TICAM REPORT 98-06March, 1998

Finite Element Gradient Superconvergencefor Nonlinear Elliptic Problems

A.I. Pehlivanov, G.F. Carey, and V.K. Kantchev

FINITE ELEMENT GRADIENT SUPERCONVERGENCEFOR NONLINEAR ELLIPTIC PROBLEMS

A. 1. Pehlivanov1, G. F. Carey1, and V. K. Kantchev2

Abstract

We consider second order nonlinear elliptic boundary value problems in two-and three-dimensional domains with Dirichlet boundary conditions approxi-mated using finite element schemes based on linear and quadratic triangular(tetrahedral) finite elements. Superconvergence of the tangential derivatives atGauss points on edges of elements is proved for these nonlinear problems andquasiuniform partitions of the domain.

1. Introduction

In finite element applications the derivative of the approximate solution is frequentlya quantity of primary interest. This is the case, for instance, in determining theheat and mass fluxes, velocities and starins or stresses. Simply differentiating theapproximate solution on an element will generate an approximation to the gradientbut the accuracy and asymptotic rate will be inferior to that of the primary vari-able. However, it is now well known that certain superconvergence properties of thisapproximate gradient may hold at special points in the element or that simple post-processing schemes such as locally averaging can yield superconvergent extraction

ITexas Institute for Computational and Applied Mathematics (TICAM), Univer-sity of Texas at Austin, Austin, Texas 78712-1085, USA, atanas«lticam·.utexas.edu,carey«lcfdlab.ae.utexas.edu

2Numerical Analysis Laboratory, Institute of Mathematics, Acad. G. Bonchev str. block 8, 1113Sofia, Bulgaria, veselin«liscbg. acad. bg

1

formulas. In turn, this implies that good gradient approximations can be computedwithout resorting to the expense of computations on finer grids. Moreover, these su-perconvergent properties may be useful in constructing a posteriori error indicatorsfor adaptive refinement.

Superconvergence has been the subject of theoretical investigations by several au-thors primarily for linear elliptic problems - see e.g. Douglas and Dupont [13, 14],Douglas, Dupont and Wheeler [15], Nitsche and Schatz [30], Bramble and Schatz [3].Superconvergence of the gradient for bilinear and biquadratic quadrilateral elementsis proved by Zlamal [35, 36]. Generalization of these results for polynomials of higherdegree can be found in Lesaint and Zlamal [27]. Linear triangular elements are con-sidered by Andreev [1], Levine [28] and linear tetrahedrons by Kantchev and Lazarov[25]. The results of Andreev, Lazarov [2] and Pehlivanov [31,32] extend the previousresults for quadratic triangular and tetrahedral isoparametric elements in ~2 and ~3

respectively, see also Schatz, Sloan, and Wahlbin [33, 34].Numerical experiments strongly suggest that most of the superconvergence results

for the linear problems may be carried out over to reasonably well behaved nonlinearproblems. Recently, a number of studies related to superconvergence in nonlinearelliptic problems have appeared: for a class of minimal surface problems - Chen [4];for semilinear problems - Chow and Lazarov [7], Chow, Carey, and Lazarov [6]; forsecond and fourth order equations in rectangular 2-D domain and rectangular finiteelements - Dautov, Lapin, and Lyashko [12], Dautov and Lapin [11], Dautov [10].Since simplex elements do not have the cartesian or tensor-product "symmetry" of therectangular elements, analysis is more difficult and the theoretical results concerningsimplicial elements are rare. However, super convergence of linear triangular elementsfor strongly nonlinear problems and uniform partitions has been studied by Kantchev[24].

Here we consider second order nonlinear elliptic boundary value problems in two-and three-dimensional domains with Dirichlet boundary conditions. The finite ele-ment solutions of problems of this type are studied in Ciarlet, Schultz, and Varga [9],Frehse and Rannacher [19,20], Glowinski and Marrocco [22,23], Melkes [29], Korneev[26], Feistauer and ZeniSek [18], Chow [5].

In the present paper finite element schemes, based on linear and quadratic tri-angular and tetrahedral finite elements, are investigated and we extend the gradientsuperconvergence results in the following directions:

2

(i) for quadratic simplicial elements in Iftn, n = 2,3;

(ii) for isoparametric elements;

(iii) for quasiuniform partitions.

We prove superconvergence of the gradient under assumptions of regularity of thenonlinear coefficients, ellipticity and monotonicity. Note that these assumptions arequite natural and close to the assumptions in Feistauer and Zenisek [18]. Excellentexamples of practical problems which satisfy our assumptions can also be found inFeistauer [16, 17], Glowinski [21], Glowinski and Marrocco [22], and Chow [5].

An outline of this paper is as follows: In Section 2 we introduce some notationsand state our main assumptions. Some results from Ciarlet [8] and Feistauer andZenisek [18] are listed as preliminary results in Section 3. Several estimates for theinterpolant that are crucial to the superconvergence analysis are proved in Section 4.First, we consider the corresponding linear problem and prove the estimate

for any function Vh from the finite dimensional space (see Theorem 4.1 and Theorem4.2). Here UI is the standard finite element interpolant. Next, we prove the followingestimate for nonlinear problems (Theorem 4.4)

where k equals 1 for linear elements and 2 for quadratic elements.All results are derived for regular and k-quasiuniform partitions. The error due

to the numerical integration is investigated in Theorem 4.5. Finally in Section 5, weobtain the main result of this paper:

where I . I~,h is a discrete semi norm constructed by using the tangential derivatives atGauss points of order k on the element edges.

3

2. Notations and Problem Formulation

Let n be a bounded domain in ~n, n = 2,3, with Lipschitz boundary r. Denoteby W;(n), 1 :S; P :S; 00, mEN, the Sobolev space on n, provided with the usualnorm II· Ilm,p,n and seminorms I· li,p,n, 0 ::; i ::;m, and by Hm(n) the Sobolev spaceWr(n) with norm II . Ilm,n and seminorms I . li,n, 0 :S; i :S; m. The symbol Pk( G)stands for the space of polynomials of degree at most k over G. Let

V=H6(n)= {vEH1(n) : v=O on an} . (2.1 )

We consider the following nonlinear problem with Dirichlet boundary conditions:Find u E V such that

where

a(u,v)=(f,v) forall vEV, (2.2)

n

a(u,v) = 1Lai(X, \7U)OiV dx, (J,v) = 1fv dx. (2.3)n i==1 n

Here the nonlinear coefficients ai depend on x E nand \7u. All results below hold inthe general case when ai depend on x, u and \7u (see Remark 5.1).

Notations a and (3 are used for any multi-indices such that

aE{(a1,a2'0,0) : ai~O}, (3E{(0,0,(31,(32) : (3i~O}

aE{(a1,a2,a3'0,0,0) : ai~O}, (3E{(0,0,0,(31,(32,(33) : (3i~O}.

Throughout the paper k is the degree of polynomials in the finite element spaces(see (2.8)).

The following assumptions will be made concerning a(·, .):

(HI) The functions ai(x, e), x E n, e E Iftn, are continuous in n x ~n; there exists aconstant C > 0 such that for 1 :S; i :S; n

for all x E nand e E ~ n . (2.4)

(H2) The derivatives aa(a/3ai), 1 :S; i :S; n, for lal + 1(31 :S; k + 1, laJ ::; k, 1.81 ~ 1, arecontinuous and bounded in n x ~n; there exists a constant C > 0 such that

(2.5)

4

(H3) The following inequality holds

for all x E nand e, 'fl E ~n , (2.6)

where µ > 0 is a constant independent of x, e and 'fl.

(H4) The functions aaai, 1 :S; lal :S; k + 1, 1 :S; i :S; n, are continuous in n x ~n; thereexists a constant C > 0 such that

for all x E nand e E ~ n . (2.7)

Remark 2.1 It can be proved that under assumptions HI, H2 with lal = 0 and1(31 = 1, H3, and f E £2(n) there exists a unique solution of problem (2.1) (seeFeistauer and Zenisek [18]). 0

Remark 2.2 In the general case of nonconstant coefficients it is necessary to in-troduce some quadrature formulas to evaluate a(-,·) and (f,.) approximately. Theassumption H4 will appear only in the analysis of the error due to the numericalintegration. 0

Consider the natural partition of ~2 into unit squares. Then let each of thesesquares be divided into two triangles using one of the diagonals. These diagonals mustbe oriented identically. Hence we obtain a partition of ~2 into isosceles triangles (seeFig. 2.1). In the three-dimensional case, we start from the natural partition of IR. 3

into unit cubes. Then, for each cube, we project one of the cube's celestial diagonalsonto each face. Thus each cube is divided into six tetrahedra. Again, as in ~2,

these celestial diagonals must be oriented identically (see Fig. 2.2). We denote thesepartitions by T.

Next, let the whole domain n be covered by isoparametric linear or quadratictriangular (tetrahedral) elements [{ E Th (Th is the triangulation of n). We assumethat for each [{ E Th there exists k E t and invertible mapping FK : x E k --+FK(X) E [{ such that FK(k) = [{ and the components of FJ{ are polynomials ofdegree k; i.e. (FJ{) j E Pk (k) , 1 ::; j ::; n, where

{

I for linear elementsk = (2.8)

2 for quadratic elements

5

Figure 2.1: Partition of ~2 (pattern)

As commonly used, we have the correspondence v(x) = v(F[{(x)). Define the finiteelement space

(2.9)

Since we allow the use of isoparametric elements, some of the elements may becurvilinear. Denote by di, 1 :S; i :S; n + 1, the vertices of any element [{ and byF[{ the unique affine mapping, which satisfies F[{(di) = di, 1 :S; i :S; n + 1. Here di,

1 :S; i :S; n + 1, are the vertices of the corresponding element k. Let dij = (di + dj) /2,dij = (di + dj)/2, 1 :S; i < j :S; n + 1. The family of partitions {Th} is termed regularif the following holds for all [{ E Th (see Ciarlet [8]):

,

(i) there exists a constant (J" such that

where h[{ is the diameter of [{ and PK is the diameter of the inscribed circle(sphere) in [{;

(ii) the quantities h[{ approach zero;

(iii) for quadratic elements. - 2dlst (dij , dij) ::; C h K ,

where dij = FK( dij), 1 :S; i < j :S; n + 1.

Define

6

Figure 2.2: Partition of 1ft3 (pattern)

where, = (,1,,2) for n c ~2 ( , = (,1,,2,,3) for n c ~3). We say that thepartition Th of n is 1-quasiuniform if for every two elements [{I and [{2 which havea common side (face) the following inequality is fulfilled:

max max W'I(F1<Ji - a"l(FK2)ilo,oo,aktnak

21::;i::;n 1"11==1

< Ch2, (2.10)

where the constant C does not depend on h. The partition Th is 2-quasiuniformwhen, in addition to (2.10), we have

(2.11)

Obviously, any uniform mesh satisfies these assumptions. Also, any mesh with nodeswhich are obtained by a smooth mapping of uniform mesh nodes satisfies (2.10)and (2.11) (see Zlamal [36], Levine [28], Pehlivanov [32]). Of course, such 1- or 2-quasiuniforill meshes impose a restriction on the domain n. Later we relax (2.10)

7

and (2.11) to obtain results for more general domains but at the expense of a lowerorder of convergence.

Now we can formulate the discrete problem which corresponds to the problem(2.2): Find Uh E Vh such that

(2.12)

where the index h in ah(·,·), (f, ')h indicates that we have applied some quadratureformula with positive coefficients in order to calculate the integrals in a( Uh, Vh) and(f, Vh)'

3. Preliminary Results

We shall use the following notations:

for any element K E Th.

Theorem 3.1 (Ciarlet [8, Theorems 3.1.3 and 4.3.3]) Let the partition ofn beTegular. Then the following estimates hold:

IFK111,OO,K:S; Chl./, IFKI12,oo,K:S; Chl/, IFKI13,oo,K:S; Chl/ ,

IJFKlo,oo,k :S; C meas(K), IJFI~llo'OO,K:S; C (meas(K)t1. 0

Define the following bilinear form

n

a1in(ry,v) = 1L aij(X) airy ajv dxn i,j==1

(3.1 )

Suppose that the matrix of coefficients A = (ai]"(X) r "-1' x E n, is positive definitet,]-

and the coefficients aij are bounded; i.e. there exist constants al and a2 such that

8

for all vectors ( E ~ n and all x E n.For any element I< we have

where(3.3)

Here we have used the notations

Theorem 3.2 Let aij E w~+I(n). Then:

IBis 00 k :S; Chs-2 meas(I<) for 0:S; s ::; k + 1, (3.4), ,

Ibij(x) - bij(xo)1 :S; Ch-1 meas(I<) , (3.5)

for all x E i<, Xo E k.

Theorem 3.3 (Feistauer and Zenisek [18]) The following implications hold:

(i) H2 =?- the form a(-,·): H1(n) x Hl(n) ----+ ~1 is Lipschitz continuous; i.e.there exists a constant C such that:

la(v, z) - a(w, z)1 :S; Cllv - wlknllzlkn for all v, wE H1(n) . (3.6)

(ii) H3 =?- the form a(·,·): H1(n) x H1(n) ----+ ~1 is H1(n) strongly monotone

with respect to the seminorm 1·ll,n; i. e. there exists a constant µ > 0 such that

la(v,v-w)-a(w,v-w)I~µlv-wl~n forall v,wEH1(n). (3.7),

(iii) H2 and the condition that the quadrature formula with positive coefficients is

exact for p E Po =?- the forms ah (-, .): Vh X Vh ----+ ~ 1 are uniformly h-Lipschitz continuous; i. e. there exists C > 0 such that:

9

(iv) H3 and the condition that the quadrature formula with positive coefficients is

exact for p E Po :::} the forms ah (', .): Vh X Vh ---+ ~ 1 are uniformly Vh-strongly

monotone with respect to the seminorm 1·ll,ni i.e. there exists a constant µ > 0such that:

lah( Vh, Vh - Wh) - ah( Wh, Vh - wh)1 ~ µIvh - whl; n for all Vh, Wh E Vh .,(3.9)

4. Basic Estimates

Several estimates are now developed that will be used subsequently in Section 5 to ob-tain the desired superconvergence results. First, we construct interpolation estimatesfor the bilinear functional alin

(-,.) where k-quasiuniform partitions of the domain nare required (Theorems 4.1 and 4.2). This condition is relaxed in Theorem 4.3 topiecewise k-quasiuniform partitions at the expense of a lower order of convergence.Finally, quadrature formula error estimates are established in Theorem 4.5.

Denote

Theorem 4.1 Let Th be a regular and l-quasiuniform partition of linear finite ele-

ments, the coefficients aij E CO(n) and Ilaijlll,oo,Th :S; C. Let u E H3(n) and UI bethe standard finite element interpolant of u. Then the following estimate holds:

(4.1 )

Proof The case n c ~2 is studied in Andreev [1], Levine [28], and the three-dimensional case for uniform partitions in Kantchev and Lazarov [25]. Here we con-sider the three-dimensional case for l-quasiuniform partitions.

Since COO(O) is dense in H3(n), it is sufficient to prove (4.1) for U E COO(O).Denote W = U - UI. We have

n

a1in(w,Vh) - ?= L [aij(x)8iW8jVhdx',J==1 KETh

n

L L 1,bij(x) ai'W ajVh dxi,j==1 K ETh h

10

where bij are defined in (3.2). We shall prove that

L (.bp1(x) apw a1Vh di; :S; Ch21IuI13,nlvhll,nKETh JA.

(4.2)

for 1 :S; p ::; 3. The remaining cases are treated in the same way.Consider any tetrahedron in T. Then it has an edge parallel to the xI-axis. Denote

by 1<1,1<2,' .. , 1<6 the tetrahedra which share this edge, and by U their union. Letdi, 1 :S; i :S; 8, be the vertices of U (see Fig. 4.1). Then dm, dm+1, d7, d8 are the verticesof I<m, 1 :S; m :S; 6. Further all indices with m are interpreted as modulo 6.

Let G be an arbitrary point from edge d7d8 and for notational convenience let usdenote bp1,km = gm' We have

L:m (gm(x) - gm(G))apw a1Vh dx

+ gm(G) a1Vh(G) tm apw dx , (4.3)

since Vh E PI (I<m). From Theorem 3.2 we get

Hence

L (.bpl,kJx)apWalVh dx = II + L gm(G)a1Vh(G) f,. apw dx, (4.4)KmETh JAm KmETh J Km

where(4.5)

It remains to estimate the second term on the right in (4.4). For each tetrahedronI<m, 1 :S; m :S; 6, we construct a functional H (m; u) such that the functional

L(m; u) = a1Vh(G) f, apw dx + H(m; u)Jkmvanishes for polynomials of second degree, i.e.

11

(4.6)

(4.7)

/ II I

II

Figure 4.1: The tetrahedra 1<1,' .. ,1<6

12

First, define the following functionals

(4.8)

for each multi-index ,= (,1,,2,,3),111 =,1 +'2 +'3 = 2, where (xP == xTx;2xj3.Let

For 2 ::; m :S; 6 we define recursively:

µm(,; m)

µm(,; m + 1)

Further we set

-µm-l(,;m) ,

<Pm(,)+ µm-l (,; m)

,!meas( T1(m 7 8)) r al'ulk dT,,,1-r(m,7,8) m

,!meas(T(~+178)) r o"lulk dT,,1-r(m+l,7,8) m

where T(m, 7, 8) is the triangle with vertices dm, d7, d8, 1 :S; m :S; 6. Let us note thatif u = (x)8, IJI = 2, then

{

I for , = J'm+1 =5m(,; m) = 5m(" ) 0 for , =I- J

Finally, setting

8

Rm(,;m)

Rm(,; m + 1)

-OIVh(G) ,

85m(,; m)µm(,; m)

85m(,; m + 1)µm(,; m + 1)

we can construct the desired functional

H(m; u) = L (Rm(,; m) + Rm(,; m + 1))11'19

13

(4.9)

Then property (4.7) holds. This property can now be utilized in (4.4) as follows

L ( bpl,kJx)apWalVh dxKmEThJkm

II + L gm(G) (OIVh(G) (. apwdx + H(m;it))KmE~ JAm

- L gm(G)H(m; it)KmETh

II + 12 - L gm(G)H(m; it) ,KmETh

where

Using the quasi uniformity of the partition, the last term can be estimated by

L gm(G)H(m; it) ::; Ch21IuI13,nllvhlll,nKmETh

(4.10)

(4.11 )

(4.12)

Substituting (4.5), (4.11) and (4.12) into (4.10) we get the desired estimate (4.2).If one of the six tetrahedra of U is not an image of a finite element in Th then

d7ds is an image of a part of the boundary. But then 0 = 0, since v E Vh C HJ(n).This completes the proof. 0

Theorem 4.2 Let Th be a regular and 2-quasiuniform partition of quadratic finiteelements, the coefficients aij E CO(n) and lIaijlll,oo,Th :S; C. Let u E H4(n) and UI bethe standard finite element interpolant of u. Then the following estimate holds:

(4.13)

Remark 4.1 The case n E ~2 is considered by Andreev and Lazarov [2] and thecase n c ~3 by Pehlivanov [31, 32]. 0

Remark 4.2 The fact that Vh = 0 on r is crucial for the validity of (4.1) and (4.13).Suppose that Vh does not vanish on r, i.e.

(4.14 )

14

Then we have the estimate

L gm(G)H(m; u) :S; Chk+l/2I1ullk+3/2,nllvhlll,n + l/r ,KmETh

where vr denotes terms defined on r. Since the dimension of r is n - 1, the terms onthe boundary elements can be estimated by Chk+1/2 (see Pehlivanov [32]). Hence

and

L gm(G)H(m; u) ::; Chk+l/21Iullk+3/2,nllvhllt,nKmETh

(4.15)

(4.16)

We say that the partition Th of n is piecewise k-quasiuniform if the domain ncan be divided into subdomains and each of these sub domains has a k-quasiuniformpartition (see (2.8), (2.10), and (2.11)). Obviously, the class of domains with piecewisek-quasiuniform partitions is much larger than the class of domains with global k-quasi uniform partition. We have the following result:

Theorem 4.3 Let Th be a regular and piecewise k-quasiuniform partition of n. Letu E Hk+3/2(n). Then estimate (4.16) holds. 0

Theorem 4.4 Let Th be a regular and k-quasiuniform partition of n. Let u EHk+2(n) and Hi, H2 be satisfied. Then

for all Vh E Vh, where

Proof Let w = u - UI. We have

a(u, Vh) - a(uI, Vh)n

L L 1.(ai(X, \7u) - ai(X, '\lUI)) aiVh dxKETh i==1 K

n?= .L k(ai(x, \7uDF-1(x)) - ai(X, \7uIDF-1(x))) CijajVh dx . (4.18)K ETh ',)==1

15

(4.21 )

(4.19)

(4.20)

where C = (Cij)7,j==1= (DF-l(x){ JF(x). Let x E nand P = (PI,'" ,Pn),q =(ql, ... , qn) belong to ~n. Let gi(t) == ai(x,p + t(q - p)), t E [0,1]. Then

gi(O) = ai(x,p), gi(l) = ai(x,q) ,

g;(t)=t~~i(x,p+t(q-p)) (ql-PI) ,/=:1 r.,,1

ai(X, q) - ai(X,p) =11

g;(t) dt .

Let (J"ijbe independent oft. Multiplying (4.21) by (J"ijwe get

(ai(x,q) - ai(x,p))(J"ij =11

g;(t)(J"ij dt. (4.22)

Substituting q = \7uDF-l(X), P = \7uIDF-l(x) and (J"ij = CijajVh in (4.22) weobtain:

where

n (11aA51 = L a~i(x, \7uIDF-l(x) + t\7wDF-l(x)) dt1==1 0 r.,,1

- ~~;(X, \7UDF-l(x))) (ql- pL)CijajVh ,

52 = t ~ai(x, \7uDF-l(x))(ql- PI)CijajVh .1==1 el

Let us set (for fixed i and I)

aa·r(t,s) = ae; (x, \7uDF-l(x) + S (\7uIDF-l(x) + t\7wDF-l(x) - \7uDF-l(x))).

Then

r(t,O)

r( t, 1)

aai (A ~ADF-l( ))= ael x, vU x,

aa= ae; (x, \7udJF-l(x) + t\7wDF-l(x))

16

and we can express 51 as

n 1

51 = L r (r(t, 1) - r(t, 0)) dt (ql - PI)CijajVh .1==1 Jo (4.24)

Furthermore,

r(t, 1) - r(t, 0)

Using (4.24), (4.25), and assumption H2, we get

(4.26)

C (L la13ailo,oo,kXlRn)1131==2

.tit ((t - l)\7(u - uI)DF-l(x)(ql - PI)CijajVh dx dtl1==1 Jo J k

< Ch-3meas(K)lu - uII;,4,ilvhll,k

< Ch-3meas(K)lul~+1,4,klvhll,k

< Ch2kllull~+1,4,[{lvhl1,I{'

( 51 dx :S;Jk

Then

(4.27)

where the continuous embedding wk+2,2(n) C wk+l,4(n) was used. From (4.18),(4.23), and (4.27),

a( u, Vh) - a( UI, Vh)

::; C h2k lIulI~+2,n IVh 11,n

+ I L ttL ~~;(x, \7uDF-1(x))(ql - PI)CijajVh dxl (4.28)KETh ',)==1 1==1

17

Now we consider the second term on the right in (4.28). Define (for fixed i and l)

g(x) aa·ae;(x, \7u)

aa·ae; (x, \7uDF-l (x))

§(x) .

Note that g(x) E CO(n) for 'U E H2(n). Following the same steps as in Theorem 4.1,we have to estimate terms of the form

and

(4.29)

(4.30)

where xo corresponds to a fixed point in I< (see (4.3)). Using the quasiuniformity ofthe partition and assumption 1-12,

In order to estimate 53 we will use the same technique as in the estimation of 51.More specifically,

53 :S; L Ch-lmeas(K)II(§(x) - §(xo))(q[ - PL)lIo,klvhll,kKETh

:S; L Ch-lmeas(K)II§(x) - §(xo)llo,4,kllq/ - ptllo,4,klvhll,kKETh

:S; L Ch-2meas(K)I§ll,4,klu - ulll,4,k!vhll,kKETh

:S; L Ch-2meas(I{)h(meas(K)tl/4IgI1,4,Klulk+l,4,klvhll,kKETh

::;L Ch-2meas([{) (h + h(meas(K))-1/4IuI2,4,K) lulk+l,4,klvhll,kKETh

< Chk+l L (meas(K))1/41Iullk+l,4,I{lvhI1,I{KETh

18

+ Chk+l L IluII~+1,4,[(lvhll,K

[(ETh

( )

1/2

< Chk+l L (meas(J{))1/21Iull~+1,4,K IVhll,nKETh

+ Chk+lllull~+1,4,nlvhll,n< Chk+

l(1 + Ilullk+l,4,n)IIullk+l,4,nlvh!t,n

We obtain the desired estimate (4.17) from (4.28), (4.31), and (4.32).

(4.32)

o

Remark 4.3 A similar result to that in Theorem 4.4 also holds for piecewise k-quasiuniform partitions but with 1/2 order of convergence less. 0

Theorem 4.5 Let Th be a regular partition of n and the quadrature formula on [{

be exact for P2k-l(k). Let UI E h be the nodal interpolant of u. If u E Hk+l(n))f E Hk+l(n)) and assumptions Hi) H2) H4 are satisfied then

(4.33)

and

(4.34)

for all Vh E Vh) where

Q2 = 1 for linear elements (k = 1)

and

Q2 = 1 + IIull3,n + IIull~,n for quadratic elements (k = 2) .

Proof. Let us estimate the quadrature formula error over an element J{ E Th:

EK t,(i ai(x, VUI)8iVh dx - twI,K (ai(', VUI)8iVh)(bl,K))

it, (i ai(X, VuIDr')c;j8jVh dx - tWI (ai(" VuIDp-l)C;j8jVh) (hi)) ,

where (Cij)~j==1 = (DF-l(X){ !F(X). Here WI and bl are the weights and the nodesof the quadrature formula on J{,

19

We haven

IEKI :S;Clvhll,oo,k L laicijlo,oo,k .i,j==1

(4.35)

From the embedding H2(k) C CO(k),

laicij lo,oo,k :S;CllaiCij 112,k

Then using the exactness of the quadrature formula and the Bramble-Hilbert lemmawe get

n

IEKI :S; CIVhll,l{ L laicijlk+l,ki,j==1

n k+l

< Clvhll,k L L lails,klcij Ik+1-s,oo,ki,j==1 s==o

n k+l

< Chk+llvhll,k L L Ilaills,J{ ,i==1 s==o

where the estimate

ICijlk+l-s,oo,k :S;Chk-Smeas(J{) for O:S; s :S;3 , 1 :S;i,j ::; n ,

was used. Consider linear elements first. From HI, H2, and H4 we get

lai(X, \luI)ls,K :S; CfUlll,K

< C (lUI - ull,K + lull,K)

< Cllu112,K

for 0 :S;s :S;2. Hence

(4.36)

(4.37)

IEK I :S;Ch211uI12 Klvhlll{ (4.38), ,

and estimate (4.33) follows immediately with Q2 = 1.Now we consider quadratic elements. Taking into account HI, H2, and H4, the

semi norms lai(x, \luI)!s,K are bounded as follows:

lai(x, \luI)lo,K < CIUII1,K , (4.39)

lai( X, \luI) 11,K < C (Iulll,l{ + IUII2,K) , (4.40)

lai(x, \luI)!2,K < C (lulll,K + IUII2,K + IUII~,4,K) , (4.41 )

lai(x, \lUI )13 K < C (lulll,K + IUII2,K + IUII~,4,K + IUII~,6,K) (4.42),

20

In the same way as in (4.37),

IUIll.l{ ::; ClluI12,K'IUII2 K :S; Cllull2 K ,, ,

IUII2,4,K :S; ClluI12,4,K'IUII2,6,K :S; ClluI12,6,K'

Then, from (4.36) and (4.39)-(4.46),

IEKI ::; Ch3 (1IuI12,K + llull~,4,K + 111l1l~,6,[()IVhll,K .

Summation over all elements leads to

(4.43)

(4.44)

(4.45)

(4.46)

< Ch3 L (1IuI12,K + IJull~,4,K+ llull~,6,K) IVhll,KKETh

< Ch3 (1IuI12,n+ Ilull~,4,n + Ilull~,6,n) IVhll,n

< Ch3 (lluI12,n + Ilull;,n + Ilull~,n) IVhll,n

< Ch3 (1 + Ilul13,n+ Ilull;,n) IluI13,nlvhll,n , (4.47)

where the embedding W3,2(n) c W2,6(n) was used. This completes the proof of(4.33). Estimate (4.34) follows the proof of Theorems 4.1.5 and 4.4.5 in Ciarlet [8].o

5. Superconvergence Estimates

Let go = 0 denote the Gauss point of first order in the interval [-1, 1]; i.e. the rootof the Legendre polynomial of first degree Ll (x) = x, x E [-1, 1]. Also, denote bygl = -1/-/3, g2 = 1/-/3 the Gauss points of second order; i.e. the roots of theLegendre polynomial L2(X) = (3x2 - 1)/2, x E [-1,1]. Let eij, 1 :S; i < j ::; n + 1,be the edges of K E Th (note that some of the edges may be curvilinear). Then oneach edge eij of K we have one Gauss point of first order go,ij and two Gauss pointsof second order gl,ij and g2,ij (first, we map the interval [-1,1] onto the edge E.ij ofi<, and then we use the mapping FK : I< --+ K). Let TO,ij be the unit vector along thetangent of eij at the point gO,ij. Correspondingly, Tl,ij and T2,ij are the unit tangential

21

vectors at the points gl,ij and g2,ij. Next, let us define the following seminorms: whenk = 1 (linear elements)

when k= 2 (quadratic elements)

Theorem 5.1 Let Vh E Vh. Then

Proof It is sufficient to show that

Since the norms in finite-dimensional space are equivalent we obtain:

- 2C meas(K)lvhll,oo,K

< C meas(k)h-2Ivhl; 00 k, ,

< C meas(k)h-2Ivhl; k,

- 2 - 1 2 2< C meas([{)h- (meas(K)t h IVhl1 J{,

< Clvhli,J{' 0

Now we are in a position to formulate the main result in the paper.

Theorem 5.2 Let u and Uh be the solutions of the problems (2.2) and (2.12) respec-tively. Let the following assumptions be satisfied:

(i) U E Hk+2(n) n HJ(n), f E Hk+l(n);(ii) H1-H4;(iii) Th is regular and k-quasiuniform;(iv) the quadrature formula with positive coefficients is exact JOT P2k-l (k).

22

Then the following estimate holds

wherek

Q = 1+L Ilull~+2,n .8==1

Proof Let UI be the nodal interpolant of u. From Theorem 5.1 we get:

ju -llhl~.h :S; Iu - uII~,h + lUI - uhl~,h

< Iu - uII~,h + CIUI - uhll,n

Let Vh E Vh. Then

(5.1 )

(5.2)

ah( UI, Vh) - ah( Uh, Vh)ah(uI,vh) - a(uI,vh) + a(uI,vh) - a(u,vh) + a(u,vh) - ah(uh,vh)

= ah(uI,vh) - a(uI,vh) + a(uI,vh) - a(u,vh) + (J,Vh) - (J,vhh· (5.3)

From Theorems 4.4 and 4.5

la(uI,vh) - a(u,vh)1 :S; Chk+lllullk+2,nlvhlt,n' Ql , (5.4)

la( UI, Vh) - ah( UI, Vh) 1 :S; C hk+lllullk+l,n IVhIt,n . Q2 , (5.5)

I(J, Vh) - (J, vhhl :S; C hk+lllfllk+l,n IVhlt,n . (5.6)

Estimates (5.3)-(5.6) yield

Since the forms ah(-,') are uniformly Vh-strongly monotone, we have for Vh = UI - Uh

µluI - uhl~,n :S; ah( UI, UI - Uh) - ah( Uh, UI - Uh)

< Chk+l(\Iullk+2,n' Q + Ilfllk+l,n)luI - uhll,n

Hence

23

(5.8)

Now we deal with the term Iu - uII;,h' The discrete semi norm lu - uII~,h is con-structed in such a way that the following estimate holds:

( )

1/2

lu - UII~,h = L (Iu - UII;.h,K) 2KETh

< Chk+lllullk+2,n .

The desired result follows from (5.2), (5.8) and (5.9)

(5.9)

o

Remark 5.1 The same superconvergence estimate holds when the nonlinear coef-ficients in (2.3) have the form ai(x, u, \lu), 1 :S; i :S; n, with the following changedassumptions:

(HI *) The functions ai(x,O, x E n, e = (eo, el"'" en) E ~n+l), are continuous inn x ~n+l and for 1 < i < n

for all x E nand e E ~ n+1 .

(H2*) The derivatives aa(a13ai) for lal + 1(31 :S; k + 1, lal :S; k, 1(31 ~ 1,1 :S; i :S; n, arecontinuous and bounded in n x ~n+l:

(H3*) The following inquality holds

where µ > 0 is a constant independent of x, e and 'fl.(H4*) The functions aaai, \al = 1,2,3,1 ::; i :S; n, are continuous in n x ~n+1; thereexists a constant C > 0 such that

for all x E nand e E ~n+l .

24

o

Conclusions

In this work we develop superconvergent estimates for the tangential derivatives alongedges for 2D and 3D simplex finite elements applied to nonlinear elliptic problems.The results extend previous work - in this case to include quadratic elements withcurved edges and quasi uniform partitions. These superconvergent results are provedunder assumptions of monotonicity of the nonlinear elliptic operator and regularity ofthe nonlinear coefficients and problem solution. As with the corresponding estimatesfor the linear case, there are some restrictions on the amount of local distortion inthe mesh both with respect to deviation of curved edges away from linearity andalso in the shape of adjacent elements that share a common edge or face. Further,these results can be augmented by post-processing procedures to compute continuoussuperconvergent gradient approximations.

Aknowledgements

This research has been supported in part by DARPA Grant DABT63-96-C-0061.

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