Least-squares finite element approximations to the Timoshenko beam problem

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Least-squares ®nite element approximations

to the Timoshenko beam problem

Jang Jou

a,b,*

, Suh-Yuh Yang

c,1

aDepartment of Applied Mathematics, National Chiao Tung University, 1001, Ta Hsueh Road,

Hsinchu 30050, Taiwan, ROC

bDepartment of Statistics, Ming Chuan University, Taipei 11120, Taiwan, ROC cDepartment of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung 84008, Taiwan, ROC

Abstract

In this paper a least-squares ®nite element method for the Timoshenko beam problem is proposed and analyzed. The method is shown to be convergent and stable without requiring extra smoothness of the exact solutions. For suciently regular exact

solutions, the method achieves optimal order of convergence in the H1-norm for all the

unknowns (displacement, rotation, shear, moment), uniformly in the small parameter which is generally proportional to the ratio of thickness to length. Thus the locking phenomenon disappears as the parameter tends to zero. A sharp a posteriori error

es-timator which is exact in the energy norm and equivalent in the H1-norm is also brie¯y

discussed. Ó 2000 Published by Elsevier Science Inc. All rights reserved.

AMS classi®cation: 65N15; 65N30

Keywords: Timoshenko beam problem; Least-squares; Finite element method; Locking phenom-enon; A posteriori error estimator

1. Introduction

The locking phenomenon is mainly concerned in the ®nite element analysis for parameter-dependent problems [2,5]. In this paper we shall propose and analyze a ®nite element method based on the least-squares principles for

www.elsevier.com/locate/amc

*Corresponding author. E-mail: jangjou@math.nctu.edu.tw

1The work of this author was supported in part by NSC-grant 87-2811-M-007-0017, Taiwan,

ROC.

0096-3003/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved. PII: S0096-3003(99)00139-3

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approximating the solution of the Timoshenko beam problem in its ®rst-order system formulation. The method avoids the locking problem as the thickness of the beam tends to zero, and achieves optimal order error estimates for all the unknowns (displacement, rotation, shear, moment) in the H1-norm.

Finite element analysis of the Timoshenko beam problem has been fre-quently used as a starting point for a better understanding of the much more complex problem of constructing accurate ®nite element approximations to the Reissner±Mindlin plate problem. It is well-known that some bad behaviors may occur such as the locking phenomenon when we solve these problems with standard Galerkin ®nite element methods [2,19,26,27]. That is, the convergence results of the standard ®nite element approximations deteriorate as the small parameters (which depend on the thickness of the beam and the plate for the Timoshenko beam model and the Reissner±Mindlin plate model, respectively) tend to zero. The most widely used e€ective approach to overcome the di-culty is based on the reduced integration technique. A complete analysis for the Timoshenko beam problem by using this technique has been addressed in [2]. Recently, a modi®ed reduced integration method with linear ®nite elements is proposed and analyzed by Cheng et al. [19]. The method presented in [19] uses the reduced integration technique to compute the term involving the small parameter and adds a bubble function space to the rotation to increase the solution accuracy. This method can also be applied to solve the circular arch problem and the Reissner±Mindlin plate problem. Another way to eliminate the locking phenomenon is to use the p and h ÿ p versions of the ®nite element method for which optimal error estimates are established in [25].

In the present investigation we provide an alternate way to avoid the diculty by exploiting the least-squares principles on a ®rst-order system formulation of the Timoshenko beam problem. We ®rst introduce a quadratic least-squares functional Q over a function space V consisting of functions which satisfy the boundary conditions of the problem. The functional is de®ned to be the sum of the squared L2-norms of the residuals in the di€erential

equations. Then the exact solution must be the unique zero minimizer of the functional Q over V. Therefore, the least-squares ®nite element approximate solution is de®ned to be the minimizer of Q over a ®nite-dimensional subspace Vp

hof V. Mathematical analyses show this approach can eliminate the locking

phenomenon.

Over the past decade, the use of least-squares principles in connection with ®nite element techniques has been extensively applied to the approximations in many di€erent ®elds such as ¯uid dynamics [9,12,13,16±18], elasticity [14,29,30], electromagnetism [15,24], and semiconductor device physics [8]. The approach o€ers certain advantages, especially for large-scale computations. It leads to minimization problems rather than saddle point problems led by the mixed ®nite element approach. A single continuous piecewise polynomial space can be used for the approximation of all unknowns, and accurate

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approxi-mations of all unknowns can be simultaneously obtained. The resulting alge-braic system is symmetric and positive de®nite. In addition, the value of the least-squares functional of the approximate solution provides a practical and sharp a posteriori error estimator at no additional cost [21,23,28]. This feature is very important in adaptive computations.

The layout of the remainder of the paper is as follows. In Section 2, we introduce the ®rst-order system formulation for the Timoshenko beam prob-lem and we then derive important a priori estimates for the system. The least-squares ®nite element method is given in Section 3. Then the method is proved to be convergent and stable in Section 4, where optimal error estimates in the H1-norm are also established. Finally, in Section 5, we brie¯y discuss the sharp

a posteriori error estimator for the least-squares approach. 2. The Timoshenko beam problem

In this paper we shall consider the in-plane bending of a clamped uniform beam of length L, cross-section A, moment of inertia I, Young's modulus E, and shear modulus G, subjected to a distributed load p…x†, and a distributed moment m…x†, with x 2 …0; L† representing the independent variable. According to the Timoshenko beam theory, this problem is governed by the following system of ordinary di€erential equations of ®rst-order (cf. [27]):

ÿdQdx ˆ p in …0; L†; …2:1†

ÿdMdx ÿ Q ˆ m in …0; L†; …2:2†

ÿjGAQ ‡dwdxÿ h ˆ 0 in …0; L†; …2:3†

ÿMEI‡dhdxˆ 0 in …0; L†; …2:4†

supplemented with the boundary conditions:

w…0† ˆ w…L† ˆ 0; …2:5†

h…0† ˆ h…L† ˆ 0; …2:6†

where Q…x† is the shear force; M…x† the bending moment; w…x† the transverse displacement; h…x† the cross-section rotation; and j the shear correction factor.

To explicate the dependence of this problem on a small parameter, e2ˆ EI

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we introduce the following change of variables: x ˆx L; …2:8† u1ˆwL; u2ˆ h; …2:9† r1ˆQL 2 EI ; r2ˆ ML EI ; …2:10† f1ˆpL 3 EI ; f2ˆ mL2 EI ; …2:11†

which reduces the original problem to ®nding u…x† ˆ …u1…x†; u2…x†† and r…x† ˆ

…r1…x†; r2…x††, x 2 …0; 1† satisfying ÿ r0 1ˆ f1 in …0; 1†; …2:12† ÿ r0 2ÿ r1ˆ f2 in …0; 1†; …2:13† ÿ e2r 1‡ u01ÿ u2ˆ 0 in …0; 1†; …2:14† ÿ r2‡ u02ˆ 0 in …0; 1†; …2:15†

with the boundary conditions

u1…0† ˆ u1…1† ˆ 0; …2:16†

u2…0† ˆ u2…1† ˆ 0; …2:17†

where the prime superscript denotes di€erentiation with respect to the nondi-mensional variable x. We observe that the nondinondi-mensional problem (2.12)± (2.17) depends explicitly on a parameter e. In general, e is a small parameter proportional to the ratio of the thickness to length. For instance, in the rect-angular cross-section case [26],

e2ˆ E 12jG T L  2 ;

where T represents the thickness of the beam. Thus in most realistic applica-tions, e.g., for thin beams, 0 < e  1, and the construction of accurate ®nite element approximations is delicate.

We need some function spaces throughout this paper. The classical Sobolev spaces Hr…0; 1† with their associated norms k  k

r are employed [11,20]. As

usual, we denote by …; †0and k  k0the conventional inner product and norm on the Hilbert space L2…0; 1† of square-integrable functions. The space H1…0; 1†

of functions which together with their ®rst derivatives are square-integrable, is a Hilbert space with the inner product …; †1, where

…u; v†1ˆ Z 1

0 …uv ‡ u

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The norm generated by this inner product is denoted by k  k1; that is, kvk1ˆ …v; v†1=2

1 . We denote by H01…0; 1† the subspace of H1…0; 1† consisting of

functions which vanish at the ends of the interval. For the product space ‰H1…0; 1†Š4, the corresponding inner product and norm are also denoted by

…; †1and k  k1, respectively, when there is no chance of confusion.

We shall use C with or without subscripts in this paper to denote a generic positive constant, possibly di€erent at di€erent occurrences, that is independent of the parameter e and the mesh parameter h introduced in the next section.

The following regularity result plays an important role in the theoretical analysis later.

Theorem 2.1. Let …v; x† ˆ …v1; v2; x1; x2† 2 ‰H01…0; 1†Š2 ‰H1…0; 1†Š2. Then there

exist two positive constants C1and C2both independent of the parameter e such

that C1 kv1k21  ‡ kv2k21‡ kx1k21‡ kx2k21  6 Q…v; x; 0† …2:18† and Q…v; x; 0† 6 C2 kv1k21  ‡ kv2k21‡ kx1k21‡ kx2k21  ; …2:19† where Q…v; x; 0† :ˆ kx0 1k20‡ kx02‡ x1k20‡ ke2x1ÿ v01‡ v2k20‡ kx2ÿ v02k20:

Proof. Upper bound (2.19) is straightforward from the triangle and Cauchy± Schwarz inequalities. To prove the lower bound (2.18), we ®rst de®ne

g1:ˆ ÿ x01; …2:20†

g2:ˆ ÿ x02ÿ x1; …2:21†

g3:ˆ ÿ e2x1‡ v01ÿ v2; …2:22†

g4:ˆ ÿ x2‡ v02; …2:23†

in ‰0; 1Š. Then from the ®rst Eq. (2.20), we ®nd x1…x† ˆ ÿG1…x† ‡ c1;

where

G1…x† :ˆ

Z x

0 g1…t†dt; x 2 ‰0; 1Š; c1:ˆ x1…0†:

Integrating the second Eq. (2.21), we have x2…x† ˆ ÿG2…x† ‡ Z x 0 G1…t†dt ÿ c1x ‡ c2; where G2…x† :ˆ Z x 0 g2…t†dt; x 2 ‰0; 1Š; c2:ˆ x2…0†:

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Solving the last Eq. (2.23) together with the boundary condition v2…0† ˆ 0, we obtain v2…x† ˆ ÿ Z x 0 G2…s†ds ‡ Z x 0 Z s 0 G1…t†dt ds ÿ 1 2c1x2‡ c2x ‡ G4…x†; where G4…x† :ˆ Z x 0 g4…t†dt; x; s 2 ‰0; 1Š:

Finally, from the third Eq. (2.22) and the boundary condition v1…0† ˆ 0, we

®nd for x; y; s 2 ‰0; 1Š v1…x† ˆ ÿe2 Z x 0 G1…t†dt ‡ e 2c 1x ‡ G3…x† ÿ Z x 0 Z y 0 G2…s†dsdy ‡ Z x 0 Z y 0 Z s 0 G1…t†dt dsdy ÿ 1 6c1x3‡ 1 2c2x2‡ Z x 0 G4…y†dy; where G3…x† :ˆ Z x 0 g3…t†dt; x 2 ‰0; 1Š:

Utilizing the boundary conditions v2…1† ˆ 0 and v1…1† ˆ 0, we get the following

2  2 linear system of equations in the unknowns c1 and c2:

ÿ12c1‡ c2ˆ Z 1 0 G2…s†ds ÿ Z 1 0 Z s 0 G1…t†dt ds ÿ G4…1† …e2ÿ1 6†c1‡ 1 2c2ˆ e2 Z 1 0 G1…t†dt ÿ G3…1† ‡ Z 1 0 Z y 0 G2…s†dsdy ÿ Z 1 0 Z y 0 Z s 0 G1…t†dt dsdy ÿ Z 1 0 G4…y†dy:

Solving this linear system, we can ®nd the following estimates for c1 and c2,

respectively, jc1j 6 C X4 iˆ1 kgik0; jc2j 6 C X4 iˆ1 kgik0

for some constant C independent of the parameter e. Now it is easy to see that kvik216 CQ…v; x; 0†;

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for i ˆ 1; 2. Thus the assertion (2.18) follows immediately and this completes the proof. 

3. The least-squares ®nite element method

In this section we introduce the least-squares ®nite element method for problem (2.12)±(2.17). De®ne a function space V for our problem by

V ˆ H1

0…0; 1†  H01…0; 1†  H1…0; 1†  H1…0; 1† …3:1†

and then de®ne a quadratic least-squares energy functional Q: V ! R by Q…v; x; f† ˆ k ÿ x0

1ÿ f1k20‡ k ÿ x02ÿ x1ÿ f2k20‡ k ÿ e2x1‡ v01ÿ v2k20

‡ k ÿ x2‡ v02k20; …3:2†

where v ˆ …v1; v2†, x ˆ …x1; x2†, and f ˆ …f1; f2†. Note that the quadratic

en-ergy functional Q…v; x; f† is de®ned to be the sum of the squared L2-norms of

the residuals in the di€erential equations. Obviously, the exact solution …u; r† 2 V of problem (2.12)±(2.17) is the unique zero minimizer of the functional Q on V, that is,

Q…u; r; f† ˆ 0 ˆ min Q…v; x; f†: …v; x† 2 Vf g: …3:3† Applying the variational techniques, we can ®nd that (3.3) is equivalent to

B……u; r†; …v; x†† ˆ F……v; x†† 8 …v; x† 2 V; …3:4†

where the bilinear form B…; † and the linear form F…† are de®ned, respec-tively, by B……v; x†; …z; .†† ˆ Z 1 0 …x 0 1.01‡ …x02‡ x1†….02‡ .1† ‡ …ÿe2x 1‡ v01ÿ v2†…ÿe2.1‡ z01ÿ z2† ‡ …ÿx2‡ v02†…ÿ.2‡ z02††dx; …3:5† F……v; x†† ˆ Z 1 0 ÿ ÿ x0 1f1‡ … ÿ x02ÿ x1†f2dx …3:6† for all …v; x†, …z; .† 2 V.

It is evidently that B…; † is symmetric and continuous (bounded) on V  V and, for each given f 2 ‰L2…0; 1†Š2, F…† is also continuous (bounded) on V.

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C1 kv1k21  ‡ kv2k21‡ kx1k21‡ kx2k21  6B……v;x†;…v;x†† ˆ Q…v;x;0† 6C2 kv1k21  ‡ kv2k21‡ kx1k21‡ kx2k21  …3:7† for all …v; x† 2 V. Thus B…; † is coercive on V  V and

k…v; x†kBˆ B……v; x†; …v; x††… †1=2 8 …v; x† 2 V …3:8†

de®ne a new norm on V which is equivalent to the H1-norm.

For the purpose of discretization we shall use ®nite element spaces de®ned with reference to partitions of ‰0; 1Š. Let Thbe a partition of ‰0; 1Š such that the

interval ‰0; 1Š is partitioned into subintervals Iiˆ …xiÿ1; xi†, i ˆ 1; . . . ; N, with

0 ˆ x0< x1<    < xN ˆ 1. The mesh parameter h is de®ned by

h ˆ maxfjxiÿ xiÿ1j: i ˆ 1; . . . ; Ng. We denote by Pph, p P 1 integer, the space

of continuous functions on ‰0; 1Š whose restrictions to Ii, i ˆ 1;    ; N, are

polynomials of degree p, that is, Pphˆ v 2 C0…0; 1†: vj Iiis a polynomial of degree p  : De®ne ~ Pphˆ Pp h\ H01…0; 1†:

Then we will seek the least-squares ®nite element approximations in the fol-lowing ®nite-dimensional subspace of V,

Vp

hˆ ~Pph ~P p

h Pph Pph: …3:9†

By the interpolation theory, the ®nite element space possesses the following approximation property: for any …v; x† 2 V \ ‰Hp‡1…0; 1†Š4, there exists

…vh; xh† 2 Vph such that

k…v; x† ÿ …vh; xh†k16 Chpk…v; x†kp‡1; …3:10†

where the positive constant C is independent of …v; x† and the mesh parameter h.

The least-squares ®nite element method for problem (2.12)±(2.17) is then the following.

Find …uh; rh† 2 Vph such that

B……uh; rh†; …vh; xh†† ˆ F……vh; xh†† 8 …vh; xh† 2 Vph: …3:11†

Applying the Lax±Milgram theorem [11,20], we know that the approximation problem (3.11) has a unique solution. Once a basis for the space Vp

his chosen,

the matrix associated with problem (3.11) can easily be shown to be symmetric and positive de®nite.

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4. Stability, convergence and error estimates

Let …u; r† 2 V be the exact solution of problem (2.12)±(2.17) with the given function f 2 ‰L2…0; 1†Š2. We ®rst prove the stability of the least-squares ®nite

element solution …uh; rh† 2 Vph.

Theorem 4.1. The unique solution …uh; rh† 2 Vphof problem (3.11) is stable in the

H1-norm in the following sense:

k…uh; rh†k16 Ckfk0; …4:1†

where the positive constant C is independent of e and h. Proof. By (3.7), (3.8) and (3.11), we have

C1k…uh; rh†k216 k…uh; rh†k2B

ˆ B……uh; rh†; …uh; rh††

ˆ F……uh; rh††

6 kfk0k…uh; rh†kB

6 Ckfk0k…uh; rh†k1;

which implies (4.1). This completes the proof. 

Estimate (4.1) indicates that the least-squares ®nite element solution …uh; rh†

is stable with respect to the H1-norm, that is, when we change the given data

function f slightly in the L2-norm, the least-squares solution …u

h; rh† changes

only slightly in the H1-norm.

We now introduce some function spaces which will be used to prove the next theorem. Let C1

0 …0; 1† denote the linear space of in®nitely di€erentiable

func-tions with compact support in …0; 1†, and let C1‰0; 1Š denote the restrictions of

the functions in C1

0 …R† to ‰0; 1Š. Then it is obvious that the product space

S :ˆ ‰C1

0 …0; 1†Š2 ‰C1‰0; 1ŠŠ2 …4:2†

is dense in V with respect to the H1-norm.

Now utilizing the standard density argument [20], we can obtain the fol-lowing results for the convergence.

Theorem 4.2. The least-squares ®nite element solution …uh; rh† is convergent in the

H1-norm without requiring any extra regularity assumption on the exact solution

…u; r†, that is, lim

h!0k…u; r† ÿ …uh; rh†k1ˆ 0: …4:3†

Moreover, if the exact solution …u; r† 2 V \ ‰Hp‡1…0; 1†Š4, then we have the

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k…u; r† ÿ …uh; rh†k16 Chpk…u; r†kp‡1; …4:4†

where C is a positive constant independent of e and h.

Proof. Subtracting the equation in (3.11) from the equation in (3.4), we get the following orthogonality relation:

B……u; r† ÿ …uh; rh†; …vh; xh†† ˆ 0 8 …vh; xh† 2 Vph: …4:5†

Using (3.7), (4.5) and the Cauchy±Schwarz inequality, we obtain k…u; r† ÿ …uh; rh†k216C1

1k…u; r† ÿ …uh; rh†k 2 B

ˆ C1

1B……u; r† ÿ …uh; rh†; …u; r† ÿ …uh; rh††

ˆ C1

1B……u; r† ÿ …uh; rh†; …u; r† ÿ …vh; xh††

6CC

1k…u; r† ÿ …uh; rh†k1k…u; r† ÿ …vh; xh†k1

for all …vh; xh† 2 Vph. Thus,

k…u; r† ÿ …uh; rh†k16 Ck…u; r† ÿ …vh; xh†k1 8 …vh; xh† 2 Vph: …4:6†

Now, since the subspace S  V \ ‰Hp‡1…0; 1†Š4 is dense in V with respect to

the H1-norm, for any d > 0, there exists …~u; ~r† 2 S independent of h such that

k…u; r† ÿ …~u; ~r†k1< d

2C; …4:7†

where C is the same constant as in (4.6). For this ®xed suciently smooth function …~u; ~r† 2 S  ‰Hp‡1…0; 1†Š4, by the approximation property (3.10), we

can ®nd …~uh; ~rh† 2 Vph so that,

k…~u; ~r† ÿ …~uh; ~rh†k16 Chpk…~u; ~r†kp‡1;

which implies, for suciently small h,

k…~u; ~r† ÿ …~uh; ~rh†k1<2Cd ; …4:8†

where C is the same constant as in (4.6). Combining inequalities (4.7) and (4.8) with (4.6), we immediately obtain

0 6 k…u; r† ÿ …uh; rh†k16 Ck…u; r† ÿ …~uh; ~rh†k1

6 C k…u; r† ÿ …~u; ~r†k1‡ k…~u; ~r† ÿ …~uh; ~rh†k1

 < d;

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We now assume that …u; r† 2 V \ ‰Hp‡1…0; 1†Š4. By (4.6) and the

approxi-mation property (3.10) of the ®nite element space Vp

h, we can obtain (4.4)

immediately. This completes the proof. 

Since the energy norm k  kBis equivalent to the H1-norm, from (4.3), we can

conclude the following consequence.

Corollary 4.3. Let …uh; rh† ˆ …u1h; u2h; r1h; r2h† be the least-squares ®nite element

solution. Then lim h!0 k ÿ ÿ r0 1hÿ f1k0‡ k ÿ r02hÿ r1hÿ f2k0‡ k ÿ e2r 1h‡ u01hÿ u2hk0‡ k ÿ r2h‡ u02hk0  ˆ 0: …4:9†

Remark 4.4. The error estimate (4.4) is optimal in the H1-norm with

respect to the order of approximation of the ®nite element space Vph. Under

suitable assumptions (cf. [22]), it is quite possible to derive the optimal error estimates in the L2-norm for all the unknowns by using the Aubin±Nitsche

trick.

5. An equivalent a posteriori error estimator in the H1-norm

The use of a posteriori error estimators has become an accepted tool for assessing and controlling computational errors in adaptive computations. One of the most important advantageous features of the least-squares ®nite element approach is that the square root of the value of the least-squares functional of the approximate solution provides a practical and sharp a posteriori error estimator at no additional cost [21,23,28]. This is quite di€erent from the previous error estimators, see [1,3,4,6,7,10,31] for example.

In this section we shall brie¯y show that the simple estimator is indeed an exact error estimator in the energy norm, k  kB. Thus, it is an equivalent error estimator in the H1-norm. Let …u

h; rh† ˆ …u1h; u2h; r1h; r2h† be the least-squares

®nite element solution. For all subintervals Ii2 Th, let QIi…uh; rh; f† denote the value of quadratic least-squares functional of the approximate solution re-stricted on Ii, that is,

QIi…uh; rh; f† ˆ Z Ii …ÿr0 1hÿ f1†2dx ‡ Z Ii …ÿr0 2hÿ r1hÿ f2†2dx ‡ Z Ii …ÿe2r 1h‡ u01hÿ u2h†2dx ‡ Z Ii …ÿr2h‡ u02h†2dx

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for all Ii2 Th. Then QIi…uh; rh; f† ˆ Z Ii …ÿr0 1hÿ …ÿr01††2dx ‡ Z Ii …ÿr0 2hÿ r1hÿ …ÿr02ÿ r1††2dx ‡ Z Ii …ÿe2r 1h‡ u01hÿ u2hÿ …ÿe2r1‡ u01ÿ u2††2dx ‡ Z Ii …ÿr2h‡ u02hÿ …ÿr2‡ u02††2dx

ˆ k…u; r† ÿ …uh; rh†k2B;Ii;

where k  kB;Ii denotes the energy norm restricted on Ii. Therefore, the

com-putable value …QIi…uh; rh; f††1=2 de®nes an ``exact'' error indicator of the sub-interval Iiwhich assesses the quality of the approximate solution …uh; rh† in that

subinterval Ii and indicates whether the element needs to be re®ned, dere®ned

or unchanged. Taking the summation, Q…uh; rh; f† … †1=2ˆ X Ii2Th QIi…uh; rh; f† !1=2 ˆ k…u; r† ÿ …uh; rh†kB;

we get an exact a posteriori error estimator …Q…uh; rh; f††1=2, which can serve as

one of the major stopping criteria for an entire adaptive process. In general, the e€ectivity index de®ned by

H ˆk…u; r† ÿ …u…Q…uh; rh; f††1=2

h; rh†kB

is then used to quantify the quality of the estimator and hence the quality of the approximate solution …uh; rh†. In our case, H exactly equals 1.

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