A finite volume method on general meshes for a time evolution - PowerPoint PPT Presentation

A finite volume method on general meshes for a time evolution convection-diffusion equation. Konstantin Brenner Supervised by Danielle Hilhorst University Paris-Sud XI MoMaS

Model problem & Numerical scheme ; Existence of the unique discrete solution, a priori esti- mate on the approximate solution in the discrete norms L ∞ (0 , T ; L 2 (Ω)) and L 2 (0 , T ; H 1 (Ω)) ; Estimates on the differences of the space and time trans- lates, which implies the propriety of the relative compact- ness by the theorem of Fréchet–Kolmogorov ; Strong convergence in L 2 of the approximate solution to the weak solution of the problem ( P ) ; Numerical tests .

Convection-diffusion problem We consider the convection-diffusion problem  ∂ t u − ∇ · ( Λ ( x ) ∇ u ) + ∇ · ( V u ) = f ( x , t ) in Q T = Ω × (0 , T ) ,    ( P ) u ( x , t ) = 0 on ∂ Ω × (0 , T ) ,   u ( x , 0) = u 0 ( x ) for all x ∈ Ω .  Dubrovnik, 13-16 October 2008 | 2 K.Brenner, University Paris-Sud XI

Convection-diffusion problem We suppose that the following hypotheses are satisfied Ω is an open bounded connected polyhedral subset of R d , d ∈ N \ { 0 } ; ( H 1 ) ( H 2 ) Λ is a measurable function from Ω to M d ( R ) , where M d ( R ) denotes the set of d × d symmetric matrices, such that for a.e. x ∈ Ω the set of its eigenvalues is included in [ λ, λ ] , where λ, λ ∈ L ∞ (Ω) are such that 0 < α 0 � λ ( x ) � λ ( x ); V ∈ L 2 (0 , T ; H ( div, Ω)) ∩ L ∞ ( Q T ) is such that ∇ · V � 0 a.e. in Q T ; ( H 3 ) u 0 ∈ L ∞ (Ω); ( H 4 ) f ∈ L 2 ( Q T ) . ( H 5 ) Dubrovnik, 13-16 October 2008 | 3 K.Brenner, University Paris-Sud XI

The possible finite volume schemes The essential problem is related with the fact that Λ is a full matrix, in such case it is not possible to use the classical two point discretization. Possible solutions among the finite volume methods : Finite Volumes - Finite Elements (Angot A., Dolejší V., Feistauer M., Felcman j., Vohralík M.) Idea : Use the dual finite element grid in order approximate the diffusion term. Multipoint Flux Approximation Methods (MFAM) (Aavatsmark I., Eigestad G. T., Klausen, R. A.) Idea : Use several neighbors of the control volume in order to define the diffusive flux ; Hybride Finite Volume Method (Eymard R., Gallouët R., Herbin R.) Idea : Take into account the supplementary unknowns associated with the cell faces. Dubrovnik, 13-16 October 2008 | 4 K.Brenner, University Paris-Sud XI

Discretization A discretization of Ω , denoted by D , is defined as the triplet D = ( M , E , P ) , where : 1. M is a family of control volumes ; 2. E = E int ∪ E ext is a set of edges ; 3. P = ( x K ) K ∈M is a family of points, such that for all K ∈ M , x K ∈ K and K is x K -star-shaped. m ( K ) , the measure of K ∈ M ; m ( σ ) , the measure of σ ∈ E ; E K , the set of edges of K ∈ M ; M σ , the set of control volumes containing σ ∈ E ; n K,σ , the unit vector outward to K and normal to σ ∈ E K ; d K,σ , the Euclidean distance between x K and σ ∈ E K ; D K,σ , the cone with vertex x K and basis σ ∈ E K . A time discretization is given by 0 = t 0 < t 1 . . . < t N = T with the constant time step k = T/N. Dubrovnik, 13-16 October 2008 | 5 K.Brenner, University Paris-Sud XI

Discretization We associate with the mesh the following spaces of discrete unknowns X D = { (( v K ) K ∈M , ( v σ ) σ ∈E ) , v K ∈ R , v σ ∈ R } X D , 0 = { v ∈ X D such that ( v σ ) σ ∈E ext = 0 } , The space X D is equipped with the semi-norm m ( σ ) � � | v | 2 d K,σ ( v σ − v K ) 2 , X = K ∈M σ ∈E K which is the norm in X D , 0 . Dubrovnik, 13-16 October 2008 | 6 K.Brenner, University Paris-Sud XI

The finite volume scheme (i) The initial condition for the scheme � 1 u 0 K = u 0 ( x ) d x ; m ( K ) K (ii) The discrete equations K − u n − 1 � � m ( K )( u n F K,σ ( u n ) + k K,σ = m ( K ) f n V K,σ u n ) + k ∀ K ∈ M , K K σ ∈E K σ ∈E K � t n 1 � where f n K = f ( x , t ) d x dt ; m ( K ) t n − 1 K (iii) The local conservation of the total flux ( F K,σ ( u n ) + V K,σ u n K,σ ) + ( F L,σ ( u n ) + V L,σ u n L,σ ) = 0 for all σ ∈ E int , M σ = { K, L } ; (iv) The discrete analog of the boundary conditions u n σ ∈ E ext . σ = 0 for all Dubrovnik, 13-16 October 2008 | 7 K.Brenner, University Paris-Sud XI

The variational form of the finite volume scheme We put the scheme (i)-(iv) under the equivalent form Let u 0 is defined by 1 � u 0 K = u 0 ( x ) ∀ K ∈ M . m ( K ) K For each n ∈ { 1 , . . . , N } find u n ∈ X D , 0 such that for all v ∈ X D , 0 , ) + k < v, u n > F + k < v, u n > T = � K − u n − 1 � m ( K ) v K ( u n m ( K ) v K f n K , K K ∈M K ∈M where < v, u n > F = � � ( v K − v σ ) F K,σ ( u n ) , K ∈M σ ∈E K and < v, u n > T = � � ( v K − v σ ) V K,σ u n K,σ . K ∈M σ ∈E K Dubrovnik, 13-16 October 2008 | 8 K.Brenner, University Paris-Sud XI

Discretization of the convection term We define � V K,σ = V ( x ) · n K,σ ; σ let an upwind value u n K,σ be given by  u n K , if V K,σ � 0  u n K,σ = u n σ , if V K,σ < 0 .  We have completely defined the discrete version of the convective term. We give below the definition of the discret flux F K,σ ( u n ) . Dubrovnik, 13-16 October 2008 | 9 K.Brenner, University Paris-Sud XI

Discretization of the diffusion term We denote ∇ K,σ u = ∇ K u + R K,σ u · n K,σ , where 1 � ∇ K u = m ( σ )( u σ − u K ) n K,σ m ( K ) σ ∈E K and where R K,σ u = α K d K,σ ( u σ − u K − ∇ K u · ( x σ − x K )) , with some α K > 0 , which should be chosen properly. Optimization of α K has been studied by O. Angelini, C. Chavant, E. Chenier, R. Eymard. Dubrovnik, 13-16 October 2008 | 10 K.Brenner, University Paris-Sud XI

Discretization of the diffusion term We denote ∇ K,σ u = ∇ K u + R K,σ u · n K,σ , where 1 � ∇ K u = m ( σ )( u σ − u K ) n K,σ m ( K ) σ ∈E K and where √ d R K,σ u = d K,σ ( u σ − u K − ∇ K u · ( x σ − x K )) . √ The choice α K = d yields the two point scheme in case of meshes which satisfy n K,σ = x σ − x K d K,σ . Optimization of α K has been studied by O. Angelini, C. Chavant, E. Chenier, R. Eymard. Dubrovnik, 13-16 October 2008 | 11 K.Brenner, University Paris-Sud XI

The discret gradient ∇ D u We define the discret gradient ∇ D u by ∇ D u ( x ) = ∇ K,σ u x ∈ D K,σ , where D K,σ is the cone with vertex x K and basis σ ∈ E K , notice that the bilinear form � � � < v, u > F = ( v K − v σ ) F K,σ ( u ) = ∇ D v · Λ ( x ) ∇ D u Ω K ∈M σ ∈E K is symmetric. We show in what follows that it is also continuous and coercive. Dubrovnik, 13-16 October 2008 | 12 K.Brenner, University Paris-Sud XI

Advantages of the scheme Very general class of meshes ; Local conservativity ; The discretization of the convection and diffusion flux does not involve the unk- nowns outside of the cell ; One can easily eliminate the cell unknowns u K and then solve the system of card ( E ) equations. Dubrovnik, 13-16 October 2008 | 13 K.Brenner, University Paris-Sud XI

Existence and uniqueness of the discrete solution Lemma 1 Let D be the discretization of Ω . (i) There exists C 1 > 0 and α > 0 such that | < u, v > F | � C 1 | u | X | v | X and < u, u > F � α | u | 2 X . for all u, v ∈ X D . (ii) There exists C 2 > 0 such that | < u, v > T | � C 2 | u | X | v | X and < u, u > T � 0 for all u, v ∈ X D . Dubrovnik, 13-16 October 2008 | 14 K.Brenner, University Paris-Sud XI

Existence and uniqueness of the discrete solution The Lemma 1 and the Lax-Milgram Theorem implies the following result Theorem 1 The discrete problem (i)-(iv) possess the unique solution. Definition of the approximate solution Let u n ∈ X D , 0 , n = 1 . . . N , be a solution of the approximate problem, with k = T/N . We say that the piecewise constant function u h,k is an approximate solution of the problem ( P ) if u h,k ( x , 0) = u 0 for all x ∈ K K u h,k ( x , t ) = u n for all ( x , t ) ∈ K × ( t n − 1 , t n ] . K We also define its gradient by ∇ h u h,k ( x , t ) = ∇ D u n ( x ) for all ( x , t ) ∈ K × ( t n − 1 , t n ] . Dubrovnik, 13-16 October 2008 | 15 K.Brenner, University Paris-Sud XI

A priori estimates Theorem 2 (A priori estimate) Let u h,k be a solution of the discrete problem, then it is such that � u h,k ( · , t ) � L ∞ (0 ,T ; L 2 (Ω)) � � u 0 � L 2 (Ω) + 2 T � f � L 2 ( Q T ) , λ �∇ h u h,k � 2 L 2 ( Q T ) � 1 2 � u 0 � 2 L 2 (Ω) + ( � u 0 � L 2 (Ω) + 2 T � f � L 2 ( Q T ) ) T � f � L 2 ( Q T ) . We could show as well the estimates on time and space translates Theorem 3 Let u h,k be an approximate solution. There exists C > 0 and 0 < ϑ < 1 / 2 , which do not depend on h and k , such that � u h,k ( · + y , · + τ ) − u h,k � L 2 ( Q T ) � C ( √ τ + | y | ϑ ) In view of the Theorem 2, the Fréchet-Kolmogorov Compactness Theorem implies that the family { u h,k } is relatively compact in L 2 ( Q T ) Dubrovnik, 13-16 October 2008 | 16 K.Brenner, University Paris-Sud XI