mathematical analysis of zienkiewicz-zhu's derivative ...zzhang/paper/1996 - mathematical...

Mathematical Analysis of Zien kiewicz-Zhu’s Derivative Patch Recovery Technique Zhimin Zhang* and Harold Dean Victory, Jr. Department of Mathematics, Texas Tech University, Lubbock, Texas 79409

Received 1 1 January 1994; revised manuscript received 1 June 1995

Zienkiewicz-Zhu’s derivative patch recovery technique is analyzed for general quadrilateral finite elements. Under certain regular conditions on the meshes, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent for the second-order elliptic operators. For rectangular meshes and the Laplacian, the recovered gradient is superconvergent in the maximum norm at the nodal points. Furthermore, it is proved for a model two-point boundary- value problem that the recovery technique results in an “ultra-convergent” derivative recovery at the nodal points for quadratic finite elements when uniform meshes are used. o 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

Recently, a posteriori error estimates in the finite element method attract more and more attention (cf., e.g., [ 114 111 for a survey of various results), although the first article on this topic can be traced back to 1978 [12]. A closely related area is the analysis of superconvergence. This relationship is quite natural. Indeed, if we are able to recover a superconvergent solution from the post-processing, then we can use the recovered solution in place of the exact solution in computing an asymptotically exact error estimator. For the literature regarding superconvergence, the reader is referred to [ 1314 171, [30], and references therein.

The derivative recovery technique has been explored since the early 1970s (cf., e.g., [ 181 and [ 19]), and many methods have been suggested, for example, averaging techniques (cf., [20]-[23]), projection techniques (cf., [ 181, [ 19]), extrapolation techniques [24], and interpolated finite elements [25], etc. Recently, 0. C. Zienkiewicz and J. Z. Zhu proposed a new procedure [26] (Z-Z patch recovery) in which a continuous polynomial expansion of the function describing the derivatives is used on an element patch surrounding the nodes at which recovery is desired. This expansion is made to fit, in a least-squares sense, the derivatives of the finite element approximation at the Gaussian points. The

*To whom all correspondence should be addressed.

Numerical Methods for Partial Differential Equations 12,507-524 (1996) 0 1996 John Wiley & Sons, Inc. CCC 0749- 159)[/%/040507- 18

508 ZHANG AND VICTORY

nodal superconvergent recovery is observed from the numerical tests, although there is no theoretical justification for this behavior.

In a more recent work (see [4]), BabuSka, Strouboulis, et al. developed a completely numerical methodology for checking the quality of a posteriori error estimators. Results from the application of the methodology to the study of several popular error estimators show that the error estimator based on the Z-Z patch recovery is the most robust one.

In this work, the Z-Z patch recovery is investigated theoretically for a class of curved isoparametric quadrilateral finite elements of an arbitrary degree T in each variable. It is shown that for a general second-order elliptic operator, the arithmetic mean of the absolute error of the recovered gradient at the nodal points is superconvergent; and for the Laplace operator with a rectangular mesh, the recovered gradient at the nodal points is superconvergent in the maximum norm. A better result is obtained for the quadratic finite element approximation to a model two-point boundary value problem. It is shown that the recovered derivative at the nodal points converges at a rate O ( h 4 ) when uniform meshes are used. Our analysis confirms the computational results in [26].

In our analysis, we assume that the exact solution of our model problem is sufficiently smooth and the meshes of the finite element approximation are regular. In many practical problems, because of the lack of smoothness of the data and the boundary, the solution has singularity at certain points. Usually, adaptive local mesh refinement is applied to deal with solution singularities, and very likely the mesh is unstructured and nonuniform. Since the interpolation property we have obtained for the recovery technique is completely local, we believe that the results in the present work, combined with some local analysis based on interior estimates (cf., [ 15]), should be a reasonable approach when the solution has singular behavior.

II. GRADIENT SUPERCONVERGENCE OF THE FINITE ELEMENT SOLUTION FOR THE SECOND-ORDER ELLIPTIC PROBLEM

In this article, the Dirichlet problem is chosen as the model problem. In order to con- centrate on local recovery, some regularity assumptions are made on the exact solution in order to avoid the discussion of boundary behavior, corner singularities, etc. Throughout this article, C shall stand for generic constants independent of “essential” quantities that are not necessarily the same at each occurrence.

Let 0 c iw‘ be a bounded domain with a sufficiently smooth boundary r. Consider:

U I r = 0.

4 2 1 , v) = (f, v), V7J E (01,

The weak solution of (2.1 H2.2) is u E Hd (a), which satisfies

where

(2.1)

(2.2)

(2.3)

. . . ZIENKIEWICZ-ZHU'S DERIVATIVE PATCH RECOVERY TECHNIQUE 509

Assume that az3 (x)'s and b(x ) are sufficiently smooth with b 2 0, a13 (x) = ajl(x), and that a ( u , v) is H i (0)-elliptic, i.e., there exists y > 0 independent of z E 0 such that for all < = (El 1 E 2 ) E R2,

3

The aim of this article is to analyze the recovery technique. We shall not discuss the regularity of the model problem, but assume that the solution u of the model problem is sufficiently smooth to allow our analysis to be carried out.

The standard notation for Sobolev spaces is employed:

~ ' " ( 0 ) := {u E ~ ~ ( 0 1 , D*'U E ~~(~2),lal 5 m}, m = 0 , 1 , 2 , . . .

Hd(R) := {u E H'(R),ulr = O } ,

ws--(n) = {u E L"(R), D"u E L"(C2), la1 5 s}.

Denote by 1 . I , r L , ~ l and 1 1 . I l , r L , ~ l , the semi-norm and the norm in H"'(R), which are defined, respectively, by

I 4 L = c l l ~ ~ x ~ l l ; , q J l ) > l l~ll~ll,J~ = c 1 1 ~ 0 ~ 1 1 2 , ~ ( 1 2 ) ~ I C Y 1 =TI1 1<11<71l

and denote by I . I ,s,oo,c2 and 11 . ll,s,w,cl, the semi-norm and the norm in W"."(R), which are defined, respectively, by

l%GQ = max llD(t~llm,Q1 ~ l 4 ~ , ~ , Q = max l l~~'~I l rn,Q~ In I =.\ [ (1 15 s

To construct the finite element space in which the approximate solution will lie, let us cover 0 by curved isoparametric quadrilateral elements defined in the following manner. Let L,.(E) be the Legendre polynomial of degree r on [-1,1]. Then L , ( f ) has T zeroes gi:.) E ( - l , l ) , k = 1 , . . . , r , and Li.(<) has T - 1 zeros l:) E ( - l , l ) , k = 1 , . . . , T - 1. Denote $j") = -1, l!:) = 1; then

(

l;;.) < gy) < l ( 7 . ) , < . . . < $) < p < . . . < g!"' < p . ( Label gi.'",k = 1 , . . . , T , as the Gaussian points on (-1, l), and l i ) , k = O , 1 , . . . , T , as the

Lobatto points on \-l,l]. In two dimensions, denote k = 1-1, lj2, and define

(6 } T 2 - {(g(,.) g p ) } 7 . (1.+1)2 (7.) lW)} ' '

1 7=1 - 1r1 7 r i i , n = l > {Lj},=l = { ( l T i L 1 7) r n , 7 r = O ,

as the Gaussian and Lobatto points on k , respectively. Let P ( T ) denote the class of polynomials in ( E l , 5 2 ) E K, which are of degree r , and

Q ( T ) , the class of polynomials of degree T in each variable separately. In other words, any element p E p ( r ) has the form

7

51 0 ZHANG AND VICTORY

whereas any q E Q ( r ) will be of the form, T I '

1=0 J=O

Now any polynomial .i, E Q ( r ) is uniquely determined by its values at the Lobatto points L J , j = 1 , 2 , . . . , ( r + 1)2,v, = i j (L , ) , j = 1, . . . , ( r + I)', or vZ3 = . i , ( / : l ) , l : r l ) , i =

0 , l . . . . , r , j = 0 , 1 , . . . , r . Let N,(FI,E2) E Q ( T ) be the basis functions, i.e., NZ(LJ) = 6,,. To define the isoparametric mapping, let L, := (xp),zi?J)!,j = 1,. . . , ( ~ + 1 ) ~ , be ( ~ + 1 ) ~

points lying in f2 or on r and consider the mapping F K : K ---t R2 defined by

C = l

If the transformation (2.5) is one-to-one, mapping h; onto a closed domain K c R, K is called a curved quadrilateral element. The points L:, j = 1,. . . , ( r + 1)2, are the Lobatto points of K , and G: = (xl(C,),z2(G,)),j = 1 , . . . , r 2 , are the Gaussian points of K . As usual, h~ is the diameter of K , and h = nmxK h K .

Suppose R can be decomposed by such elements K , and we denote this partition by Tl . Let f11, be the interior of the union of all elements of the given partition, with rI l its boundary. Fo? the sake of simplicity, assume that R = R,, , r = r,,. The same results in this work can be obtained for the general case by using the methods of P. Lesaint and M. Z l h a l [27], [28].

The finite element space is defined as following:

S;;.'l := { I 1 E H;(f2),v(z1,2:!)(h. = . i ,(Elh(x1,22),E~(51,22)),K E T I } ,

with

, = I

Here El = < F ( x l r x 2 ) , 1 = 1,2 , is the inverse mapping of F K . Since the values vJ of u at nodes lying on r are zero, it is easy to see that u I ~ = 0. Similarly, S'." is defined without imposing zero trace on the boundary.

Definition 1. and

TI is r-strongly regular i f FK is a C' + I -diffeomorphism for every K E

(2.6) p"xy(<l r (2)1 5 Clh;', I V'lal 5 r + 1, 1 = 1,2,

C'g'hfi 5 J,<(<l,<2) 5 C;(.Lhi.. (2.7) Here Jk(El, t 2 ) is the Jacobian of the transformation described by (2.3, and the con-

stants C1 and C2 depend at most on r.

Remark 2.1. The sign of .JK ( € 1 , € 2 ) changes i f the local ordering of the nodes is taken in the opposite direction. Therefore, it is assumed without loss of generality that J K ( ( 1 , € 2 ) > 0 for any K E T,.

. . . ZlENKlEWlCZ ZHU'S DERIVATIVE PATCH RECOVERY TECHNIQUE 51 1

Condition I. For any two adjacent element K, K',

1, k = 1,2, for some constant C dependent at most on r.

Remark 2.2. Condition I is of a different nature than (2.7) in Definition I in that it involves two adjacent elements. This is satisfied, for example, if the meshes consist of elements that differ little from parallelograms having sides nearly parallel to the sides of its neighbors (cJ: [28] Remark 6 (p. 678)). For a more thorough explanation of r-strongly regular; the reader is referred to Remarks 1, 2, and 3 of [281.

For any function f defined on K, we define f on K via

f^ (€ l , E2) = f (53th t 2 ) . . ," (El , ( 2 ) ) .

We also define for v E H' ( G ) ,

where A] > 0 are the integration Gauss-Legendre weights. From 1281, I . Ih is a norm on Sh3". The finite element solution uh E Sh,h satisfies

a(u!,,v) = ( f , v ) , ~v E S;,'~. ( 1 .a The optimal error for u - uh in the H'-norm is (cf. [29])

1121 - uhllI,f2 I c h r ~ l ~ l l r + l , f 2 ~

However, P. Lesaint and M. Zlamal (cf., [27] Theorem 4.1) have shown that the arithmetic mean of the gradient error at the Gaussian points is superconvergent and, more precisely, they proved the following theorem.

Theorem 1.1. Let 5, be r-strongly regular and satisfy Condition I, and let u and U h

denote the solutions of (2.3) and (2.9), respectively. In addition, suppose u E Hr+'(G). Then

Iu - UhIh I ch'+111~IIr+3,r2,

where C is a constant independent of h and u.

Remark 2.3. In (2.9), an approximate bilinear form at, (uh, v ) could be used by employing approximate quadrature formulae to effect the integration. The resulting analysis would follow that of P. Lesaint and M . Zlamal in 1271, and would be more complicated, but the conclusions would be precisely those in the present work.

For the special case of the Laplacian with rectangular meshes, the following holds (cf. [21], Lemma 3.1).


Theorem 1.2. Let u and qt denote the solutions of (2.3) and (2.9). respectively. In addition, suppose u E WT.t:~,7L(fl) . Then for the special case all(x) = a22(x) = l,ala(x) = a2l(x) = 0 with R partitioned into quasi-uniform rectangles, there is a constant C,' independent of h and u such that,

IV(U - u , ~ ) ( G F ) ( 5 ChL?'(JuJ1,.+:3,,, i = 1,2,. . . , r 2 ,

f o r any Ii E ?;,. An essential condition of Theorem 2.1 or Theorem 2.2 is the regularity assumption on

u which is true whenever az,, 6, f, and r are sufficiently smooth. But in many practical problems. r is not smooth. A typical situation is the polygonal boundary. In this case, u has the corner singularity in general. For example, the solution of the Poisson equation with a smooth f and Dirichlet boundary conditions resides in H'trr/w-r (0) for any c > 0, where w is the largest inner angle on I'. Hence, for the square domain (w = 7r/2), u is an element of H"- ' (0 ) , and, for the L-shaped domain, u E H""-'(R) in general. We see that for polygon domain, the regularity assumption u E H'+'(Q)(u E IVr.t:3,.>o (0)) limits the application of Theorem 2.1 (Theorem 2.2) to some special situations, as in the example of Lesaint and Zlamal where f l is the unit square and u(z , y) = s( 1 - z)y( 1 - y)(l + 2s + 2y). See [27], Section 6; also see [26], Section 3.

For problems with singularities, more sophisticated analysis is required. This article considers only the situation when the global regularity assumption in Theorem 2.1 or in Theorem 2.2 is satisfied.

111. 2 2 PATCH RECOVERY

Denote by Ntl the set of element nodes, and define $ l = N,, \ r. Clearly f ih is the set of element nodes that is not on the boundary. Denote either as u, and 9 or % as o ! ~ . It is well known that u/L does not possess interelement continuity and has lower accuracy at element boundaries. The Z-Z patch recovery seeks to find, via a least-squares argument, a function u,.(x, y) defined on an element patch S ( 0 ) surrounding the particular

or

assembly node 0 E $h, where (z,y) is the local coordinate relative to 0. We represent ur(x, y) = P ( z , y)'"a' with either

P(x,y)" = ( l , z , y , . . . ,zT,x1'-'y,. . . , f ) ,

r 7

In other words, u,. is made to fit 61, at the Gaussian points in S ( 0 ) in a least-squares sense.

. . .ZIENKIEWICZ ZHU'S DERIVATIVE PATCH RECOVERY TECHNIQUE 513

In order to express matters more precisely, let us denote the Gaussian points as { (xZr yz): i = 1,. . . , n}: then the aforementioned least-squares fitting is to minimize

n

z = 1 n

n

r = l

where

1 7' a = (al,a2h,a3h,adh2,. . . ,arrlh ) .

We shall consider only the case where P ( x , y) is a linear combination of basis elements for polynomials of degree no more than T. We omit the discussion for the other case, whereby P(x ,y) is as a linear combination of basis elements for polynomials of degree T in each variable separately. The treatment for the latter case is similar and yields the same order of approximation.

The minimum a of F must of necessity satisfy ri 11

cp(E*,v?)p(Ez,vz)7 'a = ~ ~ ~ E 2 , v 2 ) ~ , , ( x ? , Y 1 ) . ? = 1 t=l

In matrix form, this condition becomes

where

7 6 L = ( ~ h ( ~ l , Y 1 ) , . . ' , . h ( ~ l l , Y 1 1 ) ) '

(The purpose of solving for 6 instead of a' is to eliminate the numerical ill-conditioning of the system (3.1) for higher T. Obviously, C is independent of h.)

( A l l

Assume that

Rank (A) = ; ( T + l)(r + 2) = m.

This is a reasonable assumption and is satisfied if enough Gaussian points are selected, or, equivalently, if the element patch is enlarged. In particular, for rectangular meshes, a natural element patch is a set of four elements that share a common vertex. It is easy to see that in such an element patch, there are n = 4r2 Gaussian points.


Remark 3.1. Since the set of 4r2 Gaussian point is Q(2r - 1)-unisolvent and, for r 2 1 , P ( r ) c Q(2r - l ) , the element patch that contains 4 elements with a common vertex satisfies (-41) automatically. Also, we see that in order to satisfy the assumption ( A l ) , we do not have to use all 4r2 Gaussian points. The minimum requirement is ( r + 1 ) x ( r + 1 ) Gaussian points.

By assumption ( A l ) , A T A is invertible. Thus, (3.1) can be uniquely solved, and the value of a,. at the assembly node 0 is

(3.2)

where 6 = ( a 1 , . . . , is the first row of the matrix (A'A)-'A7', which depends only on r and the relative positions of (<%, q7)'s. By (3.2), the least-squares fitting process defines a nodal recovery operator R such that

(RatI ) ( O ) = G ( 0 ) 7 ' 6 1 (0)

for all 0 E N,l. Clearly, this operator can be applied to any function that is well defined at the Gaussian points. In particular, when applied to a = [g] or to o = [ % I , we get

-7. - a,.(O,O) = P(O10)7a' = P(OIO),% = a1 = a

0

( 0 ) (0) (0) (0 ) 7- ( R a ) ( O ) = w)'l'w), Lw) = (4x1 1Y1 1 1 . . ' 1 4 x 7 1 l Y 7 1 ) ) .

By Holder's inequality, for any a E H r f l (0) + S",", r 2 1 ,

(3.3) (3.4)

Here 1 1 . I I l r I I . 112, and 1 1 . 1 1 % are the 1 ' , Euclidean, and lm-norms for R".

ator R. The following theorem states the interpolation property of the nodal recovery oper-

Theorem 3.1. Let ( A l ) be satisfied then for any 0 E Wr+',*(0) and 0 E 31, I(ff - Rff)(O)l I ~ ~ ( ~ ) ~ " + ' I ~ l , . + l , z , . s ( 0 ) , (3.5)

with 2"+1

(7- + l ) ! C ( r ) = ~ c bll. 1=1

Proof. To simplify the notation, we suppress the dependent index "0". Since 6'" is

(3.6)

the first row of (A '"A) - 'A7 ' , then

&"'A = (110, . . . , O ) ,

and hence I 1

Ca7 = 1, 1 = 1

(3.7)

7 1 11

7 = 1 2 = I

. . . ZlENKlEWlCZ -ZHU'S DERIVATIVE PATCH RECOVERY TECHNIQUE 515

By the Taylor expansion of u in the local coordinates, we see 71

when applying (3.7) and (3.8). To estimate the residual term, note that since 1x21 < h, ly21 < h, and

(3.9)

we have

(3.10)

The conclusion follows by combining (3.9) and (3.10). rn Corollary. vertex, then for any u E W 7 + 2 "(0) with T even,

If S ( 0 ) contains 4 uniform rectangular elements that have 0 as a common

(3.1 1) I(. - Rff)(O)l I C(r + l )h7+Lla17+2 .~ ,S (0 ) , where C ( r ) is dejned as same as in Theorem 3.1.

Proof. By hypothesis, S ( 0 ) contains 4 uniform rectangular elements, and assumption ( A l ) is automatically satisfied (see Remark 4). Furthermore, we know that (-x3, -9,) E S ( 0 ) whenever (x , y,) E S ( 0 ) . Hence, if xty,k-' appears in (3.9), there exists a j such that 1 5 3 < n a n d s : & ' = (-xl)'(-y7)'.-'alsoappearsin(3.9),1 5 k s r + l , l = O , l , . . . , k. Moreover, they have the same coefficient by symmetry, i.e., a] = a,. Consequently,

I1

- C Q 7 ( - x 2 ) ' ( - y 2 ) ' + l - ' = ( - l ) '+ l C a , x t y : + ' - ' . z = 1 1 = 1 7 = 1


When T is an even number, 71

1=1 1 = 1

Following the same arguments as used in Theorem 3.1, we will have a residual term r+2 a 1 1

~ 1 (1 - t ) “+ ’& (..- ax + yl”) a y a(tx,,ty,)dt. + 1Y 0 1=1

The estimate results in exactly the same manner as in Theorem 3.1 by changing T to T + 1. . Next a stability result for the recovery operator R will be shown. Let us denote R =

(R, R) , J?h = { O J } y J l , where Nil is the number of internal element nodes, and assume that

( A 2 ) r I laX1<j<Nf , IlG(o)lll 5 C I , R O I I N D , maxl<j<N), 11G(O)112 5 C 2 . 1 3 0 I i N D ,

where C l , n 0 r l N D and C ~ , M I ( I N I ) are constants independent of Nh or h. The following assumption is the standard quasi-uniformity of the meshes:

(A31 h 5 rrnink h k , for some T > 0.

Remark 3.2. For regular partitions, (A2) is a reasonable assumption. For example. when the partition is uniform, all ~ ~ G ( O J ) ~ ~ l (or ~ ~ G ( O l ) \ ~ 2 ) , j = 1 , 2 , . . . , N,,, are the same and will remain so ifthe uniform mesh refinement is applied. Ifthe partition is not uniform, but rejnement follows in a uniform way, ~ ~ G ( O J ) ~ ~ l (or ~ ~ G ( O J ) ~ ~ ~ ) will be uniformly bounded independently of N,, (or h).

As a consequence of (A3) and (2.7), there exists C,, > 0 independent of h such that

(3.13) C,,’A,J,&) L - 1 I C,3AlJ&,), 3 = 1 , 2 , . . . , T 2 . Nh

Theorem 3.2. Under the assumptions of (A l ) , (A2), (A3), and (2.7),

where

. . . ZlENKlEWlCZ- ZHU'S DERIVATIVE PATCH RECOVERY TECHNIQUE 517

From (A2) ,

Consequently,

(3.14)

+ - (GJ) - -(GJ) [t EG d Y 13 (3.15) - 2 2

- C ; L , B O U N D C ~ ~ ~ - u h l h ,

by virtue of (3.14), (3.13), and the definition of I . Ih. From Holder's inequality,

We now show the two main results of this section.

Theorem 3.3. and (A3). Let the element patch be chosen so that (Al ) is satisfied. Then

Let l h be r-strongly regular and satisfy Condition I , assumptions (A2),

where C is the same constant as in Theorem 2.1 and {Ol}sv_"l is the set of all interior element nodes.

Proof. By the triangle inequality,

Il(vU - RVUh)(OJ)l 5 I(vU. - RVU)(OI)I 4- I(RV(.tl- U h ) ( 0 3 ) 1 . (3.16)

and o = %, respectively, we have Using Theorem 3.1 twice by setting 0 = 0, , ~7 =


Hence,

( (Vu - RVU)(O,)( L J Z C ( T ) ~ ' + ' ( U ( , . + ~ . ~ .

Summing up (3.16) over j = 1 , 2 . . . N I , , and dividing by N,, , we have

(3.17)

Theorem 3.4. element node O,, j = 1,2,. . . , N I , ,

Under the same assumptions as in Theorem 2.2, we have for any interior

I(Vu - RVW,)(O,)I 5 JZ(C ' (T ) Iu I r+2 z + C 1 . I ~ O I ~ N / ~ ~ l l ~ l l r + . l , z ) ~ r + ' r

where C' is the same constant appearing in Theorem 2.2.

3.1). By using (3.3), we have as in (3.14), Proof. With uniform rectangular meshes, (A 1) is automatically satisfied (see Remark

( J ) ( J ) IRV(U. - W,)(O,)\ 5 C I , ~ O I ~ N D J Z I ( l ( 7 l max (V(u - U I , ) ( Z ~ , Y, ) I

5 Jzc:, , D O l J N DCh' + l l ~ l l I +3,% 1 (3.18)

where C is the same constant as in Theorem 2.2. Inserting (3.17) and (3.18) into (3.161, we obtain the conclusion.

Conclusion and Remarks

1.

2.

3.

4.

Theorem 3.3 states that the arithmetic mean of the absolute error of the recovered gradient at the nodal points inside fl is superconvergent for a class of regular curved quadrilateral meshes. Theorem 3.4 states that, for uniform rectangular meshes, the recovered gradient is superconvergent in the maximum norm at every nodal point inside 0. This has been observed by practical computation [261. Theorem 3.1 describes the interpolation properties of the Z Z patch recovery. It clearly shows that the interpolation feature of the Z- Z patch recovery is insensitive to the type of meshes. A parallel result of Theorem 3.1 holds for triangular meshes. These results partially explain the robustness of the method under extreme distortions of the mesh. Another feature of the analysis in this article is the concrete knowledge of the constants multiplying the rate of convergence. This is very important in the practical design of the finite element method. The generalization of the analysis to three dimensions is straightforward. Here,

1 i, y1 i, . " 5; ~ ; - l y l ~ ; - l ~ l . . . 2; 1 22 $2 i2 ' . ' x; x2 y2x2 22 . ' . i;

1 y7, i,, ' . ' x;; i ; , - ~ y , , i ; , - l i l , . . . 2;;

- r - l - A r - l -

A = . . . . . . . . . . . . . . . . . .

. . . ZlENKlEWlCZ ZHU’S DERIVATIVE PATCH RECOVERY TECHNIQUE 519

and the coefficient vector ir”’ is still the first row of the pseudo-inverse of A . 5. We would like to point out that when piecewise bilinear elements are used and

quadrilaterals are produced by a bisection scheme of mesh subdivisions, the finite element triangulation is 1 -strongly regular (see Definition 1 ) and satisfies Condition I, and Assumptions (Al), (A2), and (A3). Consequently, the Z Z recovery technique will result in a superconvergent gradient field.

IV. ULTRA-CONVERGENCE ANALYSIS IN 1-D

The authors of [26] announced an O(h“) convergent rate for the nodal values of the recovered derivative with quadratic elements when uniform meshes are used. Since this rate is two orders higher than the optimal rate, it is then termed “ultra-convergence.” In this section, we provide a theoretical justification of this celebrated phenomenon for the following two-point boundary-value problem:

(4.1 ) 4’ + b(x)u = f in I = (0, I ) ,

u(0) = u(1) = 0. (4.2)

We assume that b and f are sufficiently smooth for our analysis. The weak formulation of (4.1), (4.2) is to find u E H, f (Z ) such that

( u ’ , ~ ’ ) + (bu,u) = ( f l u ) Vv E H ( { ( Z ) . (4.3)

Let T,, 0 < h < 1/2, be a sequence of uniform subdivisions of f, N 7j, = { x ~ } ~ = ~ , z ~ = zh. h = 1/N;

let I , = ( X ~ - ~ , Z ~ ) , and set

sg = {. E H ( ; ( o , VIIt E P2(L) } .

We see that Sll is the space of piecewise quadratic polynomials on I under the uniform subdivision TL.

The finite element solution of (4.3) is to find ult E SL such that

( u ; L , 4 + (bulL,V) = ( f , V ) vu E s;. (4.4)

The first step of ultra-convergence analysis is to reduce (4.4) to a simpler problem.

(4.5)

Subtracting (4.4) from (4.3) yields

(u’ - & , 2 ) ’ ) + (b (u - u,,),2)) = 0 vv E s;. Let i i l , E SL be given by

(u’ - i&, 2)’) = 0 V V E s;;. (4.6)

Then we have the following “ultra-approximation’’ result between ulL and 711, (see [30] Remark 1.3.1).


Lemma 4.1. Let u),, satisb (4.51, (4.6), respectively. Then ’ I I I % - ~ / l l l L % ( / ) 5 Ch4Il4lrr.:(I,? (4.7)

where C is a constant independent of h and u. By virtue of Lemma 4.1, we can reduce our discussion to a simple case:

-u” = f u(0) = u(1) = 0.

in I = (0, I ) ,

or

(u’, 4 = (f, .) wu E 4; ( 0 1 (4.8)

since the finite element solution of (4.8) satisfies (4.6). In the following, we will construct the finite element solution u ~ , E SE for (4.8) and

prove the ultra-convergence property of Ru’,, at the nodal points. Here Ru’,, is the recovered derivative by the Z Z patch recovery technique.

We characterize Sii by the following basis functions:

s;: =Span {~,(z),i = 1 , . . . , N - 1; 4 J ( z ) , j = I , . . .,N}, where

(4.9)

(4.10)

It is well known that the finite element solution of the variational problem (4.8) is exact at the nodal points, i.e., u/ , (z , ) = u(z , ) ,z = 0 ,1 , . . . , N. We then have on Z,,

W I ( 5 ) = 4z,-l)N,-l(z) + 4 z , ) N ( z ) + c,dh(z), (4.1 1 )

where

c, = (fl41)/(4:$4;). Associated with any interior node z,, z = 1,:. . , N - 1, there is an element patch J , =

(z, - h, z, + h), and a linear mapping F, from I = (- 1 , l ) onto J , defined by z = z, + ht. Given any function u on J t , we define

6 = v o F,, or 6(<) = u(F , (<) ) = v(z, + ht ) .

Now, consider h

~ ’ (z , ) - Ruk(z,) = (u’ - Ru’,,, 6,) = h(u’ - R U ; , ~ ~ ) = hE(C’). (4.12)

Here 8, = 6 o F, and 6 is the Delta function. Obviously, E(C’) is a linear functional that is bounded in W&(i ) . We shall show that E(C’) vanishes when C’ is a polynomial of degree no more than 3. To this end, we first introduce the following result from [31].

. . . ZIENKIEWICZ-ZHUS DERIVATIVE PATCH RECOVERY TECHNIQUE 521

Lemma 4.2. Let u be the solution of (4.8), and let Uh be i t s jn i te element approximation on 5’:. Assume that u is a polynomial of degree no more than 3 on J,, then Rui, = u’ on J,.

Note that uiL is a piecewise linear function on J , with a possible jump at x,, but Ru’, is a quadratic polynomial on J,. We see that E ( 2 ) vanishes when 6’ is a quadratic polynomial.

Next, we examine the case when

on J,. Note that u’(x,) = 0. Since u(x) is symmetric with respect to L C , , we have

c, = ( -U’’ ,42)/(4: ,4;) = ( -u”,4,+1)/(4t+l ,4t+~) = G+1.

Recall (4.1 1 ), and we have on J , ,

Observe that

we then have

Hence, by the Z -Z patch recovery procedure

Ru;,(a,) = c , a [u;, ( 5 . - 5 (1 - A)) + u’, (x. + 5 (1 - A))] + c,P [.it ( X I - ; (1 + 5)) +u;, ( 5 , + 5 (1 + $))I

= 0 = u’(5,). (4.13)

Here

5, f 5 (1 - 5) > 5 2 f 5 (1 + 5) 2 2

are the four Gauss points on the element patch J , and a , /3 are weights of the least-squares fitting.

Since any u E P4(J,) can be decomposed into 2

u(5) = a ( ” - ; . - ’ ) 2 ( 2 - z,+1 ) + W ( X )

for some a E R’ and w E P7(Jz), we see from Lemma 4.2 and (4.13) that

Rui , (~z) = u’(5,) th E P,(JL), (4.14)


i.e., the linear functional E ( 2 ) vanishes for all u‘ E P 3 ( i ) . Therefore, by the Bramble- Hilbert Lemma, we have

P@’)I I Cll& llLv;J(j) I%/$( i)

I Ch- l l ~ * l l w ~ J ( ~ , ~ ~ ~ l ~ ’ l w ~ ~ . l , ~

= Ch“ lul w: (.I, ) . (4.15)

Note that (15, I l c t . ~ ~ ( I ) = 1.

this section.

I , ‘ Combining (4.12), (4.15) with Lemma 4.1, we have proved the following main result of

Theorem 4.1. Let u be the solution of problem (4.1), (4.2), let ul, E SE be itsJinite element approximation, and let Ru;~ be the recovered derivative from the Z Z patch recovery technique. Then

IU’(Z1) - R4,(zt)l I c~1(14w:(.I ,) + I l 4 W ~ ( / ) ) > (4.16)

where C is a constant independent of h and u.

Remark 4.1. Since the above argument is completely local, the result of Theorem 4.1 holds for globally nonuniform but locally uniform meshes. The only requirement is on the element patch, namely that two elements have the same length.

Remark 4.2. The conclusion for general two-point boundary-value problems

-(u2(s)u’)’ - (ul(x)u)’ + uo(x)u = f in I , (4.17)

with the boundary condition (4.2) is unknown at present. The reason is that we have only the “super-approximation” between uil and i& in this case, i.e.,

lM1 - ~ ; , l l L , ~ ( l ) I Ch”ll~llu,;(/)?

instead of (4.7). See [30], Theorem 1.3.1, for the details.

Remark 4.3. The generalization to the higher-order polynomials is possible. If the finite element subspace 5’;: contains piecewise polynomials of degree 2r, ( T 2 l) , then we conjecture that under local uniform meshes,

Iu’(z1) - Ru;,(xc,)l 5 ch2’+L(IuIM.:‘t’(.,,) + l l ~ l l w y ( / ) ) ,

where u is the solution of the two-point boundary problem (4.1), (4.2).

The authors are grateful to Professor Ivo Babuika and Dr. J. Z. Zhu for many help- ful discussions. The work of these authors was partially supported by National Science Foundation Grant DMS-90-23063.

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