Finite volume method for linear and non linear elliptic problems with discontinuities Franck BOYER and Florence HUBERT L.A.T.P. - Marseille, FRANCE Paris December 2007 1/ 46
O UTLINES I NTRODUCTION The classical finite volume scheme Anisotropic operator Nonlinear operator Operator with discontinuous coefficients References T HE standart DDFV SCHEME Assumptions on the continuous problem The meshes Construction of the scheme Convergence of DDFV scheme T HE M -DDFV SCHEME The method in 1D The method in 2D A NUMERICAL ALGORITHM N UMERICAL RESULTS 2/ 46
O UTLINES I NTRODUCTION The classical finite volume scheme Anisotropic operator Nonlinear operator Operator with discontinuous coefficients References T HE standart DDFV SCHEME Assumptions on the continuous problem The meshes Construction of the scheme Convergence of DDFV scheme T HE M -DDFV SCHEME The method in 1D The method in 2D A NUMERICAL ALGORITHM N UMERICAL RESULTS 3/ 46
T HE SCHEME FV4 Approximate the solution (for Dirichlet BC for instance) of − ∆ u = f (*) in an open bounded set Ω discretized by control volumes K (ex : triangles). The finite volume scheme principle ◮ Integrate (*) overall control volumes : � � � � f = − ∆ u = − ∇ u · n K σ . σ K K σ ⊂ ∂ K σ ◮ Approximate normal fluxes � L K ∇ u · n K σ n K σ τ σ x K x L ◮ Taylor expansion for σ = K | L � | σ | u ( x L ) − u ( x L ) x L x K � ∼ ∇ u · τ KL where τ KL = � � x L x K � d KL σ 4/ 46
T HE FV4 SCHEME Approximate the solution (for Dirichlet BC for instance) of − ∆ u = f (*) The classical FV4 scheme � � � | σ | u L − u K � � f = − ∆ u = − ∇ u · n K σ ≈ . d KL σ K K σ ⊂ ∂ K σ ⊂ ∂ K C ONSISTENCY : YES if [ x K , x L ] ⊥ σ . σ L K x K x L ⇒ Such meshes are called admissible. 5/ 46
T HE FV4 SCHEME Approximate the solution (for Dirichlet BC for instance) of − ∆ u = f (*) The classical FV4 scheme � � � | σ | u L − u K � � f = − ∆ u = − ∇ u · n K σ ≈ . d KL σ K K σ ⊂ ∂ K σ ⊂ ∂ K σ x K C ONSISTENCY : NO if [ x K x L ] ⊥ σ . L K x L ⇒ Such control volumes are said to be non admissible. 5/ 46
E RROR ESTIMATES FOR THE FV4 SCHEME A DMISSIBLE M ESHES T HEOREM The error of the FV4 scheme in case of admissible meshes, is bounded by C size ( T ) . 6/ 46
E RROR ESTIMATES FOR THE FV4 SCHEME N ON ADMISSIBLE MESHES T HEOREM If non admissible control volumes are located along a curve Γ , the error of the FV4 1 2 . scheme is bounded by C size ( T ) Example of non admissible meshes. 6/ 46
A NISOTROPIC OPERATOR Can we only use admissible meshes ? NO 1. To few meshes satisfy the admissibilty condition (triangles, Vorono¨ ı, ...). 2. For the anisotropic operator − div ( A ∇ u ) = f the admissibility condition becomes : A [ x K x L ] � n ... 3. How write these geometrical condition in case of variable diffusion tensor ? − div ( A ( z ) ∇ u ) = f . A solution is to approximate the two componants of the gradient. 7/ 46
N ONLINEAR DIFFUSION OPERATOR Example : p -laplacian Approximate in “ W 1 , p 0 (Ω) ” the unique solution of − div ( |∇ u | p − 2 ∇ u ) = f , 1 < p < + ∞ . Finite volume approach requires a consistant approximation of � |∇ u | p − 2 ∇ u · n . σ Impossible to obtain only with the two values u K and u L . � We still need an approximation of the whole gradient . 8/ 46
T HE PROBLEM OF DISCONTINUOUS COEFFICIENTS σ L K − div ( k ( z ) ∇ u ) = f , x K x L k ( z ) ∈ R . 9/ 46
T HE PROBLEM OF DISCONTINUOUS COEFFICIENTS σ L K − div ( k ( z ) ∇ u ) = f , x K x L k ( z ) ∈ R . If k is smooth, the finite volume FV4 writes : � � � u L − u K � � f = − div ( k ( z ) ∇ u ) dz = − ( k ( s ) ∇ u ) · n ds ≈ | σ | k σ , d KL � �� � σ K K σ ⊂ ∂ K σ ⊂ ∂ K = flux where k σ is an approximation of k on the edge σ � k σ = k ( x σ ) where k σ = 1 k ( s ) ds . | σ | σ 9/ 46
T HE PROBLEM OF DISCONTINUOUS COEFFICIENTS σ L K − div ( k ( z ) ∇ u ) = f , x K x L k ( z ) ∈ R . If k is discontinuous across σ : k K ans k L on K et L : How to write the scheme ? We look for k σ such that � u L − u K | σ | k σ ≈ ( k ( s ) ∇ u ( s )) · n ds . d KL σ The simple choices of k σ = k K , k σ = k L where k σ = 1 2 ( k K + k L ) lead to non consistent fluxes. Indeed ∇ u · n is discontinuous across σ ! 9/ 46
T HE PROBLEM OF DISCONTINUOUS COEFFICIENTS σ L K − div ( k ( z ) ∇ u ) = f , x K x L k ( z ) ∈ R . T AKE A NEW UNKNOWN u σ ON THE EDGE σ : Write the continuity of the approximate fluxes across σ . u L − u σ u σ − u K def = | σ | k L = | σ | k K . F KL d L σ d K σ Eliminate the fictive unknown u σ : u σ = k L d K σ u L + k K d L σ u K k L d K σ + k K d L σ u L − u K , with k σ = k K k L ( d K σ + d L σ ) = ⇒ F KL = | σ | k σ , harmonic mean value . k L d K σ + k K d L σ d KL 9/ 46
T HE PROBLEM OF DISCONTINUOUS COEFFICIENTS In presence of discontinuities, the scheme converges but the order of convergence depends on the choice of k σ : � k + if x > 0 . 5 � � C AS 1D : − d k ( x ) d = f , with k ( x ) = dx u e k − if x < 0 . 5 dx ◮ k σ arithmetic mean value : order 1 2 ◮ k σ harmonic mean value : order 1 16 mailles, avec lambda=1si x<0.5 et 10 sinon 0.045 exact moyenne arithmétique 0.04 moyenne harmonique 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10/ 46
R EFERENCES ◮ The finite volume scheme FV4 ◮ Eymard, Gallou¨ et, Herbin (00) ◮ Gradient reconstructions ◮ MPFA schemes. Aavatsmark (98/04), Lepotier (05),... ◮ Gradient FV schemes. Eymard, Gallou¨ et, Herbin (06), ... ◮ Mixte FV scheme Droniou, Eymard (06) ◮ Diamond schemes, DDFV schemes. Coudi` ere (99), Hermeline (00), Domelevo & Omn` es (05), Pierre (06), Delcourte & al (06), ABH (07), ....... ◮ Anisotropic problems with discontinuities ◮ Hermeline (03) ◮ BH (07) ◮ Benchmark - FVCA5 Aussois june 2008 11/ 46
O UTLINES I NTRODUCTION The classical finite volume scheme Anisotropic operator Nonlinear operator Operator with discontinuous coefficients References T HE standart DDFV SCHEME Assumptions on the continuous problem The meshes Construction of the scheme Convergence of DDFV scheme T HE M -DDFV SCHEME The method in 1D The method in 2D A NUMERICAL ALGORITHM N UMERICAL RESULTS 12/ 46
A IM AND NOTATIONS ◮ The DDFV scheme (D ISCRETE D UALITY F INITE V OLUME ) for � − div ( ϕ ( z , ∇ u e ( z ))) = f ( z ) , in Ω , u e = 0 , on ∂ Ω , ◮ Ω in an open bounded polygonal set R 2 . ◮ u �→ − div ( ϕ ( · , ∇ u )) is an monotonic and coercitive (of Leray-Lions type) operator. 13/ 46
A SSUMPTIONS ON ϕ ◮ Let p ∈ ] 1 , ∞ [ , p ′ = p − 1 and f ∈ L p ′ (Ω) . ◮ p ≥ 2 to simplify. p ◮ ϕ : Ω × R 2 → R 2 is a Caratheorory function such that : ( ϕ ( z , ξ ) , ξ ) ≥ C ϕ ( | ξ | p − 1 ) , ( H 1 ) � | ξ | p − 1 + 1 � | ϕ ( z , ξ ) | ≤ C ϕ . ( H 2 ) ( ϕ ( z , ξ ) − ϕ ( z , η ) , ξ − η ) ≥ 1 | ξ − η | p . ( H 3 ) C ϕ 1 + | ξ | p − 2 + | η | p − 2 � � | ϕ ( z , ξ ) − ϕ ( z , η ) | ≤ C ϕ | ξ − η | . ( H 4 ) ◮ ϕ is lipschitz continuous with respect to z . 14/ 46
T HE DDFV MESHES primal, dual and “diamond”. x L x K L K mesh M Primal control volumes � ( u K ) K ∈ M 15/ 46
T HE DDFV MESHES primal, dual and “diamond”. x L∗ L∗ x L x K L K∗ K x K∗ mesh M ∗ mesh M Primal control volumes Dual control volumes � ( u K ) K ∈ M � ( u K∗ ) K∗ ∈ M ∗ 15/ 46
T HE DDFV MESHES primal, dual and “diamond”. x L∗ L∗ x L x K L K∗ K x K∗ mesh M ∗ mesh D mesh M Primal control volumes Dual control volumes Diamond cells � ( u K ) K ∈ M � ( u K∗ ) K∗ ∈ M ∗ � Discrete gradient 15/ 46
T HE DDFV SCHEME T HE DISCRETE GRADIENT � u L − u K � n K σ + u L∗ − u K∗ 1 T = T ∇ D u , ∀ diamond cell D . n K∗ σ | σ ∗ | | σ | sin α D x K∗ x K∗ u K + u K∗ 2 n K σ τ K∗L∗ u L + u K∗ 2 x K x K τ KL n ∗ K∗ σ ∗ x L x L u K + u L∗ 2 u L + u L∗ 2 x L∗ x L∗ � T · ( x L − x K ) = u L − u K , T ∇ D u Equivalent definition T · ( x L∗ − x K∗ ) = u L∗ − u K∗ . ∇ T D u 16/ 46
T HE DDFV SCHEME T HE DISCRETE GRADIENT � u L − u K � n K σ + u L∗ − u K∗ 1 T = T ∇ D u , ∀ diamond cell D . n K∗ σ | σ ∗ | | σ | sin α D T HE STANDARD DDFV SCHEME Classical finite volume formulation : � � T ) , n K σ ) = − | σ | ( ϕ D ( ∇ T f ( z ) dz , ∀ K ∈ M , D u K σ ∈E K � � T ) , n K∗ σ ) = K∗ f ( z ) dz , ∀ K∗ ∈ M ∗ , | σ ∗ | ( ϕ D ( ∇ T − D u σ ∗ ∈E K∗ with � ϕ D ( ξ ) = 1 ϕ ( z , ξ ) dz , approximate flux on the diamond cell | D | D 16/ 46
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