A posteriori error analysis for the normal compliance problem

Applied Numerical Mathematics 60 (2010) 64–73

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Applied Numerical Mathematics www.elsevier.com/locate/apnum

A posteriori error analysis for the normal compliance problem J.R. Fernández a,∗ , P. Hild b a b

Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, Escola Politécnica Superior, Campus Universitario s/n, 27002 Lugo, Spain Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon, France

a r t i c l e

i n f o

Article history: Received 29 April 2008 Received in revised form 20 July 2009 Accepted 11 September 2009 Available online 15 September 2009 MSC: 74M15 74B05 65N30 65N15

a b s t r a c t In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator. © 2009 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Normal compliance problem Finite element method A posteriori error estimates Residuals

1. Introduction and notation The finite element method is currently used in the numerical approximation of contact problems occurring in several engineering applications (see [10,11,14,17,25]) and the a posteriori error estimators are efficient tools for evaluating numerically the quality of these finite element computations. An extensive review of different estimators used in various contexts can be found in [2,3,9,23,24]. Several error estimators have been chosen and studied for frictionless or frictional contact problems, in particular in [4,26] (residual approach using a penalization of the contact condition), in [12,13] (residual approach for the variational inequality and the corresponding mixed formulation for unilateral contact), in [6,7] (error in the constitutive relation for unilateral contact with or without Coulomb friction), in [19] (duality approach for the compliance model with friction) and finally in [8] (residual approach for BEM-discretizations). In the present work we are interested in developing residual estimators for the two-dimensional normal compliance contact model in linear elasticity initially introduced in [20,21] and studied in [15,16]. As a consequence, we consider a similar study as the ones in [4,26] since the compliance law could be seen (to simplify) as a kind of penalization with a fixed penalty parameter depending on the material characteristics. In [4,26] the authors propose a first error estimator for a penalization of the unilateral contact law and prove (in [4]) that the discretization error is bounded by the estimator. In our work we also prove that the local indicators are bounded by the local discretization error for the compliance model. To our knowledge, the asymptotic equivalence between an estimator and the discretization error is a new result for the contact model with normal compliance.

*

Corresponding author. Tel.: +34 981 563 100 (Ext. 23244); fax: +34 982 285 926. E-mail addresses: [email protected] (J.R. Fernández), [email protected] (P. Hild).

0168-9274/$30.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2009.09.002

J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

65

The paper is organized as follows. In Section 2 we introduce the equations modeling the contact problem between an elastic body and a deformable foundation. We write the problem using a nonlinear variational equation which is well posed. In Section 3, we choose a classical discretization involving continuous finite elements of degree one, we write the discrete problem and we recall the corresponding a priori error estimates. Section 4 is concerned with the study of an a posteriori error estimator: we propose a residual error estimator and we prove that the discretization error is bounded by the estimator. In addition we prove that the local estimators are bounded by local discretization errors. As usual, we denote by ( L 2 (.))d and by ( H s (.))d , s  0, d = 1, 2 the Lebesgue and Sobolev spaces in one and two space dimensions (see [1]). The usual norm of ( H s ( D ))d is denoted by  · s, D and we keep the same notation when d = 1 or d = 2. For shortness the ( L 2 ( D ))d -norm will be denoted by  ·  D when d = 1 or d = 2. In the sequel the symbol | · | will either denote the Euclidean norm in R2 , or the length of a line segment, or the area of a plane domain. Finally the notation a  b means that there exists a positive constant C independent of a and b (and of the mesh size of the triangulation) such that a  Cb. The notation a ∼ b means that a  b and b  a hold simultaneously. Next, bold letters like u, v indicate vector valued quantities, while the capital ones (e.g., V, Vh , . . .) represent functional sets involving vector fields. 2. The normal compliance problem in elasticity Let Ω represent an elastic body in R2 where plane strain assumptions are assumed. The boundary ∂Ω is supposed to be polygonal, i.e., it is the union of a finite number of linear segments. Moreover we suppose that the boundary consists in three nonoverlapping parts Γ D , Γ N and ΓC with meas(Γ D ) > 0 and meas(ΓC ) > 0. The normal unit outward vector on ∂Ω is denoted ν = (n1 , n2 ) and we choose as unit tangential vector t = (−n2 , n1 ). In its initial configuration, the body is in contact on ΓC with normal compliance conditions. The body is clamped on Γ D , for the sake of simplicity, and it is subjected to volume forces f = ( f 1 , f 2 ) ∈ ( L 2 (Ω))2 and to surface forces g = ( g 1 , g 2 ) ∈ ( L 2 (Γ N ))2 . Let us denote by u = (u i )1i 2 the displacement vector and by σ = (σi j )1i , j 2 the stress tensor such that

σi j (u ) = ai jhk

∂ uh , ∂ xk

i , j , h, k ∈ {1, 2},

where the summation convention of repeated indices is adopted. The functions ai jhk ∈ L ∞ (Ω) are the coefficients of a fourth order tensor, representing the elastic properties of the material. As usual we assume that ai jhk = a jihk = ahki j and the ellipticity condition ai jhk ξi j ξhk  α |ξ |2 , ∀ξi j = ξ ji , for some α > 0. The frictionless contact problem with normal compliance in elastostatics is to find the displacement field u such that Eqs. (1)–(4) hold (see [15,16,20,21]):

div σ (u ) + f = 0

in Ω,

(1)

u = 0 on Γ D ,

(2)

σ (u )ν = g on ΓN ,

(3)

where div denotes the divergence operator of tensor valued functions. For each displacement field v and for each density of surface forces σ ( v )ν defined on ∂Ω , we adopt the following notation:

v = vnν + vt t

and

σ ( v )ν = σn ( v )ν + σt ( v )t ,

where the normal and tangential components of the displacement field are given by v n = v · ν and v t = v · t, respectively, while the normal and shear components of the stress field are defined as σn ( v ) = σ ( v )ν · ν and σt ( v ) = σ ( v )ν · t. Then the conditions of normal compliance without friction on ΓC are written as follows: n σn (u ) = −cn (un )m + ,



(4)

σt (u ) = 0,

where (.)+ stands for the positive part so that (un )+ represents the penetration of the body into the foundation. The constant mn  1, as well as the nonnegative function cn in L ∞ (ΓC ), stand for interface parameters characterizing the contact behavior between the body and the deformable foundation. Remark 1. When mn = 1 and cn = ε −1 (where ε > 0 is a small positive parameter) we recover the classical penalty method (see, e.g., [4,14]) used in the approximation of unilateral contact problems. The set of admissible displacements





2

V = v ∈ H 1 (Ω) : v = 0 on Γ D



is endowed with the norm of ( H 1 (Ω))2 . We denote by a(·,·) the standard bilinear form of linear elasticity



a( u , v ) =



σ (u ) : ε ( v ) dx = Ω

ai jhk Ω

∂ ui ∂ vh dx, ∂ x j ∂ xk

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J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

where ε = (εi j )1i , j 2 stands for the strain tensor where εi j (u ) = (∂ u i /∂ x j + ∂ u j /∂ xi )/2. If Γ D has positive superficial measure it is well known that the bilinear form a(·,·) is V-elliptic,

  2 ∂ v i 2 2 a( v , v )  α ∂ x dx  β v 1,Ω , j

Ω i , j =1

∀ v ∈ V, β > 0.

The previous boundary value problem leads to the following nonlinear variational equation (see, e.g., [15,16]),

u ∈ V,

a(u , v ) + jn (u , v ) = L ( v ),

∀ v ∈ V,

(5)

where the linear form L is given by



L( v ) =



f (x) · v (x) dx + Ω

g (x) · v (x) dγ (x)

ΓN

and the normal compliance functional jn : V × V → R is defined as



jn (u , v ) =

m

cn (un )+n v n dγ (x). ΓC

Afterwards we use the embedding (see, e.g., [1])

H 1 (Ω) → L q (ΓC )

(6)

for each q ∈ [1, +∞[. It is obvious that problem (5) admits a unique solution (see, for instance, [14]). Moreover, it is easy to check that n σn (u ) ∈ L 2 (ΓC ). Since σn (u ) = −cn (un )m + , we find that



σn (u )

L 2 (ΓC )



n

= cn (un )m + L 2 (ΓC )



n

 cn L ∞ (ΓC ) (un )m + L 2 (ΓC )

n

2  cn L ∞ (ΓC ) um n L (Γ ) C

n = cn L ∞ (ΓC ) un m L 2mn (Γ

 

cn L ∞ (ΓC )

C)

n u m 1,Ω ,

where we use (6) and we obtain the desired regularity. 3. Finite element approximation and a priori error estimate We approximate problem (5) by a standard finite element method. Namely we fix a family of meshes T h , h > 0, regular in Ciarlet’s sense (see [5]), made of closed triangles and assumed to be subordinated to the decomposition of the boundary ∂Ω into Γ D , Γ N and ΓC . For K ∈ T h we recall that h K is the diameter of K and h = max K ∈ T h h K . The regularity of the mesh implies in particular that for any edge E of K one has h E = | E | ∼ h K . Let us define E h (resp. Nh ) as the set of edges (resp. nodes) of the triangulation and set E hint = { E ∈ E h : E ⊂ Ω} the set of interior edges of T h (the edges are supposed to be relatively open). We denote by E hN = { E ∈ E h : E ⊂ Γ N } the set of exterior edges included into the part of the boundary where we impose Neumann conditions, and similarly E hC = { E ∈ E h : E ⊂ ΓC } is

the set of exterior edges included into the part of the boundary where we impose the contact conditions. Set NhD = Nh ∩ Γ D (note that the extreme nodes of Γ D belong to NhD ).

For an element K , we will denote by E K the set of edges of K and according to the above notation, we set E int K = C C N E K ∩ E hint , E N = E ∩ E , E = E ∩ E . For an edge E of an element K , introduce ν = ( n , n ) the unit outward normal K K K ,E 1 2 K K h h vector to K along E and the tangent vector t K , E = ν ⊥ K , E = (−n2 , n1 ). Furthermore, for each edge E, we fix one of the two normal vectors and denote it by at a point y ∈ E is defined as

ν E and we set t E = ν ⊥E . The jump of some vector valued function v across an edge E ∈ E hint

J v K E ( y ) = lim v ( y + αν E ) − v ( y − αν E ), α →0+

∀ E ∈ E hint .

Note that the sign of J v K E depends on the orientation of ν E . Finally we will need local subdomains (also called patches). of all elements having a nonempty intersection with K . Similarly for a node x and an edge E, As usual, let ω K be the union  let ωx = K : x∈ K K and ω E = x∈ E ωx . The finite element space used in Ω is then defined by





2



2



Vh = v h ∈ C (Ω) : ∀ K ∈ T h , v h | K ∈ P1 ( K ) , v h |Γ D = 0 .

J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

67

Thus, the discrete formulation of the normal compliance problem (5) is the following:

uh ∈ Vh ,

a(u h , v h ) + jn (u h , v h ) = L ( v h ),

∀ v h ∈ Vh .

(7)

Using the same arguments as in the continuous case (see [15,16]), it is straightforward that problem (7) admits a unique solution. We now consider the quasi-interpolation operator πh : for any v ∈ L 1 (Ω), we define πh v as the unique element in V h = { v h ∈ C (Ω): ∀ K ∈ T h , v h | K ∈ P1 ( K ), v h |Γ D = 0} such that:



πh v =

αx ( v )λx ,

(8)

x∈Nh \NhD

where for any x ∈ Nh \ NhD , λx is the standard basis function in V h satisfying λx (x ) = δx,x , for all x ∈ Nh \ NhD , and is defined as follows:

αx ( v ) =

1

|ω x |

αx ( v )



v ( y) d y,

∀x ∈ Nh \ NhD .

ωx

The following estimates hold (see, e.g., [24]). Lemma 2. For any v ∈ V = { v ∈ H 1 (Ω): v = 0 on Γ D } we have

 v − πh v  K  h K ∇ v ω K , 1/2 h v  E  h E ∇ v ω E ,

v − π

∀K ∈ Th, ∀E ∈ Eh.

Since we deal with vector valued functions we can define a vector valued operator (which we denote again by the sake of simplicity) whose components are defined above. Consequently we can directly state the following.

πh for

Lemma 3. For any v ∈ V we have

 v − πh v  K  h K  v 1,ω K , 1/2 h v  E  h E  v 1,ω E ,

v − π

∀K ∈ Th,

(9)

∀E ∈ Eh.

(10)

We next recall a standard result dealing with the a priori error estimate (see [18] for the early studies). Theorem 4. Let u be the solution to problem (5) and let u h be the solution to discrete problem (7). If we assume that u ∈ ( H 2 (Ω))2 , then we have

u − uh 1,Ω  hu 2,Ω . Proof. Let v h ∈ Vh . From the V-ellipticity of a(·,·) and the equations in (5) and (7) we obtain:

u − uh 21,Ω  a(u − uh , u − uh ) = a( u − u h , u − v h ) + a( u − u h , v h − u h ) = a( u − u h , u − v h ) + j n ( u h , v h − u h ) − j n ( u , v h − u h ) = a(u − uh , u − v h ) + jn (uh , v h − u ) − jn (u , v h − u ) + jn (uh , u − uh ) − jn (u , u − uh ).

(11)

Using a standard monotonicity argument, we get

 jn (u h , u − u h ) − jn (u , u − u h ) = −



m

m



cn (un )+n − (u hn )+n (un − u hn ) dγ (x)  0.

(12)

ΓC

From the inequality

m     (a) − (b)m  m (a)+ − (b)+ (a)m−1 + (b)m−1  m|a − b| |a|m−1 + |b|m−1 a, b ∈ R, m  1, + + + + we deduce for each u 1 , u 2 , v ∈ ( H 1 (Ω))2

(13)

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J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

jn (u 1 , v ) − jn (u 2 , v ) 



  |u 1n − u 2n | |u 1n |mn −1 + |u 2n |mn −1 | v n | dγ (x)

ΓC

  n −1 n −1  v n Lr (ΓC )  u 1n − u 2n Lr (ΓC ) u 1n m + u 2n m L q(mn −1) (ΓC ) L q(mn −1) (ΓC )   mn −1 n −1  u 1 − u 2 1,Ω u 1 m  v 1,Ω . 1,Ω + u 2 1,Ω

(14)

In the previous bounds we use Hölder inequalities with 1/r + 1/q + 1/r = 1 and q(mn − 1)  1 together with embedding (6). Note that the case mn = 1 is straightforward. Combining (11), (12) and (14), we get

 mn −1  n −1 u − v h 1,Ω u − uh 21,Ω  u − uh 1,Ω u − v h 1,Ω + u − uh 1,Ω u m 1,Ω + u h 1,Ω  u − uh 1,Ω u − v h 1,Ω . In the last expression we used the straightforward property resulting from (5) and (7) that u 1,Ω and u h 1,Ω are bounded by constants depending on the loads. Therefore, we conclude the proof of the theorem by choosing v h = I h u, where I h stands for the Lagrange interpolation operator mapping onto Vh . 2 4. A residual a posteriori error estimator 4.1. Definition of the residual error estimator The element residual of the equilibrium equation (1) is defined by

div σ (u h ) + f = f

on K .

As usual this element residual can be replaced by some finite-dimensional approximation, called approximate element residual (see, e.g., [2])



2

f K ∈ Pk ( K ) .



A current choice is to take f K = K f (x)/| K | dx since for f ∈ ( H 1 (Ω))2 , scaling arguments yield  f − f K  K  h K  f 1, K and it is then negligible with respect to the estimator η defined hereafter. In the same way g is approximated by a computable quantity denoted g E on any E ∈ E hN . Definition 5. The local error estimators

ηK =

4 

η K and the global estimator η are defined by

1/2

η

2 iK

,

i =1

η1K = h K  f K  K , η

1/2 2K = h K







J E ,n (uh ) 2

E

1/2 ,

N E ∈ E int K ∪E K

η3K = h1K/2

  

1/2

σt (uh ) 2 , E

E ∈ E CK

η4K = h1K/2 η=

  

1/2

cn (uhn )mn + σn (uh ) 2 , + E

 

E ∈ E CK

1/2

η2K

,

K ∈T h

where J E ,n (u h ) means the constraint jump of u h in the normal direction, i.e.,



J E ,n (uh ) =

Jσ (uh )ν E K E ,

∀ E ∈ E hint ,

σ (uh )ν E − g E , ∀ E ∈ E hN .

The local and global approximation terms are given by

(15)

J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

69

1/2    ζ K = h2K  f − f K 2K + h E  g − g E 2E , K ⊂ω K

ζ=

 

E⊂E N K

1/2 ζ K2

.

K ∈T h

4.2. Upper error bound We first study the upper error bound of the discretization error. Theorem 6. Let u be the solution to nonlinear variational equation (5) and let u h be the solution to the corresponding discrete problem (7). Then we have

u − uh 1,Ω  η + ζ. Proof. Afterwards we adopt the following notation for the displacement error term:

e = u − uh . Let v h ∈ Vh . From the V-ellipticity of a(·,·) and the equations in (5) and (7) we obtain:

e 21,Ω  a(u − uh , u − uh ) = a( u , u − u h ) − a( u h , u − u h ) = L ( u − u h ) − j n ( u , u − u h ) − a( u h , u − u h ) = L ( u − v h ) − j n ( u , u − u h ) − a( u h , u − v h ) + j n ( u h , v h − u ) + j n ( u h , u − u h )  L (u − v h ) + jn (uh , v h − u ) − a(uh , u − v h ), where we use the monotonicity argument (12). Integrating by parts on each triangle K and using the definition of J E ,n (u h ) in (15) yields:

e 21,Ω 

 

f · (u − v h ) dx +

K ∈T h K

−



 

m

cn (u hn )+n ( v hn − un ) dγ (x) −

E ∈ E hC E



J E ,n (u h ) · (u − v h ) dγ (x) +

 

  



σ (uh )ν · (u − v h ) dγ (x)

E ∈ E hC E

( g − g E ) · (u − v h ) dγ (x).

E ∈ E hN E

E ∈ E hint ∪ E hN E

Splitting up the integrals on ΓC into normal and tangential components gives:

e 21,Ω 

 

f · (u − v h ) dx +

K ∈T h K

+

 

E ∈ E hC

+



m

cn (u hn )+n + σn (uh ) ( v hn − un ) dγ (x)

E ∈ E hC E



σt (uh )( v ht − ut ) dγ (x) −

E ∈ E hint ∪ E hN

E

 

  

 J E ,n (uh ) · (u − v h ) dγ (x) E

( g − g E ) · (u − v h ) dγ (x).

(16)

E ∈ E hN E

We now need to estimate each term of this right-hand side. For that purpose, we take

v h = u h + πh (u − u h ), where

πh is the quasi-interpolation operator defined in Lemma 3.

(17)

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J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

We start with the integral term. Cauchy–Schwarz’s inequality implies

 



f · (u − v h ) dx 

K ∈T h K

 f  K u − v h  K ,

K ∈T h

and it suffices to estimate u − v h  K for any triangle K . From the definition of v h and (9) we get:

u − v h  K = e − πh e  K  h K e 1,ω K . As a consequence

 f · (u − v h ) dx  (η + ζ )e 1,Ω . Ω

We now consider the interior and Neumann boundary terms in (16). As we previously noticed, the application of Cauchy– Schwarz’s inequality leads to











J E ,n (uh ) · (u − v h ) dγ (x) 

E ∈ E hint ∪ E hN E



J E ,n (uh ) u − v h  E . E

E ∈ E hint ∪ E hN

Therefore using expression (17) and estimate (10), we obtain 1/2

u − v h  E = e − πh e  E  h E e 1,ω E . Inserting this estimate in the previous one we deduce that









J E ,n (uh ) · (u − v h ) dγ (x)  ηe 1,Ω .

E ∈ E hint ∪ E hN E

Moreover,

   ζ e 1,Ω . ( g − g ) · ( u − v ) d γ ( x ) h E E ∈ E hN E

The following two terms are handled in a similar way as the previous ones so that

    mn cn (u hn )+ + σn (u h ) ( v hn − un ) dγ (x)  ηe 1,Ω , E ∈ E hC E

and

   ηe 1,Ω . σ ( u )( v − u ) d γ ( x ) t t h ht E ∈ E hC E

Putting together the previous estimates we come to the conclusion that

u − uh 1,Ω  η + ζ.

2

4.3. Lower error bound We now consider the local lower error bounds of the discretization error terms. Theorem 7. For all elements K ∈ T h , the following local lower error bounds hold:

η1K  u − uh 1, K + ζ K ,

(18)

η2K  u − uh 1,ω K + ζ K .

(19)

For all elements K such that K ∩

E hC

= ∅, the following local lower error bounds (with p > 2) hold:

η3K  u − uh 1, K + ζ K , η4K 



E ∈ E CK

1/2 C ( p )h K un

− uhn 

L p (E)

1/2

+ h K n (u

σ

− uh ) E .

(20) (21)

J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

71

Proof. The estimates of η1K and η2K in (18) and (19) are standard (see, e.g., [23]). We now estimate η3K . Writing w E = w En ν + w Et t on E ∈ E CK and denoting by b E the edge bubble function associated with E (i.e., b E = 4λa1 λa2 , when a1 , a2 are the two extremities of E; we recall that λx is the standard basis function at node x in V h satisfying λx (x ) = δx,x for any node x , see (8)), we choose w En = 0 and w Et = σt (u h )b E in the element K containing E (here we made a slight abuse of notation to simplify the writing) and w E = 0 in Ω \ K . Therefore,



σt (uh ) 2 ∼ E



σt (uh ) w Et dγ (x) E

 =

σ (uh ) : ε ( w E ) dx K

 =



σ (uh − u ) : ε ( w E ) dx + K

K



= L( w E ) +

σ (u ) : ε ( w E ) dx

σ (uh − u ) : ε ( w E ) dx K

  f  K  w E  K + u − uh 1, K  w E 1, K 1   f  K  w E  K + h− K u − u h 1, K  w E  K ,

where we use an inverse inequality in the last bound. Another inverse inequality and estimate (18) imply





1/2 h K σt (u h ) E  u − u h 1, K + h K  f  K

 u − uh 1, K + ζ K . This estimate gives the estimate of η3K in (20). The bound of η4K in (21) cannot be obtained as previously by choosing m w En = (cn (u hn )+n + σn (u h ))b E and w Et = 0 since we have in general (due to the positive part)



cn (uhn )mn + σn (uh ) 2  + E





m



cn (u hn )+n + σn (u h ) w En dγ (x).

E mn

So we use the identity cn (un )+ + σn (u ) = 0 and, keeping in mind that

σn (u ) ∈ L 2 (ΓC ), we write









cn (uhn )mn + σn (uh )  cn (uhn )mn − (un )mn + σn (u − uh )

+ + + E E E





mn 

n



 cn L ∞ ( E ) (uhn )m − ( u ) + σ ( u − uh ) E n + n + E





  mn cn L ∞ ( E ) (uhn − un ) |uhn |mn −1 + |un |mn −1 E + σn (u − uh ) E

 

n −1 n −1 + σn (u − uh ) E  mn cn L ∞ ( E ) uhn − un L p ( E ) uhn m + un m L q(mn −1) ( E ) L q(mn −1) ( E )



 mn −1  n −1 + σn (u − uh ) E  C ( p )mn cn L ∞ ( E ) uhn − un L p ( E ) uh m 1,Ω + u 1,Ω



 un − uhn L p ( E ) + σn (u − uh ) E ,

(22)

where 1/ p + 1/q = 1/2, q(mn − 1)  1 (the case mn = 1 is straightforward) and we have used the identity (13) and the boundedness of u 1,Ω and u h 1,Ω . 2 Corollary 8. Assume that u ∈ ( H 2 (Ω))2 and define:

ηi =



1/2

ηi2K

,

1  i  4.

K ∈T h

Then we have ηi  h for 1  i  4. So

η  h. Proof. From Theorem 7, we deduce by addition η1  u − u h 1,Ω + ζ and the latter quantity is bounded by hu 2,Ω according to Theorem 4. The bounds for η2 and η3 are the same. We now consider η4 :

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J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73

η42 







E ∈ E hC





2

h E un − u hn  L p ( E ) + σn (u − uh ) E h E un − u hn 2L p ( E ) +

E ∈ E hC





2

h E σn (u − uh ) E ,

(23)

E ∈ E hC

where p > 2. The first term is roughly bounded as follows by using (6)

h E un − u hn 2L p ( E )  h E un − u hn 2L p (ΓC )  h E u − uh 21,Ω .

(24)

By using the identity



h E = |ΓC |

(25)

E ∈ E hC

and Theorem 4 we conclude that



h E un − u hn 2L p ( E )  h2 u 22,Ω .

(26)

E ∈ E hC

The second term is bounded by using the scaled trace inequality −1/2

v E  h E

1/2

∀ E ∈ E h , ∀ v ∈ H 1 (ω E ).

 v ω E + h E ∇ v ω E ,

Hence, supposing without loss of generality that ΓC is a straight line segment parallel to the x-axis and using the latter inequality, we obtain:





2

2

h E σn (u − uh ) E = h E σ y y (u − uh ) E



2

2  σ y y (u − uh ) ω + h2E ∇ σ y y (u − uh ) ω E E



2

2  σ (u − uh ) ω + h2E ∇ σ (u ) ω . E

E

By summation, we find that





2

h E σn (u − u h ) E  u − uh 21,Ω + h2 u 22,Ω  h2 u 22,Ω .

(27)

E ∈ E hC

The corollary is proved by putting together (23), (26) and (27).

2

Remark 9. We redefine the discretization error e = u − u h 1,Ω as follows:

e˜ = u − u h 1,Ω +





2

h E σn (u − u h ) E

1/2 .

E ∈ E hC

Such an approach with a modified error has already been used successfully for the obstacle problem in [22]. Note that in this paper the additional norm term is mesh dependent contrary to [22]. With this new definition, we can prove its equivalence with the error estimator η :

e˜  η + ζ,

(28)

η  e˜ + ζ. We first prove the upper bound (28). Let E ∈

(29) E hC .

Using the estimate (a + b)  2(a + b ) gives: 2

2

2







2

2 m m 2 m h E σn (u − uh ) E  h E −cn (un )+n + cn (u hn )+n E + h E cn (u hn )+n + σn (uh ) E .

(30)

In (22) we prove

 

cn (uhn )mn − (un )mn  un − uhn L p ( E ) , + + E for any E ∈ E hC (p > 2). As a consequence we deduce from (6) that

 

cn (uhn )mn − (un )mn  un − uhn L p (Γ )  u − uh 1,Ω . + + C E

Denoting by K the element containing E, using (30) and (31) and the definition of

(31)

η4K we deduce

J.R. Fernández, P. Hild / Applied Numerical Mathematics 60 (2010) 64–73



73

2

2 h E σn (u − uh ) E  h E u − u h 21,Ω + η4K .

Therefore,





2

h E σn (u − u h ) E

1/2  u − uh 1,Ω + η,

E ∈ E hC

which together with Theorem 6 yields (28). Finally, the lower bound (29) is a straightforward consequence of the proof of Corollary 8 since 1  i  3 and η4  e˜ according to (23), (24) and (25).

ηi  u − uh 1,Ω + ζ ,

Acknowledgements The work of J.R. Fernández was partially supported by the Ministerio de Educación y Ciencia (Project MTM2006-13981) and the work of P. Hild was supported by l’Agence Nationale de la Recherche (Project ANR-05-JCJC-0182-01). We are also thankful to the anonymous referee for its suggestions which led to improvements in this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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