconvergent points for function values and derivatives in the case of a locally uniform mesh for IJ- = 1 and r even (with a remark about smoothest cubics). 1.9. Superconvergence in function values at points about which the meshes are locally symmetric (r odd). 1.10. Finding all superconvergent points for function values and derivatives in the case of a locally uniform mesh for IJ- = 1 and r odd. 1.11. First order difference quotients of Uh as superconvergent approximations to u' on locally uniform meshes. 1.12. Two examples of superconvergence by "iteration". 1.13. A graphical illustration of superconvergence.

Chapter 2. Remarks about some of the tools used in Chapter 1. 2.1. 2.2. 2.3. 2.4.

Inverse estimates. On approximation theory, and duality. Superapproximation. A typical combination of inverse estimates and approximation theory used in Chapter 1.

Chapter 3. Local and global properties of Lrprojections. 3.1. Assumptions. 3.2. Estimates for L 2-projections. Chapter 4. Introduction to several space dimensions: some results about superconvergence in L 2-projections. 4.1. Negative norm estimates and existence of general superconvergence points for function values. 4.2. Superconvergence in Lrprojections on n-dimensional tensor product spaces. 4.3. Superconvergence by symmetry in L 2-projections.

1

1

3 4 8 8

10

13

14 16

19 22 25 26 28 28 30

32 35 36 36 38 42 42 43 44

x

Chapter 5. Second order elliptic boundary value problems in any number of space dimensions: preliminary considerations on local and global estimates and presentation of the main technical tools for showing superconvergence. 5.1. Introduction. 5.2. Existence of superconvergence points in general: an example (also an example of a multi-dimensional duality argument). 5.3. General comments on local a priori error estimates. 5.4. General comments on L oo estimates. 5.5. The main technical tools for proving superconvergence in second order elliptic problems in several space dimensions. Chapter 6. Superconvergence in tensor-product elements. 6.1. 6.2. 6.3. 6.4. 6.5.

Introduction. Superconvergence in derivatives for the case of the Laplacian. Negative norm estimates for U - Uh: Examples. Superconvergence in derivatives for the case of (6.1.1.). Superconvergence in function values for the Laplacian and r 2: 3.

Chapter 7. Superconvergence by local symmetry.

48 48 49 52 58 62 65 65 65 69 70 72 74

Introduction. The case of a symmetric form with constant coefficients. The general case of (5.1.3) with variable smooth coefficients. Historical remarks.

74 74 78 79

Chapter 8. Superconvergence for difference quotients on translation invariant meshes.

84

7.1. 7.2. 7.3. 7.4.

8.1. 8.2. 8.3. 8.4.

Introduction. Constant coefficient operators and unit separation, d 1. Constant coefficient operators and general separation d. Variable coefficients.

Chapter 9. On superconvergence in nonlinear problems. Chapter 10. Superconvetgence in isoparametric mappings of translation invariant meshes: an example.

84 86 89 89 93 98

10.1. Introduction. 10.2. Superconvergence in difference quotients for first derivatives.

98 101

Chapter 11. Superconvergence by averaging: mainly, the K -operator. 11.1. Introduction 11.2. Preliminaries on Fourier transforms and multipliers. 11.3. The K -operator in general. 11.4. The K -operator applied to finite element approximations in second order elliptic problems. 11.5. Boundary integral equations and the K -operator: an example. 11.6. Remarks, including some other averaging methods. 11.7. A superconvergent "global" averaging technique for function values.

107 107 107 111 115 116 121 123

xi

Chapter 12. A computational investigation of superconvergence for first derivatives in the plane. 12.1. Introduction. 12.2. Proof of (12.1.5), (12.1.6) and precise definition of the principal error term ib, 12.3. Results of computational studies, with comments. References. Subject index.

125 125 128 132 136 165

Chapter 1. Some one-dimensional superconvergence results. We shall move briskly into the subject of superconvergence. In Chapters 2 and 3 we shall then backtrack and elucidate some of the techniques and results used in this chapter. 1.1. Introduction.

Consider the following two-point boundary value problem: Find u that

{

(1.1.1 )

-(a2(X)u')' - (al(x)u)' u(O) = u(l) = O.

+ ao(x)u =

= u(x) such

f in 1= (0,1),

We shall assume throughout this chapter that the coefficients a; and the right hand side f are as smooth as necessary on I for our analyses to carry through. Furthermore, we demand that (1.1.2) o

The weak formulation of (1.1.1) is to find u E HI(I) such that

(1.1.3)

A(u, X)

Let T h , 0

=1(a2 U'x' + alUX' + aoux)dx = 1fxdx,

for all X E 1JI(I).

< h < 1/2, be a sequence of subdivisions of I, Th =

(1.1.4)

0

= Xo < Xl < ... < XN = 1, N(h)-l

Ii=(Xi,Xi+l),I=

U

1;.

i=O

Let hi = Xi+l - Xi. With, perhaps, a slight abuse of notation, we shall let h = max, hi. With integers 0 ::; f.-l < r - 1, we set (1.1.5)

o

Sh =

0

= {X(x) : X E CI-'(I) n Com, x(O) = x(l) = 0,

xiIi

E IIr-I(Ii ) }

where II r- 1 (Ii) denotes the polynomials of degree j, r - 1 on 1;. When no confusion o

0

0

HI(I) since f.-l

can arise we use the shorter notation Sh. Note that Sh Well known examples are: f.-l

= 0, r = 2:

O.

piecewise linears

= 1, r = 4: Hermite cubics f.-l = r - 2: smoothest splines. f.-l

o

The finite element solution is sought as follows: Find Uh E S h such that (1.1.6)

o

A(Uh, X) = (j, X), for all X E Sh.

(We shall not take into account numerical integration in (1.1.6).) In our analyses below we shall need to use various results from the literature. These can be found with varying degrees of generality, in particular with respect to dependence on the distribution of mesh sizes hi in a subdivision. To keep things

2

simple in this chapter we make the following explicit assumption of quasi-uniformity of meshes, except in the case of continuous elements (Ji = 0) when no such restriction is made. If Ji :::: 1, then there exists a positive constant C Qu independent

(1.1.7)

of h such that in any subdivision

Th,

we have h::; CQuminh i . l

We shall assume that (1.1.3) has a unique solution u for any f in L 2 (1) , say. Following [Schatz 1974], d. also [Hildebrandt and Weinholz 1964] and [Schatz and Wang 1994], we then know that there exists h a > 0 such that for h ::; h a, given any f E L 2 (1) there is a unique solution Uh to (1.1.6). Furthermore, under our general conditions on smooth enough data, we have the following estimates for the error

e=

U -

Uh:

(1.1.8) and

(1.1.9) Here sup (v, w). wEH'(I)

(1.1.10)

IlwIIH'(I)=l

Correspondingly we have in maximum-norm,

(1.1.11) and

(1.1.12)

IleIIW';;;S(I) ::;

for s ::; r - 2,

where now

(1.1.13)

IlvIIW';;;'(I)

=

sup (v, w). wEwt(I)

Ilwllw{(I)=l

The constants C occurring are independent of hand u. They do depend on the coefficients ai and on Ji and r; in the case of Ji :::: 1 they may also depend on CQU in (1.1. 7). The HI-estimate in (1.1.8) can be found in this generality in [Schatz 1974], after use of standard approximation theory. The L 2-estimate and the negative norm estimates (1.1.9) follow by standard duality arguments, d. (2.2.4) below. The L oo estimate (1.1.11) for Ji = 0 is in [Wheeler, M. F. 1973]. The case of general Ji, with the quasi-uniformity condition (1.1.7), is in [Douglas, Dupont and Wahlbin 1975]. From this the -estimate follows in the quasi-uniform case. The -estimate in the case Ji = 0, although "well-known folklore" , we have not been able to locate without restrictions on the meshes. We will therefore give a proof, in Remark 1.3.2 below. The negative norm estimates in (1.1.12) follow in a standard duality fashion from (1.1.11).

3

It is not hard to convince one-self that the powers of h occurring in (1.1.11) are the best possible in general. E.g.,

(1.1.14)

min

xEII r- 1 (O,h)

= hr = h

r

Ilxr

XIILoo(o,h)

-

min

XEIIr-J{O,h)

min

XEIIr-J{O,h)

= h" XEII _ min

r_ 1(0,1)

h

lie -

- hr

Loo(O,h)

hr

IW - x(OIIL

L oo (O,l ) oo

(O,l )

= Ch":

A superconvergent "point" for function values of order such that

(J

=

is now a family of points

le(OI :::; cv>,

(1.1.15)

where (J > 0 and C = C(u, a2, a1, ao) (and possibly also depending on GQu in (1.1.7)). Similarly, 'T/ = 'T/(h) is superconvergent of order (J for first derivatives if

le'('T/)I

(1.1.16)

< cte >»,

with (J > O. In principle, we could talk about superconvergence points for a particular problem or for a particular solution. Generally what we have in mind, though, is some class of problems with (locally) smooth coefficients and solutions, and we then wish to determine points which are superconvergent for the whole class. Furthermore, we point out that what is described in (1.1.15) and (1.1.16) is only so-called "natural" superconvergence. That is, Uh (or u;,) is simply evaluated at the point (or 'T/) and then compared with (or u' ('T/)). In these notes we shall also see many examples of superconvergence involving postprocessing of Uh (trivial, or not so trivial). In fact, in terms of implementation on a computer, many postprocessing methods are simpler to implement than is evaluating Uh at a (non-node) point. 1.2. Nodal superconvergence for function values in continuous elements (Jl = 0).

We shall give the argument of [Douglas and Dupont 1974]. Let be the Green's function for (1.1.1), cr. e.g. [Birkhoff and Rota 1969, Theorem 10, p.52-53], so that

= A(e,

(1.2.1)

.)). o

From (1.1.3) and (1.1.6), A(e, X) = 0, for X E Sh and thus (1.2.2)

= A(e,

o

- X), for any X E Sh'

Let now = Xi, a meshpoint. Since is continuous and (uniformly smooth on both sides of X = and since Jl = 0, we have from standard approximation theory that, for a suitable X, (1.2.3)

4

Since by (1.1.8) also IleIIH1(I) ::; Chr-11Iullw(I) we thus obtain from (1.2.2),

le(xi)1 < Ch2r-21Iullw(I)'

(1.2.4)

for Xi a meshpoint. Thus, provided r 2: 3 (i.e., continuous piecewise quadratics or higher elements are used), we have superconvergence at the knots. There is no restriction on the geometries of the meshes allowed. We state the above as a theorem.

Theorem 1.2.1. Under the assumptions of Section 1.1, for /1 = 0 (with no mesh restrictions) , (1.2.5) Let us mention that [Douglas and Dupont 1974] gives an explicit example (with a2 E C-1(I) only, though) showing that the power h2r - 2 is sharp.

1.3. Reduction to a model problem. In Sections 1.5-1.11 we shall investigate superconvergence of, typically, order one (i.e., (J = 1 in (1.1.15) or (1.1.16)). It turns out to be convenient to reduce the investigations to the case a2 == 1, = ao == O. The argument follows [Wahlbin 1992] which in turn was based on [Douglas, Dupont and Wahlbin 1975]. Let thus

al

(1.3.1)

A(u - Uh, X)

= 0,

o

for X E Sh,

o

(cf., (1.1.3) and (1.1.6)) and let Uh E Sh be another approximation to u given by

(u' -

(1.3.2) Then, with (1.3.3)

= 0,

X')

o

for X E Sh·

o

e = Uh - Uh E Sh, W, X') = ((Uh =-

u)', X') - ((Uh - u)', X')

(a 2(uh -

u)',

:2 X')

(:2 X) ') + (Uh - U, X) ') u)', (:2 X)' - 1/J') + (al (Uh - u),1/J')

= - ( a2(Uh - u)',

= - ( a2(Uh + (aO(Uh

-

- u),1/J)

+ (Uh -

U,

c.»

for x,1/J E Sh'

Let us now consider (...LX)' - 1/J', and let us treat in detail the continuous case U2 (/1 = 0). Let then 1/J be the natural (Lagrange) interpolant (at Xi, Xi+l; and the appropriate number of equispaced points in the interior of Ii) to ...LX. By standard U2 approximation theory, (1.3.4)

1 )' -1/J' I 11( -X a2

< etc:' ,

L 1 (I;) -

11(1-X )(r)11 a2

LdI;) ,

5

and since X(T) == 0 we have by Leibniz' formula and inverse estimates 2.1),

(cr.

Section

(1.3.5) where C depends on a2. Thus, summing over all intervals, (1.3.6) This result is a special case of "superapproximation", cr. [Nitsche and Schatz 1974]. Employing a quasi-interpolant ala [deBoor and Fix 1973] (see e.g. [Wahlbin 1991,

Sh

Lemma 3.2, p.367]), (1.3.6) is true for any considered by us, cf. (1.1.7). We shall give some detail on this in Section 2.3 below. From (1.3.3) we now have using also (1.1.11),

I(O',x')I::; ChTllullw;;'(I)IIXllw,'(I)' We proceed to estimate 110'IIL oo(1)' We have

(1.3.7)

1It1'IIL oo(1) =

(1.3.8)

(0', 'Ij;).

sup

11t/JII L l ( l) =

l

Let (1.3.9) and let P denote the LTProjection into $h. For each (0', 'Ij;)

(1.3.10) since 0' E $h. Now set 1>(x)

(1.3.11)

=

l

'Ij; occurring in (1.3.8) we have

= (0', P'Ij;)

x

P'Ij;(y)dy - xMV(P'Ij;)

1

0

where MV(P'Ij;) fa P'Ij;dy. Then 1> E Sh, and (0',1) = 0 we have from (1.3.10) and (1.3.7)

1/ = P'Ij; - MV(P'Ij;). Since

I(O','Ij;)1 = 1(0',1>')1::; Ch ullw;;'(1)II1>llw,'(I)' Tll

(1.3.12)

Since 111>llw,'(I) ::; C1IP'Ij;IIL,(I) and the L 2-projection into $h is bounded in L 1 (easy if fL = 0 so that P is completely local; for the cases fL 2: 1 see [Douglas, Dupont and Wahlbin 1975] or [Wahlbin 1991, Lemma 3.5] or, Theorem 3.2.3 below) we thus have 111>llw,'(I) ::; CII'Ij;IIL,(I) = C. Hence

1It1'IIL oo(1) ::; ChTllullw;;'(I).

(1.3.13)

We state this result as a theorem. o

Theorem 1.3.1. Let Uh be the projection into Sh of u based on the form A, which o

satisfies the assumptions of Section 1.1. Let Uh E Sh be given by o

for XES h· For fL 2: 1 assume quasiuniformity as in (1.1.7). Then

(1.3.14)

- u',X') = 0,

6

We shall next prove an analogue of this for function values.

Theorem 1.3.2. With assumptions as in Theorem 1.3.1 and, in addition, r 2: 3, we have

Iluh -

(1.3.15)

uhIILoo(I) :::; Chr+lllullwi,;,(I)'

Proof: In the case M= 0 we have from Theorem 1.2.1 that B(Xi) == (Uh -Uh)(Xi) = 0(h 2r - 2 ) :::; cir». For x Eli, B(x) = B(Xi) + J:, B'(y)dy and (1.3.15) follows in this case from Theorem 1.3.1. For the general case of M2: 1 we write

IIBIILoo(I) =

(1.3.16) For each such v, let -w"

=v

in I, w(O) = w(l)

where P is the L 2-projection into $h

= J; P(w')

(1.3.18)

= O.

Then

(B, v) = (B', w') = (B',P(w'))

(1.3.17)

Q

sup (B, v). II v IIL Jt I) = 1

o

E Sh with Q'

(B,v)

=

= P(w')

Since MV (P( w'))

so that from (1.3.3), with e

= MV (w') = 0,

= Uh -

u,

= (B',Q') =-

(a2e',

(:2 Q)' -1/;') + (ale, 1/;') + (aoe, 1/;) + (e, Q)')

==h+I2 + h + h Integrating by parts, and using (1.1.11) and superapproximation, (1.3.19)

(a2 C1 Q)' - 1/;') ') I :::; CIIellL II a2 - 1/;'11

IhI = I(e,

2

(I)

00

wl(I)

:::; Chr+IIIQllwf(I)' Similarly, using the

error estimate of (1.1.12),

(1.3.20) Under our quasi-uniformly assumption (1.1.7) it is easy to see that, since P is stable in L I , it is also stable in Wl(I). Thus IIQllwN) :::; CIIwllwf(I) :::; CllvllL 1 = C. It follows from this and (1.3.16)-(1.3.18) that (1.3.21)

IIBIILoo(I) :::; cv:',

This completes the proof of the theorem. 0 Remark 1.3.1. In various special cases, better results can be had. E.g., if r 2: 3, if a2 == 1 and al == 0, then (1.3.14) in Theorem 1.3.1 may be replaced by

II(Uh - uh)'ll t..; (I)

:::;

Ch"+lllullwi,;,(I)'

7

This is easily seen from the proof. Similarly we then have Ch r +2 in Theorem 1.3.2 for r ?: 4. D Remark 1.3.2. In the case of J.l = 0 we will now briefly point out how the error estimate in (1.1.11) follows without any mesh restrictions. With e = Uh - U we have as in (1.3.3)

1 ') = - (' a2e, (1 a2X)') + (' e, a2X)

(0", X ) = - ( e, a2X a2'

=-

(a 2e', (:2 X)' - 7jJ')

+ (ale, 7jJ') + (aoe, 7jJ) + (e',

== h + 12 + h + 14 , for x,7jJ

X)

o

E

Sh·

Using the H l estimate (1.1.10), Ihl :::; Chr-lllullw(I) II

a2

-7jJ'11

.

L 2 (I)

By the L 2 analogue of the superapproximation result (1.3.6), and using inverse estimates, cf. Section 2.1 below (recall that J.l = 0 now),

II

a2

L 2 (I;)

:::;

2

,


0 on .:J; (a B-spline basis function). Let now (1.4.1)

'lj;i(X)

=

l

x

Bi(y)ds - xMV(B i)

o

which belongs to Sh. Then (1.4.2) The theorem follows since B, > 0 on Ji (note that if Ji = 0, Ji A corresponding result for function values is as follows.

= Ii)'

o

Theorem 1.4.2. a) For Ji = 0, (u - Uh)(Xi) = 0, for all meshpoints Xi.

b) For r > 3, let k f = There exists a point

E

and J, = (Xi,Xi+kj)' Ji such that (u = O. o

Proof: a) follows as in Section 1.2 by noting that now G(Xi;') E Sh. For b), as in the proof of Theorem 1.4.1, we may find B, E $JL-2,r-2 such that B, is supported in Ji and positive in the interior. Letting _'Ij;" = B, on I, 'Ij;(0) = 'Ij;(I) = 0 we have o

'Ij; E Sh and so

(1.4.3)

o

Of course, Theorems 1.4.1 and 1.4.2 give no information as how to nail down what these points may be. They do, however, give us a hunting license. 1.5. Superconvergence for interior points of mesh-intervals for continuous elements (Ji = 0). For the results in this section we refer to [Chen 1979], [Lesaint and Zlamal 1979] and also to [Bakker 1982, 1984], where many results were rediscovered but also extended to 2m t h order problems. We point out that no mesh restrictions will be imposed. We shall first show superconvergence for derivatives at certain points. For a mesh interval Ii, Li,k(X) shall denote the k t h Legendre polynomial normalized to Ii, i.e.,

9

if I, = (Xi, Xi+l), then Li,k(X) = L k(2(x - (Xi + Xi+l)/2)/h i) so that Xi corresponds to -1, the midpoint to 0 and Xi+1 to 1. Theorem 1.5.1. For

f..L

= 0,

(1.5.1 )

where TJi is any zero of Li,r-I. We note that L'i,"-l has r - 1 zeroes inside I, and that is a polynomial of degree r - 2. The number of superconvergence points found is thus "maximal". For, if we had r stable points TJj on each Ii where (u - Uh)'(TJj) = O(hr), then we would have II(u - uh)'liLoo(I) = O(hr ) which is not true in general. Proof: By Theorem 1.3.1 it suffices to consider Uh instead of Uh. Now with P the into $h = $h"I,r-l, i.e., discontinuous piecewise polynomials of degree :S r - 2, we have = Pu', or, that == u' satisfies

e:

-

(1.5.2) or,

L

(1.5.3)

e'1jJ

= 0,

for all 1jJ E IIr -

2(Ii ) .

e:

Taylor-expanding around the midpoint (Xi + Xi+l)/2 and writing the expansion in terms of the Legendre polynomials, we have (1.5.4) e'(x) = cOLi,o(x) + clLi,I(X) + ... + cr-ILi,r-I(X) + O(hT), where the O(hr ) term now involves (r T Co, Cl,'" , Cr - 2 are all O(h ) , so that

+ 1)th

derivatives of

U

on h

By (1.5.3),

(1.5.5) The desired result follows. The corresponding superconvergence result for function values is as follows. Theorem 1.5.2. For

f..L

= 0,

r :::-:

0

3,

(1.5.6)

where

is any zero of L:,r-I (or, a mesh point, where we have 0(h 2T- 2))).

Again, the number of superconvergence points found is maximal. Proof: From Theorem 1.3.2 it is again enough to consider Uh. By (1.5.5) and Theorem 1.2.1, (1.5.7)

e(x)

= e(xi) + = 0(h

1:

e'(iJ)dy

2r- 2) + Cr-l

Since (r -l)rL r- l = -( (1follows.

l

x

x;

Li,r-l(y)dy + O(hT+ l).

by Legendre's differential equation, the result 0

10

1.6. Superconvergence in derivatives at points about which the meshes are locally symmetric (1' even). In this section we shall consider general I-" and since Theorem 1.5.1 gives a maximally possible number of superconvergence points when I-" = 0, identifiable a priori to boot, we have in mind here I-" ? 1. Our arguments in this section follow [Wahl bin 1992J and are restricted to one­dimensional problems. In return, we then obtain more sharply delineated results than in several dimensions, in particular with respect to mesh­geometry outside an immediate neighborhood of the point of interest and with respect to how close to the boundary of I the results are valid. By Theorem o

1.3.1 it turns out to be sufficient to treat only iLh E Sh given by o

«u ­ Uh)', X') = 0,

for XES h; this is because only superconvergence of order 1 will be at issue. The quasi­uniformity condition (1.1.7) is assumed throughout. Let W = u' and Wh = E $h = $Jt­l,r­l; then as we have seen before, (1.6.1)

Wh

= pew)

where P is the L 2­projection into $h. Fixing a point such that (1.6.2)

X(1])

= (X,o),

1],

there exists 0

=

E $h

for all X E $h·

We shall refer to 0 as a "discrete delta­function" centered at 1]. In case I-" = 0, 0 is of course a function with support on I, :3 1]. For I-" ? 1 this is no longer true; suffers some influence from outside Ii. However such influences are small. In fact we have:

Lemma 1.6.1. Assume (1.1.7). There exist constants C and c of hand 1], such that

> 0,

independent


s there is nothing to prove. Since X(k) to consider £ = O. Now, by Holder's inequality,

(2.1.10) Letting (2.1.11)

Ilx(k)IILp(.1)

x=

('131)' x(x) =

s::

p

s::

::k-_l/ (X(i))

p

s::

00

and

it is enough

IJI1jPllx(k)IILoo(J)'

x(x), we have -!l;.X(x) =

IIx(k) IlL (.1)

=

s::

IJl1jp-k II

dx k

W

so that

xii Loo(I) .

and again appealing to the equivalence of any two norms on a finite dimensional space, (2.1.12)

Ilx(k) IILp(.1) < C(s) IJ11jp-kC1(s) IIXIIL (1) = C(s )C1 (s)IJlljp-kllxIILoo(.J) < C(s )C 1 (s) IJI- k IIXIILp(J) oo

where we used Theorem 2.1.1 in the last step. This proves the theorem. 0 Corresponding to Theorem 2.1.2 we now have in the quasi-uniform situation:

30

Theorem 2.1.4. There exists a constant C(s) S.t. for k >

(2.1.13)

Ilx(k)IIL

(1) P

< C(S)(C h )-(k-f)llx(f)11L QU

(1),

e,

1 ::; P ::; 00,

for X

E

Sr,S+l(I).

P

As a final comment to this section, the reader will have noticed that care has been taken to avoid constants depending on p and q in Theorems 2.1.1 and 2.1.2. It is important to, at least, keep uniform control of such constants. As we shall see in the multi-dimensional situation, one may want to apply the results with p depending on h. 2.2. On approximation theory, and duality. We shall merely give some scattered remarks. Of course, of central importance (for us) is the approximation of smooth functions. Thus, with $h = (I) we want to ascertain that, e.g.,

(2.2.1) where C is a constant which does not depend on h or v. The case of J-l = 0 (or, = -1, of course) is simple: On each Ii, we choose the two endpoints Xi,l = Xi and Xi,r = Xi+1 and then r - 2 additional points Xi,j, j = 2, ... ,1' - 1 forming an equispaced patition o(!;. Letting I nt( v) be the Lagrange interpolant of polynomial degree r - 1 to v at these points, almost any elementary text in Numerical Analysis tells us that there is a constant C = C (1') such that J-l

(2.2.2) Since the mesh points

Xi

are interpolation points, I nt( v) will belong to

(i) when

o

= 0 (or J-l = -1). Also, if $h is replaced by S h there is no additional problem if o v(O) = v(l) = 0; the piecewise Lagrange interpolant will be in Sh. J-l

A similar classical procedure which takes into account only what happens on I, for the approximation occurs in the Hermite case, (or, rather "a" Hermite case), when r = 2(J-l+1). Then we find Int(v) E II r - 1 (Ii ) to match V(j)(Xi) and v(j)(xi+d, for j = 0, ... , u, The combined result will lie in If 2(J-l + 1) < 1', throw in a few equispaced interpolation points in between .... However, if 2(J-l + 1) > l' we run out of free parameters in this game. The construction now becomes more elaborate, but the result (2.2.1) is still valid, also if o

$h is replaced by Sh, i.e., boundary conditions are taken into account. We refer the

reader to, say, [Powell 1981, Chapters The interpolant used can be "global" in the sense that the value used at one point influences the interpolant all over I (but with exponentially diminishing influence away from the point) or it can be "local", i.e., the value of the interpolant at a point depends only on the values of the function over a few mesh-Intervals around the point. For such local interpolants, "quasi-interpolants", see in particular [deBoor and Fix 1973] and also [Powell 1981, Section 20.3]. However, there is a need not only for high-order approximation as in (2.2.1) but also for low-order approximation. Let us motivate this need via a duality argument. Let us then say we know, for e = U - Uh where Uh is the solution to (1.1.6), that (2.2.3)

31

Now we wish to show that, for r

3,

(2.2.4) where (2.2.5)

=

sup

Ilwll wi (I) =1

(v, w)

(cf. Remark 1.3.2 and Section 1.12 where (2.2.4) = (1.1.12) was used.) A typical duality argument now proceeds as follows:

=

(2.2.6) For each such w, let 1/J E weak form,

sup

IIwll w i (l ) = l

(e, w).

H (1) be the solution of the adjoint problem to (1.1.1), in 1

o

A(X,1/J) = (X,w), for any X E H 1 (1).

(2.2.7)

By Fredholm's alternative, if (1.1.1) has unique solutions, so does (2.2.7). Furthermore, in this one­dimensional case, it is not hard to show the a priori estimate that (2.2.8) Then o

(e, w) = A(e, 1/J) = A(e, 1/J ­ X), for any X E Sh,

(2.2.9)

and so by Holder's inequality, and (2.2.3),

l(e,w)l::; cie :'

(2.2.10)

mion 111/J ­ Xllwl(I)' XESh

Thus, to complete the program of deriving (2.2.4) it would suffice that, d. (2.2.8),

111/J ­ Xllwl(I) ::; Ch2111/Jllwf(I)'

(2.2.11) XESh

where 1/J(0) = 1/J(I) = O. We note that the Hermite interpolant given above would not work in general: it requires point­values at mesh­points for certain derivatives of 1/J and such would in general not be available if 1/J is merely in Wl(I). A solution lies in first smoothing 1/J before applying the "interpolant". This is easy if one disregards the boundary conditions X(O) = X(I) = O. Let K be a "smoothing function with certain properties to be given below. Assume that 1/J kernel", a is extended over the boundary 81 in some suitable way, stable in for m as large as needed, cf. [Necas 1967, p. 75] or [Friedman 1969, p. 10] where the rather well­known and simple Lions­Nikolskii­Peetre extension operator is given. Setting then

Co

(2.2.12)

W;

Smh(1/J)(x) =

IJ

h

(X- y

1/J(y)K -h- ) dy

32

it is not hard to see that a "high-order" interpolant applied to Sttu;('IjJ) (X) does the trick, provided "moment conditions" such as (2.2.13) for L sufficiently high are fulfilled. (I.e., the smoothing operator reproduces polynomials up to a certain order.) We refer to [Hilbert 1973] or [Strang 1973] for details. Finally, if one must respect boundary conditions (as in (2.2.11) as stated!), one uses instead of Int(Smh('IjJ)) as an approximation a perturbed one, Int(Smh('IjJ)) , which equals 'IjJ(0) at x = 0 (or 1/J(l) at x = 1). In the case of (2.2.11), e.g., one checks that (for Int being the local quasi-interpolant, d. [deBoor and Fix 1973], e.g.)

(2.2.14)

IInt(v)(x) _ Int(v)(x)1 < {CiIn

Since, if 'IjJ(0)

= 0,

ot(,v)(O)I,

IInt(Smh('IjJ))(O)1

(2.2.15)

for for

< Ch211'IjJllwJo(I) < Ch2111/Jllwt(I)'

we obtain upon using inverse estimates (there is no problem if (2.2.16)

O:S; x:S; Ch, x> Ch.

fJ,

= 0 or -1), that

11'IjJ-Int(Smh('IjJ)) Ilwl(I)

:s; 11'IjJ - Int(Smh('IjJ))llwl(I)

+ II(Int -

Int)(Smh('IjJ))llwl(o,Ch)

2 C :s; Ch 11'ljJllwt(I) + h!l(Int - Int)(Smh('IjJ))IIL1(O,Ch)

:s; Ch211'IjJllwt(I) + CII(Int - Int)(Smh('IjJ)IILoo(o,Ch) :s; Ch211'IjJllwtu) + Cllnt(Smh('IjJ)) (0) I :s; Ch211'IjJllwt(I) + Ch211'IjJllwJo(I) (since 'IjJ(0) = 0) :s; Ch211'IjJllwt(I) (Sobolev in one dimension). Applying a cut-off argument at x = 1/2 and similar techniques at x = 1, this then shows (2.2.11) in general even when respecting the boundary conditions X(O) = X(l) = O. 2.3. Superapproximation. Recall that this played a major role in Section 1.3. We shall assume that we have at our disposal a high order local approximation operator h into $h = st,S(I) o

(or, into S h if boundary conditions are to be respected). This operator is a linear operator such that (2.3.1)

hx == X for X E $h.

33

It is a high-order local approximation in the sense that there exist two constants C 1 and C 2 such that (2.3.2)

Ilv -

h(v) Ilw.{;, (I;)

for k

:s; C 1hs- k llv(s)IILoo (I;) ,

= 0, 1, ... , s - 1, any

v E

where (2.3.3)

II

=

{x E I: dist(X,Ii)

:s; C 2h} n I.

Furthermore, (2.3.4)

h(v) has support in (Supp(v)

where for any set B and real d > 0, B We shall assume that, d. (1.1.7),

+d =

+ C 2h) n I,

{x E R: dist(x, B)

:s; d}.

the meshes are quasi-uniform. The above approximation operator h is in general the quasi-interpolant of [de Boor and Fix 1973]. Remark 2.3.1. The case of J1 = 0 (or, J1 = -1) is much simpler and is left to the reader. The operator h can then be taken completely local on each Ii, namely, as the Lagrange interpolation operator on equidistanced points including Xi and Xi+l' The quasi-uniform condition is then not necessary, as the reader should check. (Actually, the quasi-uniformity is not necessary in any case in which the approximation operator is completely local on each 1;; this hapens e.g. in Hermite cases.) 0 We shall use piecewise norms with respect to the partitions T, N-l

(2.3.5)

/

Ilvllw;,h(I) = ( L Ilv11fv;(I;)) 1 P, i=O

with the usual modification if p = 00. We have the following as the major (very complete!) result on superapproximation in one space dimension. Theorem 2.3.1. There exist constants C3 and C4 depending on C 1 , C 2 above and on sand CQu (in (1.1.7)) such that the following holds. Let w be a smooth function such that

(2.3.6)

Ilwllw.{;,(I) :s;

Ad- k , for k = 0,1, ... , s,

where d > h. Then for any X E st,S(I) there exists 'IjJ E k < s - 1 and any 1 :s; p :s; 00,

such that for any

(2.3.7) Furthermore,

(2.3.8)

Supp('IjJ) o} and {v E Sh odsupp(v) , Sl) > O}. We first state our superapproximation assumption. Assumption 3.1.1. (Superapproximation) There exist constants c and C and a number L, independent of h, such that the following holds: Let Slo Sll "Ph with d = 0< (Slo, Sll) > ch. Let further w E (no) with (3.1.2)

IlwllwJ...,(QoJ ::::: Are, P = 0, ... ,L.

37

Then for any X E Sh there exists 'I/J E S;; (0 1 ) such that

o

(3.1.3)

We shall also need an inverse estimate: Assumption 3.1.2. (Inverse estimate) There exists a constant C independent of h such that for any element T ih, for any 1 p 00, 1 q 00, (3.1.4)

IlxIILp(Tih)

for X E

s;

o

We note that for q < p, the estimate (3.1.4) is valid with Lp(Tih ) and Lq(Tih ) replaced by Lp(V h ) , respectively Lq(V h ) , cf. the proof of Theorem 2.1.2. An inverse estimate in general is proven essentially as in Section 2.1 by mapping to a standard (reference) element of unit size and appealing to finite-dimensionality. In the form we have stated (3.1.4), one has to avoid degeneracies in meshes such as triangles getting thin. We refer to [Ciarlet 1991, Theorem 17.2, p. 135] for more details. Our superapproximation Assumption 3.1.1 can often be proven by following the lines of Section 2.3. E.g., in the Lagrange case described above, there is an interpolation operator h locally determined on each T ih which preserves polynomials of total degree r - 1 such that (3.1.5) in case certain geometry constraints are not violated, see [Ciarlet 1991, Theorem 16.2, p. 128]. Here 1·lwr denotes the r t h semi-norm involving only r t h derivatives. and noting that r t h derivatives of X vanish, and using Applying this to v = inverse estimates, one easily obtains our Assumption 3.1.1. The systematic derivation of error estimates such as (3.1.5) using polynomial invariance of h goes back to [Bramble and Hilbert 1970], the "Bramble-Hilbert Lemma". (In this respect, cf. also (2.2.13).) For the rather laborious derivation of error estimates before the Bramble-Hilbert Lemma, the reader may ponder the following quote from [Prenter 1975, p. 127] (who did not employ the Bramble-Hilbert Lemma): "Waning sadism forbids us to go further." A key idea in the Bramble-Hilbert Lemma, namely that inf"EII s _ Ilv - 7rllw;(I1) ' is equivalent to Ivlw;(I1)' can be traced at least as far back as [Deny and Lions 1953-4, Theorerne 5.1]. Of course, there it had nothing to do with finite element appoximation theory! In other cases, Assumption 3.1.1 is less trivial. E.g., if (in some piece over Vh CC R 2 ) the mesh is rectangular and Sh then consists of continuous bilinear functions ao + a1x + a2Y + a3XY on the rectangle Tih, then the counterpart of (3.1.5) would not be enough since i- 0 in general, for X E Sh. What is needed then is the socalled "sharp" form of the Bramble-Hilbert Lemma [Bramble and Hilbert 1971], d. also [Nitsche and Schatz 1974, Appendix] and [Bramble, Nitsche and Schatz 1975, Appendix]. In our present case of bilinears this means that

w';:

(3.1.6)

38

where (T ih) = I x IL (h) + 1 Y IL (h)' i.e., only those derivatives which Ti T i annihilate X E Sh are involved. Then superapproximation is easily proved assuming suitable inverse properties. 00

00

00

3.2. Estimates for L2-projections. Assuming that the superapproximation Assumption 3.1.1 holds, we have first the following: Lemma 3.2.1. There exist positive constants Co, CI and G such that the following holds. Let flo fl i o; with d = 0< (flo, fl l ) :2: coho Assume further that Vh E Sh is such that (3.2.1) Then

(3.2.2) Of course, if Sh consists of discontinuous elements this is trivial! Proof: Let 0 = ch with c as in Assumption 3.1.1, i.e., the "separation" beflo fl 20 with 0 = 0< (flo, flo) = tween domains required there. Let further flo odflo, flo)· Let W E be nonnegative with

(3.2.3)

W

== 1 on flo,

and

:s ss:', e = 0, ... ,L.

(3.2.4)

(It is well known how to construct such a function.) Then

Ilvhlli,(!!o) :S

(3.2.5)

(Vh' WVh)'

For any X E S;;-(fl l ) , we have

(3.2.6)

(Vh,WVh)

=

(Vh,WVh ­ X)·

Assuming that fl 20 fl l , we choose X according to superapproximation, in particular with X E S;;-(fl 20 ) and obtain from (3.1.3), (3.2.7)

Ilvh 11i,(!!o) :S Gllvh IIL 2(!!28)A :S

IlvhIIL 2(!!28)

Ilvhlli,(!!28)'

Repeat the argument with domains fl 20 ... fl 2N 0 as long as they remain inside fl l , i.e., since 0 = ch, for N :2: canst. d/h. Increasing 0 in Assumption 3.1.1 if necessary, we may assume GAh/o < 1 and set GAh/o = e- c ; ; then (3.2.8)

Ilvhlli,(!!o) :S e-constc;d/hllvhlli,(!!,j'

The desired result follows. We next investigate the discrete o­function centered at o= E Sh such that (3.2.9)

(X,o)

= X(xo),

for all X E Sh.

Xo E 1)h,

0 i.e., the function

39

(Here we assume that Sh 0, (4.1.6)

IlwIILJ(Bd)

cd":

Thus, since by (4.1.5) Ilwllw[(Dh) S; Ilwllw[(Bd) S; Cd":"; (4.1.7)

I(v - Pv, w)1 S; Ch 2r dn- r .

Assuming now that v - Pv does not vanish on Ed we have since w >

v - Pv > 0, say), (4.1.8)

min (v - Pv)(x):::::

xEBd

r (v - Pv)(y)w(y)dy/ll wIIL

JBd

1(Bd

°

43

(taking

)

< Ch 2rd- r = Thus, in any ball of radius d = h 1 - al r, there is a supercovergent point of order (7 (at least). (Here we have taken h" to be the general best order of approximation in L p , the so-called "optimal order".) Of course, the above analysis gives no clue as to how to locate such superconvergence points a priori. They do, again, give us a hunting license. 4.2. Superconvergence in L 2-projections on n-dimensional tensor product spaces. Our second theme will be to show that any superconvergence result we know for one-dimensional L 2-projections will "automatically" carryover to n-dimensional L 2-projections on finite element spaces which are locally tensor-products. We shall give the argument in detail for n = 2. Let D h be plane domains and 0 0 = 0 0 (h) 0 d. (1.1.15) and (1.1.16). Again, we shall also consider postprocessing. The main technical tools for proving superconvergence will be presented in Section 5.5. The intervening sections are devoted to elucidating these tools.

5.2. Existence of superconvergence points in general: an example (also an example of a multi-dimensional duality argument). Let D be a bounded domain in R" with sufficiently smooth boundary. Consider the Neumann problem of finding u such that

{

(5.2.1)

-

+u = f = 0 on

in

D,

EJV.

The weak formulation of this problem is to find u E HI (D) such that (5.2.2)

A(u, X) ==

1

(V'u· V'x

+ ux)dx = (1, x) ==

1tx,

for all X E HI(D).

(Note that the "natural" boundary conditions = 0 on aD are not seen in the weak formulation, cf. e.g., [Necas 1967, Example 2.8, p. 33].) In our case the form A is coercive over HI (D), indeed, (5.2.3)

A(v,v) =

for v E HI(D)

so that the Riesz representation theorem guarantees a unique solution for any f in L 2(D) (say). One may now introduce finite elements partitions of D; if numerical integration is not taken into account we may assume that the elements that meet aD are curved to fit aD exactly. Under conditions on quasi-uniformity (and other general standard conditions that we shall not give here) it may be shown (see [Schatz and Wahlbin 1994, Theorem 4.1], d. also [Scott 1976] for an original, somewhat weaker, result when n = 2), that with Uh E Sh the finite element approximation given by (5.2.4) we have for e (5.2.5)

=u-

Uh,

Ilellw;,,(D) :::; Ch r - l, smoothness of u (i.e., I).

where C depends on the the coefficients and various parameters describing Sh; however, it can be taken independent of h. We now make explicit the approximation hypothesis that (5.2.6)

50

where C is independent of w, hand p (see [Ciarlet 1991]; in certain cases smoothing a la [Hilbert 1973] or [Strang 1973], cf. also Section 2.2, may be required). We shall estimate, via a typical duality argument, the following negative norm,

(5.2.7) where p is close to let

00

and'!p

+ 1, = 1, and thus pi is close to p

{

(5.2.8)

-bow

aw an

+w =v

= 0

on

We then have the following a priori estimate,

V,

av. for av smooth enough,

1

Ilwllw p'r (V)

(5.2.9)

in

1. For any v as above,

vll wr-2(D)' C--ll pi _ 1 p!

We refer the reader to [Gilbarg and Trudinger 1983, Chapter 9] for tracing constants in a priori estimates to see that (5.2.9) is true. (The argument there is for a Dirichlet problem but easily adapted to our present Neumann problem. The reader who pursues this will connect with the constant in the Calderon-Zygmund Theorem and, ultimately, with that in Lp-estimates for the Hilbert transform.) The point is that, in contrast to the one-dimensional situation, we cannot take pi = 1. We now have

(5.2.10)

(e, v) = A(e, w) = A(e, w - X), for any X E Sh(V),

Thus, by Holder's inequality and (5.2.5), (5.2.6), (5.2.9), for a suitable choice of X E s.:

(5.2.11)

I(e, v)1

Chr-11lw - Xllw1,(D) p

< Ch

2r- 21I wllw;,(D)

Ch 2r- 2

< (pi -1) Ilv11w;,-2(D)

Ch 2r- 2 (pi _ 1)" Thus

(5.2.12) for p < 00. Analogously to our investigation in Section 4.1, d. (4.1.4) we shall find that, now for r 2: 3, the negative norm estimate (5.2.12) leads to an abundance of (nonidentifiable) superconvergence points. Let thus Ed be a ball of radius d, inside V, and v a smooth nonnegative function with support in Ed such that

(5.2.13) and

(5.2.14)

v

=1

on

Bd/ 2

51

Then assuming that e does not vanish on Ed, say e > 0 there, with 1p

+ 1, = p

1,

(5.2.15) By (5.2.13), (5.2.16) From (5.2.12) we then obtain

(5.2.17)

min e(x) :::; Cd-nh2r-2pllvllwr_2(B )

xEB d

p'

d

and using (5.2.14) and Holder's inequality, this is bounded as

(5.2.18)

min e(x) < Cd-nh2r-2pd-(r-2)dn/p' xEB d

= ci:

o:

pd- n / p

< where we have chosen p = In(l/d). We conclude the following upon choosing d

=

h 1-

a/(r-2):

Proposition 5.2.1. For r :2: 3, under the various assumptions above, any ball of radius d = h I-a /(r-2) contains a point where (5.2.19)

le(OI < Ch r + a

In(l/h),

i.e., a point of superconvergence of (almost) order

0-.

A similar investigation considering any first derivative aae and II aae Ilw- 0, corresponding superconvergence results hold but with min(Cf, T).

Cf

replaced by 0

69

6.3. Negative norm estimates for u - Uh: Examples. The term Ilu - uhllwq-8(rltl in the previous section is the only one that takes into account what happens outside of n 1 : all other considerations are local to n 1 . We shall give four brief examples of how to estimate it. For future purposes we shall also give estimates of higher order than immediately necessary in the previous section. Note that with our definition, Ilvllwq-s(rltl ::; Ilvllwq-s(rl2) for n1 n2 . Example 6.3.1. Homogeneous Neumann problems in smooth domains in R", In Section 5.2 we considered a model Neumann problem, + u = f in D, au/an = a on aD, with f and aD smooth. Assuming a perfect match for elements meeting aD, we showed that

(6.3.1)

Ilu -

u h IIW q- '" ,f3n) with all f3j with O2 = 0 1 +Mh, M = M(f3,p),

Ilv -

(11.3.9)

K.!f'p)

> 0, and let p ::::: 1. Then for 0 1 ,

* vllLoo(lld :S

Proof: Set

(11.3.10)

kj(a-)

= k{3J,p(u)X(u){3j.

From (11.3.2) and since k{3j,p(u) and X(u) are bounded together with their first derivatives, we find using Lemma 11.2.1 that the analytic function E M oo(R 1 ) . Thus, by (11.2.4), using also (11.2.12) in one dimension, and (11.2.13), 1IJ:'-1 (1 - k j

(11.3.11)

)vIILoo(Rn)

= h2PII.F- 1((1_ :S Now write by elementary algebra n

(11.3.12)

=

1-

1-

IT

j=l

n

=

L IT

-

j=l l(hO,

(11.3.22)

1+

< C(1 +

=

Next note that the function follows that

L

1

n

j=l

2

).

(sin

1-

is increasing. Since

y2 it

(11.3.23) Thus, since sin;u) is decreasing for 0 (J" denoting the largest component of (for (11.3.24)

(h 2 < 'I/J(fm) 0 -

1, and bounded by 1, we have with

< 1),