Finite volume methods for dissipative problems Claire - PowerPoint PPT Presentation

Finite volume methods for dissipative problems Claire Chainais-Hillairet CEMRACS summer school Marseille, July 2019, 15-20

Lecture 3 : Finite volume schemes and long time behavior

Outline Discrete functional inequalities 1 Results for the porous media equations 2 Results for the Fokker-Planck equations 3

Outline Discrete functional inequalities 1 Results for the porous media equations 2 Presentation of the schemes Long time behavior Results for the Fokker-Planck equations 3 Presentation of the B-schemes and first results Long time behavior About nonlinear schemes

Some references ❑ Herbin, 1995 ❑ Coudi` ere, Vila, Villedieu, 1999 ❑ Eymard, Gallou¨ et, Herbin, 1999, 2000, 2010 ❑ Gallou¨ et, Herbin, Vignal, 2000 ❑ Coudi` ere, Gallou¨ et, Herbin, 2001 ❑ Droniou, Gallou¨ et, Herbin, 2003 ❑ Andreianov, Gutnic, Wittbold, 2004 ❑ Filbet, 2006 ❑ Glitzky, Griepentrog, 2010 ❑ Andreianov, Bendahmane, Ruiz Baier, 2011 ❑ Bessemoulin-Chatard, C.-H., Filbet, 2015

Space of approximate solutions and norms � � � ⊂ L 1 (Ω) , X ( T ) = u T = u K 1 K K ∈T ∈ H 1 (Ω) but X ( T ) / L q -norms For 1 ≤ q < + ∞ , � 1 /q �� | u T ( x ) | q dx � u T � 0 ,q = Ω � 1 /q � � m( K ) | u K | q = . K ∈T � u T � 0 , ∞ = max K ∈T | u K | .

About the mesh Regularity of the mesh Each control volume K is star-shaped with respect to x K . There exists ξ > 0 such that ∀ K ∈ T , ∀ σ ∈ E K , d( x K , σ ) ≥ ξ d σ . L σ • x L d σ x K • K Remark Admissibility assumption not necessary.

Discrete W 1 ,p -norms General framework Discrete W 1 ,p -semi-norm : | u L − u K | p | u T | p � 1 ,p, T = m( σ )d σ . d p σ σ = K | L Discrete W 1 ,p -norm : � u T � 1 ,p, T = � u T � 0 ,p + | u T | 1 ,p, T . With homogeneous Dirichlet boundary conditions on Γ 0 ⊂ Γ ( D σ u ) p | u T | p � 1 ,p, Γ 0 , T = m( σ )d σ d p σ σ ∈E  | u K − u L | si σ = K | L,  si σ ⊂ Γ 0 , | u K | where D σ u = si σ ⊂ Γ \ Γ 0 . 0 

Relations between the norms For 1 ≤ s ≤ p , for all u T ∈ X ( T ) , p − s ps � u T � 0 ,p , � u T � 0 ,s ≤ m(Ω) and � p − s � d m(Ω) ps | u T | 1 ,s, T ≤ | u T | 1 ,p, T ξ Proof older inequality with p ′ = p p s and q ′ = H¨ p − s Due to the regularity of the mesh : m( σ )d σ ≤ 1 m( σ )d( x K , σ ) = d m(Ω) � � � . ξ ξ K ∈T σ ∈E K σ = K | L

The space L 1 ∩ BV (Ω) Total variation Let Ω be an open set of R N and u ∈ L 1 (Ω) . We define : �� � u ( x )div ϕ ( x ) dx ; ϕ ∈ C 1 c (Ω , R N ) , � ϕ � ∞ ≤ 1 TV Ω ( u ) = sup Ω L 1 ∩ BV (Ω) L 1 ∩ BV (Ω) = u ∈ L 1 (Ω); TV Ω ( u ) < + ∞ � � . L 1 ∩ BV (Ω) is endowed with the norm : � u � BV (Ω) = � u � L 1 (Ω) + TV Ω ( u ) .

Relation between X ( T ) and L 1 ∩ BV (Ω) Total variation of u T ∈ X ( T ) � TV Ω ( u T ) = m( σ ) | u K − u L | = | u T | 1 , 1 , T . σ = K | L Inclusion For all u T ∈ X ( T ) , � u T � 1 , 1 , T < + ∞ and X ( T ) ⊂ L 1 ∩ BV (Ω) .

Starting point for the discrete functional inequalities ❑ Ambrosio, Fusco, Pallara, 2000 ❑ Ziemer, 1989 Theorem Let Ω be a bounded Lipschitz domain of R N , N ≥ 2 . There exists C > 0 , depending only on Ω such that � N − 1 �� N N ∀ u ∈ L 1 ∩ BV (Ω) . | u | ≤ C � u � BV (Ω) N − 1 Ω L 1 ∩ BV (Ω) ⊂ L N/ ( N − 1) (Ω) with continuous embedding.

Discrete Poincar´ e-Sobolev inequality Theorem Let Ω be a polyedral bounded domain of R N , N ≥ 2 . Let ( T , E , P ) be a regular mesh of Ω , with regularity ξ . pN If 1 ≤ p < N , let 1 ≤ q ≤ p ∗ = N − p . If p ≥ N , let 1 ≤ q < + ∞ . There exists C > 0 , depending only on p , q , N and Ω such that C � u T � 0 ,q ≤ ξ ( p − 1) /p � u T � 1 ,p, T ∀ u T ∈ X ( T ) .

A crucial lemma Lemma Let Ω be an open bounded polyhedral domain of R N , N ≥ 2 . Let ( T , E , P ) a regular mesh of Ω , with regularity parameter ξ . For all s > 1 , p > 1 , we have : C ξ ( p − 1) /p � u T � ( s − 1) � u T � s 0 ,sN/ ( N − 1) ≤ 0 , ( s − 1) p/ ( p − 1) � u T � 1 ,p, T ∀ u T ∈ X ( T ) . Proof ➟ Application of the Theorem on L 1 ∩ BV to v T = | u T | s . � � ξ ( p − 1) /p | u T | 1 ,p, T � u T � ( s − 1) 1 0 , ( s − 1) p/ ( p − 1) + � u T � s ➟ lhs ≤ C 0 ,s ➟ Interpolation : � u T � 0 ,s ≤ � u T � 1 /s 0 ,p � u T � ( s − 1) /s 0 , ( s − 1) p/ ( p − 1) .

The key points of the proof of (PSdis) p = 1 Direct consequence of the embedding Theorem : � u T � 0 ,N/ ( N − 1) ≤ C � u T � 1 , 1 , T . N p ∗ = ⇒ result sill holds ∀ 1 ≤ q ≤ p ∗ . N − 1 = 1 < p < N Let s = ( N − 1) p N − p . Then, s > 1 , ( s − 1) p sN sN Np = N − 1 and N − 1 = N − p . p − 1 Application of the lemma : C � u T � 0 ,pN/ ( N − p ) ≤ ξ ( p − 1) /p � u T � 1 ,p, T . Result ∀ 1 ≤ q ≤ p ∗ = pN N − p .

The key points of the proof of (PSdis) p = N Application of the lemma with p = N : C ξ ( N − 1) /N � u T � ( s − 1) � u T � s 0 ,sN/ ( N − 1) ≤ 0 , ( s − 1) N/ ( N − 1) � u T � 1 ,N, T . But L sN/ ( N − 1) (Ω) ⊂ L ( s − 1) N/ ( N − 1) (Ω) , so that C � u T � 0 , ( s − 1) N/ ( N − 1) ≤ ξ ( N − 1) /N � u T � 1 ,N, T s = 1 + ( N − 1) q/N . p > N C We have : � u T � 1 ,N, T ≤ ξ ( p − N ) / ( pN ) � u T � 1 ,p, T . We apply the result for p = N .

Discrete Poincar´ e-Sobolev inequality, Dirichlet case Theorem Let Ω be a polyedral bounded domain of R N , N ≥ 2 . Let Γ 0 ⊂ Γ , m(Γ 0 ) > 0 . Let ( T , E , P ) be a regular mesh of Ω , with regularity ξ . pN If 1 ≤ p < N , let 1 ≤ q ≤ p ∗ = N − p . If p ≥ N , let 1 ≤ q < + ∞ . There exists C > 0 , depending only on p , q , N , Γ 0 and Ω such that C � u T � 0 ,q ≤ ξ ( p − 1) /p | u T | 1 ,p, Γ 0 , T ∀ u T ∈ X ( T ) .

Discrete Poincar´ e-Wirtinger inequality Theorem Let Ω be a polyedral bounded domain of R N , N ≥ 2 . Let ( T , E , P ) be a regular mesh of Ω , with regularity ξ . For all 1 ≤ p < + ∞ , there exists C > 0 , depending only on p , N and Ω such that C � u T − ¯ u T � 0 ,p ≤ ξ ( p − 1) /p | u T | 1 ,p, T ∀ u T ∈ X ( T ) , where 1 � ¯ u T = u T . m(Ω) Ω

Outline Discrete functional inequalities 1 Results for the porous media equations 2 Presentation of the schemes Long time behavior Results for the Fokker-Planck equations 3 Presentation of the B-schemes and first results Long time behavior About nonlinear schemes

FV scheme for the evolutive equation ∂ t f = ∆ f β , in Ω × R +     f = f D on Γ D × R + , ∇ f · n = 0 on Γ N × R +   f ( · , 0) = f 0 > 0 .  The scheme m( K ) f n +1 − f n  τ σ D K,σ ( f n +1 ) β = 0 K K � − ∀ K ∈ T   ∆ t   σ ∈E K 1 1 � �  f D f D , f 0  σ = K = f 0   m( σ ) m( K ) σ K u L − u K if σ = K | L    u D if σ ⊂ Γ D σ − u K with the notation : D K,σ u =  if σ ⊂ Γ N  0

Hypotheses and first result Hypotheses Admissibility and regularity of the mesh E D ext � = ∅ f 0 K ≥ 0 ∀ K ∈ T ∃ m D and M D such that 0 < m D ≤ f D σ ≤ M D ∀ σ ∈ E D ext . Proposition The scheme has a unique nonnegative solution ( f n K ) K ∈T ,n ≥ 0 . ❑ Eymard, Gallou¨ et, Hilhorst, Na¨ ıt Slimane, 1998

Scheme for the steady state � ∆ f β = 0 , in Ω × R + f = f D on Γ D × R + , ∇ f · n = 0 on Γ N × R + The scheme τ σ D K,σ ( f ∞ ) β = 0 , � ∀ K ∈ T . σ ∈E K Proposition The scheme has a unique nonnegative solution ( f ∞ K ) K ∈T , which satisfies : m D ≤ f ∞ K ≤ M D ∀ K ∈ T .

Outline Discrete functional inequalities 1 Results for the porous media equations 2 Presentation of the schemes Long time behavior Results for the Fokker-Planck equations 3 Presentation of the B-schemes and first results Long time behavior About nonlinear schemes

At the continuous level f β +1 − ( f ∞ ) β +1 � − ( f ∞ ) β ( f − f ∞ ) E ( t ) = β + 1 Ω � f β − ( f ∞ ) β � � | 2 D ( t ) = |∇ Ω Relation between entropy and dissipation : D ( t ) ≥ ( m D ) β − 1 E ( t ) . C P Exponential decay of the entropy : E ( t ) ≤ E (0) e − λt , with λ = ( m D ) β − 1 . C P

At the discrete level Discrete relative entropy K ) β +1 − ( f ∞ � ( f n K ) β +1 � E n = � − ( f ∞ K ) β ( f n K − f ∞ m( K ) K ) . β + 1 K ∈T Discrete dissipation � 2 D n = � D K,σ (( f n +1 ) β − ( f ∞ ) β ) � τ σ σ ∈E Discrete entropy-entropy dissipation property E n +1 − E n + D n +1 ≤ 0 ∀ n ≥ 0 . ∆ t

Exponential decay towards the steady-state Discrete Poincar´ e inequality K ) β � 2 ≤ C P � ) β − ( f ∞ � ( f n +1 ξ D n +1 . m( K ) K K ∈T Elementary inequality � x β +1 − y β +1 � ( x β − y β ) 2 ≥ y β − 1 − y β ( x − y ) ∀ x, y ≥ 0 . β + 1 Consequences C P E n +1 ≤ ξ ( m D ) β − 1 D n +1 � − 1 � 1 + ∆ t ξ ( m D ) β − 1 E n +1 ≤ E n C P