Multi-level adaptive vertex-centered finite volume methods for - PowerPoint PPT Presentation

Multi-level adaptive vertex-centered finite volume methods for diffusion problems Fayssal Benkhaldoun supervising: Tarek Ghoudi - PhD Joint work with Imad Kissami Postdoc July 3, 2018 F. Benkhaldoun cluster MAGI 1 / 27

Motivations and mathematical model Adaptive FE-FV are now widely used in the numerical solution of (PDEs) to achieve better accuracy with minimum degrees of freedom. We first solve the PDE to get the solution on the current mesh. The error is estimated using the solution, and used to mark a set of triangles that are to be refined. Triangles are refined in such a way to keep mesh regularity and conformity. F. Benkhaldoun cluster MAGI 2 / 27

Motivations and mathematical model A typical loop of (AFE-FVM ) through local refinement involves: F. Benkhaldoun cluster MAGI 3 / 27

Motivations and mathematical model � Conformity of the mesh � Prevent the propagation of refinement levels � Efficiency of estimator � Convergence of error � Performance of CPU time F. Benkhaldoun cluster MAGI 4 / 27

Motivations and mathematical model Let a : R + → R be a given nonlinear function. Typically, a ( x ) = x p − 2 for some real number p ∈ (1 , + ∞ ). Let σ such that ∀ ξ ∈ R d σ ( ξ ) = a ( | ξ | ) ξ (1) where | . | is the Euclidean norm in R d . Then, for a given source function f : Ω → R , the nonlinear Laplace problem consists in looking for u : Ω → R such that − div ( σ ( ∇ u )) = f in Ω (2) = g on ∂ Ω u F. Benkhaldoun cluster MAGI 5 / 27

Linear problem � ⊂ R d =2 , 3 − div ( K ∇ p ) = f in Ω Problem : find p ∈ H 1 0 (Ω), ( S ) p = g on ∂ Ω (3) Unicity and Existence Assumptions: ( H 1) K ∈ L ∞ (Ω). ( H 2) f ∈ L 2 (Ω) . ( S ) has a unique solution. Remark : The problem (2) represents,for instance, the extension of the problem (3) which takes into account the nonlinear dependence of the Darcy velocity on the pressure head gradient ∇ p . Note that (2) and (3) coincide, for a ( x ) = x p − 2 , when p = 2. F. Benkhaldoun cluster MAGI 6 / 27

Linear problem FV-FE scheme Figure: Dual cell S 1 , S 2 , and S 3 are the vertices of a triangle T, B its barycentre, Σ opp , Σ opp and Σ opp The edges[ S 2 S 3 ], [ S 1 S 3 ] et 1 2 3 [ S 1 S 2 ] ; − → , − → and − → n opp n opp n opp outgoing unit normals such that 1 2 3 n pq ⊥− − − → n pq . − − → − → M pq B and − → S p S q > 0 F. Benkhaldoun cluster MAGI 7 / 27

Linear problem FV-FE scheme The approximation of the diffusive flux is based on an implicit scheme: � � K ∇ p . − → − n d σ = f ( x ) dx (4) ∂ D h D h � � � K T ∇ p . − → − n d σ = f ( x ) dx (5) ∂ D h ∩ T h D h T ∩ D h � = ∅ We note the elementary diffusion terms by: | Σ opp | Σ opp | | − → − → n opp n opp k flow 1 2 12 ( T ) = | T | K T 1 2 2 | T | 2 | T | | Σ opp | Σ opp | | − → − → n opp n opp k flow 1 3 13 ( T ) = | T | K T 1 3 2 | T | 2 | T | Finally, the finite volume scheme for the flow equation is written: � � k flow 12 ( T )( p 2 − p 1 ) + k flow 13 ( T )( p 3 − p 1 ) = f ( x ) dx (6) D h T ∈ D h F. Benkhaldoun cluster MAGI 8 / 27

Linear problem FV-FE scheme Figure: Primal mesh T h , Dual mesh D h and the fine simplicial mesh S h Remark : the flux − K ∇ p ∈ H(div, Ω) but − K ∇ p h / ∈ H(div, Ω) Flux reconstruction (exploits the local conservativity): t h ∈ RTN 0 ( S h ) ⊂ H(div, Ω) ∀ D ∈ D int ( div t h , 1) D = ( f , 1) D , h F. Benkhaldoun cluster MAGI 9 / 27

Linear problem FV-FE scheme Construction of t h by Direct Prescription : We solved the following system ( S ′ ):  t h · − N 1 = − K ∇ p h · − → → N 1     t h · − → � K | K ∇ p h · − → � � K | L ∇ p h · − → � ( S ′ ) N 2 = − w K , s N 2 − w L , s N 2   t h · − N 3 = − K ∇ p h · − → →   N 3 n2 K | K ( K | L ) is an L e Th approximation of the tensor N1 Dh flux1=th.N1 nC of permeability on the flux2 =th.N2 nG N2 triangle K ( L ) G_tsh(xG,yG) K tsh N 1 , − − → N 2 and − → → n1 N3 N 3 : unit flux3 =th.N3 normal vectors. Sh Harmonic averaging : K K K L w K , s = K K + K L , w L , s = K K + K L F. Benkhaldoun cluster MAGI 10 / 27

Linear problem FV-FE scheme Error estimator : � � � � � 2 1 2 ∇ p + K − 1 1 ||| p − p h ||| 2 � � � � 2 t h ) 2 Ω = 2 ∇ ( p − p h ) Ω = ( K (7) � K � � � Ω   2 � � � � � ||| p − p h ||| 2 ≤ 2 ∇ p h + K − 1 1  � � � �   m D || f − div t h || D + � K 2 t h � � �  D � �� � D ∈D h � �� � residual error flux error m D , a = C P , D h 2 m D , a = C F , D h 2 D if D ∈ D int D if D ∈ D ext , h h c a , D c a , D C P , D is equal 1 π 2 if D is convexe, C F , D is equal to 1 on general. � 1 � � 2 ( η R , D + η DF , D ) 2 D ∈ D h Effectivity index: ||| p − p h ||| Ω F. Benkhaldoun cluster MAGI 11 / 27

Linear problem Mesh Adaptation T : = Triangulation of Ω, for all τ ∈ T we define v ( τ ) the ”newest vertex”. E ( τ ): = Is the longest edge of τ , v ( τ ) is the vertex opposite to E ( τ ). (R1): The first step consists in dividing the elements by joining v ( τ ) to the middle I of E ( τ ). (R2): I becomes the ”newest vertex” of each of the two created triangles. (R3): Neighbor refinement by R 1 and conformity. Figure: Bisect a triangle and Completion by Newest-Vertex-Bisection strategy F. Benkhaldoun cluster MAGI 12 / 27

Linear problem Mesh Adaptation T1 0 3 1 3 1 3 T2 0 2 2 T3 0 1 1 T4 T5 Figure: Mesh refinement with ADAPT and conformity with propagation levels F. Benkhaldoun cluster MAGI 13 / 27

Linear problem Mesh Adaptation T1 0 3 T2 0 T3 0 T4 T5 Figure: Mesh refinement with ADAPT-NEWEST First we proceed in a first time by refining our mesh by the ADAPT strategy, then for the conformity one uses the method Newest vertex bisection. There is no more propagation of the refinement on triangles T 1 , T 2 et T 4 . F. Benkhaldoun cluster MAGI 14 / 27

Linear problem Numerical Tests Test with an analytical solution, α = 0 , 127 Problem: in Ω = ( − 1 , 1) 2 − div ( K ∇ p ) = f = 0 on ∂ Ω p heterogeneous permeability � 1 . I 2 if x ∈ Ω 1 , 4 K = 100 . I 2 else . Solution p ∈ H 1+ α (Ω) , a i , b i = const. p ( r , θ ) = r α ( a i sin( αθ )+ b i cos( αθ )) F. Benkhaldoun cluster MAGI 15 / 27

Linear problem Newest-Vertex-Bisection strategy Regular mesh: α = 0 . 127 F. Benkhaldoun cluster MAGI 16 / 27

Linear problem Newest-Vertex-Bisection strategy 10 3 9 error adapt. effectivity ind.adapt. estimate adapt. 8 7 Energy error effectivity 10 2 6 Energy error 5 4 10 1 3 2 10 0 1 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4 Number of vertices Number of vertices Figure: Energy Error, Estimator, Efficiency NewestVB approach iter DoFs η ǫ 1 ǫ 2 f η CPU 1 128 103.3915 15.836 0.30586 6.5289 0.679237 6 436 67.4077 10.0475 0.13689 6.7089 0.668070 12 942 44.08 8.8284 0.074364 4.993 0.074364 24 2170 18.4356 7.4593 0.02977 2.4715 1.426176 59 7162 7.0814 5.3987 0.021182 1.3117 4.063058 Total 75.13 F. Benkhaldoun cluster MAGI 17 / 27

Linear problem Adapt-Newest strategy Regular mesh : α = 0 . 127 F. Benkhaldoun cluster MAGI 18 / 27

Linear problem Adapt-Newest strategy 10 3 7 Erreur energie Efficiency Estimateur 6 10 2 5 4 10 1 3 2 10 0 1 10 1 10 2 10 3 10 4 10 1 10 2 10 3 10 4 Figure: Estimateur, erreur energie (gauche), efficacit´ e (droite) AdaptNVB approach iter DoFs η ǫ 1 ǫ 2 f η CPU 1 240 103.3915 15.836 0.30586 6.5289 0.671513 6 1040 42.7369 10.677 0.081534 4.0027 0.918080 13 2160 20.8044 7.846 0.035467 2.4715 1.462513 24 4920 9.3623 5.8007 0.022808 1.614 2.870125 29 7296 6.9263 5.104 0.022277 1.357 3.995373 Total 35.13 F. Benkhaldoun cluster MAGI 19 / 27

Linear problem Adapt-Newest strategy Irregular mesh : α = 0 . 127 10 2 7 Erreur energie Efficiency Estimateur 6 5 10 1 4 3 2 10 0 1 10 1 10 2 10 3 10 4 10 2 10 3 10 4 Figure: Estimateur, erreur energie (gauche), efficacit´ e (droite) F. Benkhaldoun cluster MAGI 20 / 27

Nested adaptive vertex-centered finite volume Nested adaptive vertex-centered finite volume The diffusion in a two-dimensional closed medium Ω ⊂ R 2 with boundary ∂ Ω is described by the following equation −∇ · ( K ∇ u ( x )) = f ( x ) , ∀ x ∈ Ω , (8) u ( x ) = g ( x ) , ∀ x ∈ ∂ Ω , where f is the external force, g the boundary source, and K is a piecewise constant diffusion coefficient. F. Benkhaldoun cluster MAGI 21 / 27

Numerical results Irregular mesh 7 10 2 error adapt. Effectivity ind.adapt. estimate adapt. 6.5 6 5.5 5 Effectivity Error 4.5 4 3.5 10 1 3 2.5 2 1.5 10 2 10 3 10 2 10 3 Number of vertices Number of vertices 7 10 2 error adapt. Effectivity ind.adapt. estimate adapt. 6 5 Effectivity Error 4 3 10 1 2 1 10 2 10 3 10 2 10 3 Number of vertices Number of vertices Figure: Results using conventional approach (left) and using nested approach (right). F. Benkhaldoun cluster MAGI 22 / 27