A posteriori error estimators for a model for flow in a porous - PowerPoint PPT Presentation

A posteriori error estimators for a model for flow in a porous medium with fractures Zoubida MGHAZLI (LIRNE, EIMA, Universit´ e Ibn Tofail, K´ enitra, Morocco) Jean E. ROBERTS (INRIA-Roquencourt, France) Ali SAADA (LAMSIN-ENIT,Tunisia) Scaling Up and Modeling for Transport and Flow in Porous Media Dubrovnik, 13-16 October , 2008 Supported by the project MODESS-INRIA and project AI-Morocco-Tunisia 1

I) Introdution Ω ⊂ R n , n = 2 , 3 , ¯ Ω f = fracture; Ω \ ¯ Ω f = Ω 1 ∪ Ω 2 , Ω 1 ∩ Ω 2 = ∅ ; Ω f ≡ γ : hyperplane; Γ = ∂ Ω , Γ i := ∂ Ω ∩ Ω i , i = 1 , 2 ; Z [2] is so that 1+1=2 and 2+1=1 , (cf V.Martin, J. Jaffr´ e, J. Roberts, 2005 )  u i = − K i ∇ p i in Ω i , i = 1 , 2   div u i = q i in Ω i , i = 1 , 2     u f = − K f ∇ τ p f on γ,     2    �  div τ u f = q f + u i · n i on γ,    i =1 d  [ ξu i · n i ( ξ ∈ ]1 / 2 , 1]) p i = p f +   2 K f     − (1 − ξ ) u i +1 · n i +1 ] on γ, i ∈ Z [2]      p i = ¯ p i on Γ i , i = 1 , 2     p f = ¯ p f on ∂γ.  2

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Weak Formulation  u ∈ W, p ∈ M  a ξ ( u, v ) − β ( v, p ) = − L d ( v ) , ∀ v ∈ W β ( u, r ) = L q ( r ) , ∀ r ∈ M.  { v = ( v 1 , v 2 , v f ) ∈ Π 2 i =1 H ( div ; Ω i ) × H ( div ; γ ) : v i · n i ∈ L 2 ( γ ) , i = 1 , 2 } W = { r = ( r 1 , r 2 , r γ ) ∈ L 2 (Ω 1 ) × L 2 (Ω 2 ) × L 2 ( γ ) } M = 2 � ( K − 1 u i , v i ) Ω i + ( K − 1 a ξ ( u, v ) = f u f , v f ) γ i i =1 � d 2 � � + [ ξu i · n i − (1 − ξ ) u i +1 · n i +1 ] , v i · n i , 2 K f γ i =1 � 2 2 � � � ( div u i , r i ) Ω i + ( div τ u f , r f ) γ − u i · n i , r f β ( u, r ) = , i =1 i =1 γ 2 n � � L q ( r ) = ( q i , r i ) Ω i + ( q f , r f ) γ , L d ( v ) = ( v i · n i , ¯ p i ) Γ i + ( v f · n f , ¯ p f ) ∂γ . i =1 i =1 4

Norms in M and W 2 � � r � 2 � r i � 2 0 , Ω i + � r f � 2 M = 0 ,f i =1 2 2 � � � � � v � 2 � v i � 2 0 , Ω i + � div v i � 2 + � v f � 2 0 ,γ + � div τ v f � 2 � v i · n i � 2 W = 0 ,γ + 0 ,γ . 0 , Ω i i =1 i =1 Unique Solution : a ξ ( · , · ) is ˜ W − elliptic: a ξ ( v, v ) ∃ C α > 0 , inf ≥ C α � v � 2 v ∈ ˜ W W ˜ W = { v ∈ W : β ( v, r ) = 0 ∀ r ∈ M } β ( · , · ) satisfies the inf-sup condition: β ( v, r ) ∃ C β > 0 , r ∈ M sup inf ≥ C β . � v � W � r � M v ∈ W (cf. V.Martin, J. Jaffr´ e, J. Roberts, 2005) 5

Discretization with RT 0 � Z i = H ( div ; Ω i ) , i = 1 , 2 , Z = Z i i=1 , 2 � N i = L 2 (Ω i ) , N i = L 2 (Ω) . i = 1 , 2 , N = i =1 , 2 Z h,i × N h,i ⊂ Z i × N i , i = 1 , 2 W h,γ ⊂ H ( div τ ; γ ) , Λ h = P 0 ( γ ) � � Z h,i ⊕ W h,γ , N h,i ⊕ Λ h . W h = M h = i =1 , 2 i =1 , 2  u h ∈ W h , p h ∈ M h  a ξ ( u h , v ) − β ( v, p h ) = − L d ( v ) , ∀ v ∈ W h β ( u h , r ) = L q ( r ) , ∀ r ∈ M h .  Π h : W → W h interpolation operateur in RT 0 and P h : M → M h is the L 2 -projection. 6

II) A posteriori error estimate Introduction → a priori error estimates � u − u h �≤ F ( h, u, f ) → only yield information on the asymptotic error behaviour → require regularity assumptions about u which is not satis- fied in the presence of singularity (as sharp fronts, wells,...) → overall accuracy of the numerical approximation is deteri- ored by local singularity Obvious remedy: → to refine the discretization near the critical regions = to place more grid point where the solution is less regular. → how to identify those regions, → how to obtain a good balance between the refined and unrefined regions such that the overall accuracy is ”optimal”. 7

Obtain reliable estimates of the accuracy of the computed numerical solution The need: error estimator which can, a posteriori , be extracted from the computed numerical solution and the given data of the problem Reasonable error estimator should at least satisfy three minimal requirements Reliability = estimator yields upper bounds on the error � u − u h � V ≤ G ( u h , f h , f ) u h is the computed solution. Efficiency = estimator yields lower bounds on the error G ( u h , f h , f ) ≤ C � u − u h � V Locality = information on the local distribution of the error. T ) 1 / 2 , � η 2 G ( u h , f h , f ) = ( η T ≤ C � u − u h � W ( T ) T ∈T h η T : Local error estimator 8

Principal tools e h = u − u h , ε h = p − p h . The residual equations � a ξ ( e h , v ) − β ( v, ε h ) = − L d ( v ) − a ξ ( u h , v ) + β ( v, p h ) , ∀ v ∈ W β ( e h , r ) = L q ( r ) − β ( u h , r ) , ∀ r ∈ M. The orthogonality property � a ξ ( e h , v h ) − β ( v h , ε h ) = 0 , ∀ v h ∈ W h β ( e h , r h ) = 0 , ∀ r h ∈ M h . Local interpolation error bounds on elements and faces 9

Residual error estimators Notations: T ih : for i=1,2, regular families of triangulations of Ω i by closed triangles ( n = 2 ) or tetrahedra ( n = 3 ) and T fh : regular families of triangula- tions of the hyperplane γ by closed segments ( n = 2 ) or triangles ( n = 3 ). E ih : the set of the edges or faces E of T ih ,and E 0 ih := E ih \ (Γ i ∪ γ ) , E Γ ih := E ih ∩ Γ i , E γ ih := E ih ∩ γ , E fh the set of the edges of T fh , E 0 fh := E fh \ ∂γ . For w h ∈ W h the tangential component of w h on a face E is defined by  w h · t E in 2D  ( w h ) τ,E := w h × n E in 3D  and, if [ · ] E denote the jump across E , we define the jump of K − 1 v h in the tangential direction across E by 1  2[( K − 1 u h ) τ,E ] E if E ∩ Γ i = ∅   J t,E ( u h ) :=  ( K − 1 u h + ∇ ( P h ¯ p i )) τ,E if E ∩ Γ i � = ∅ ,  10

For any T ∈ T i,h , for i = 1 , 2 , and for any t ∈ T f,h , η ( i ) h T � K − 1 η t := h t � K − 1 u ih + ∇ p ih � 0 ,T f u fh + ∇ τ p fh � 0 ,t := T i ζ ( i ) h 1 / 2 h T � curl ( K − 1 � := u h,i ) � 0 ,T + E � J t,E ( u h,i ) � 0 ,E T i E ∈ ∂T h t � curl ( K − 1 � h 1 / 2 ζ t := f u h,f ) � 0 ,t + � J t,e ( u h,f ) � 0 ,e e e ∈ ∂t ω ( i ) := � ( q i − P hi q i ) � 0 ,T ω t := � ( q f − P hf q f ) � 0 ,t T ω ( i ) h 1 / 2 ω t := h 1 / 2 := T � P h ¯ p i − ¯ p i � 0 ,∂T ∩ Γ i � P h ¯ p f − ¯ p f � 0 ,∂t ∩ ∂γ t T d δ ( i ) h 1 / 2 � p ih − p fh − ( ξu ih · n i − (1 − ξ ) u i +1 ,h · n i +1 ) � 0 ,t := t t 2 K f Upper bound on the velocity Proposition 1 : The error e u is bounded from above by the indicators 2 � � ζ ( i ) T + ω ( i ) T + ω ( i ) � � � � e u � W � + ( ζ t + ω t + ω t ) (1) T i =1 T ∈T ih t ∈T fh 11

Proof : for i=1,2,f (as in Hoppe and Wohlmuth (1999)) H ( div ; Ω i ) = H 0 ( div ; Ω i ) ⊕ H 1 ( div ; Ω i ) where H 0 ( div ; Ω i ) := { v i ∈ H ( div ; Ω i ) , s.t. div v i = 0 , and ( v i · n i ) | γ = 0 } H 1 ( div ; Ω i ) := { v i ∈ H ( div ; Ω i ) , s.t. ; � K − 1 w 0 i v i dx = 0 , ∀ w 0 i ∈ H 0 ( div ; Ω i ) } . i Ω i For n=2 we have H 0 ( div ; γ ) = { 0 } and so H ( div ; γ ) = H 1 ( div ; γ ) = H 1 ( γ ) . 12

Notations W 0 = H 0 ( div ; Ω 1 ) × H 0 ( div ; Ω 2 ) × H 0 ( div τ ; γ ) W 1 = H 1 ( div ; Ω 1 ) × H 1 ( div ; Ω 2 ) × H 1 ( div τ ; γ ) u = u 0 + u 1 u 0 ∈ W 0 , ∀ v 0 ∈ W 0 , a ξ ( u 0 , v 0 ) = − L d ( v 0 ) , (2) u 1 ∈ W 1 , β ( u 1 , r ) = L q ( r ) , ∀ r ∈ M. (3) The problems (2) and (3) are independent, and we check suc- cessively bounds of e 0 u and e 1 u . 13

• v 0 = curl φ ∈ W 0 , φ h := P C h φ , and v 0 h = curlφ h − L d ( v 0 − v 0 h ) − a ξ ( u h , v 0 − v 0 a ξ ( e u , v 0 ) = h )  � 2 �� �   � � curl ( K − 1 � = u h,i )( φ i − φ ih ) + J t,E ( u h,i )( φ i − φ ih ) i T E   i =1 T ∈T ih E ⊂ ∂T �� � � � curl ( K − 1 � + f u h,f )( φ f − φ fh ) + J t,e ( u h,f )( φ f − φ fh ) t e t ∈T fh e ∈ ∂t 2 � + ( curl ( φ i − φ ih ) · n i , (¯ p i − ¯ p ih ) Γ i i =1 +( curl ( φ f − φ fh ) · n f , ¯ p f − p fh ) ∂γ . v 0 = e 0 u gives 2 � � ζ ( i ) ω ( i ) � � � � e 0 u � 0 = � e 0 u � div � T + ¯ + ( ζ t + ¯ ω t ) . T i =1 T ∈T ih t ∈T fh 14

• From the approximate problem we have divu ih = P ih q i for i = 1 , 2 � 2 � � u ih · n i + q f divu fh = P fh i =1 � 2 � 1 / 2 2 � � � dive 1 u i � 2 0 , Ω i + � dive 1 u f � 2 e 1 u i · n i � 2 0 , Ω f + � 0 ,γ i =1 i =1 ω ( i ) � � � T + ω t T ∈T ih t ∈T fh 15

Upper bound on the pressure Proposition 2 The error ε p is bounded from above by the indicators   2  � �  η ( i ) T + ω ( i ) T + ω ( i ) δ ( i ) � � � � ε p � W � + t T   i =1 T ∈T ih t ∈T fh � ( η t + ω t + ω t ) t ∈T fh Proof : Dual problem (for the pressure)  Find v ∈ W, ¯ r ∈ M solution of ¯  a ξ (¯ v, v ) − β ( v, ¯ r ) = 0 , ∀ v ∈ W β (¯ v, r ) = − ( ε p , r ) , ∀ r ∈ M.  Regularity result ∃ C s > 0 , � ¯ v � 1 + � ¯ r � 1 ≤ C s � ε h � M . Lower bound on the error by standards arguments 17