Parallel Implementation of BDDC for Mixed-Hybrid Formulation of - PowerPoint PPT Presentation

Parallel Implementation of BDDC for Mixed-Hybrid Formulation of Flow in Porous Media Jakub ˇ ıstek 1 S´ joint work with rezina 2 and Bedˇ ık 3 Jan Bˇ rich Soused´ 1 Institute of Mathematics of the AS CR ∩ Neˇ cas Center for Mathematical Modelling 2 Technical University of Liberec 3 University of Maryland, Baltimore County T H E M A A T I M C S f o E T U s e T c c I n T e l i b S i c u N S p f o e I R y m h c e e d z a C c A International Conference on Domain Decomposition Methods XXIII Jeju Island, Korea, July 7th, 2015

A T H E M A I T Motivation M C S o f E T U Academy of Sciences T I Czech Republic T S N I Geoengineering simulations numerous examples of flow in porous media — oil and gas reservoirs, pollutant transport, nuclear waste deposits, . . . in the Czech Republic, plans to build the long-term nuclear waste deposit by 2065 – currently seven candidate sites massive granite rock with cracks Source: www.surao.cz Jakub ˇ S´ ıstek BDDC for flows in porous media 2 / 31

A T H E M A T I Motivation M C S o f E T U Academy of Sciences T I Czech Republic T S N I Subsurface flow simulations 20+ years of development of simulation tools at TUL mixed-hybrid finite element method — combined meshes of 3D, 2D and 1D elements need for robust scalable parallel solvers to handle finer models Jakub ˇ S´ ıstek BDDC for flows in porous media 3 / 31

A H T E M A I T Governing equations M S C f o E T U Academy of Sciences I T Czech Republic T S N I Darcy law k − 1 u + ∇ p = −∇ z in Ω ∇ · u = f in Ω p = p N on ∂ Ω N u · n = 0 on ∂ Ω E Ω ⊂ R 3 , ∂ Ω = ∂ Ω N ∪ ∂ Ω E ∂ Ω N , ∂ Ω E . . . natural (Dirichlet) and essential (Neumann) b. c. u . . . velocity of the fluid p . . . pressure head k . . . tensor of the hydraulic conductivity (sym. pos. def.) z . . . third spatial coordinate p h = p + z . . . piezometric head for which u = − k ∇ p h Jakub ˇ S´ ıstek BDDC for flows in porous media 4 / 31

A H T E M A I T Mixed finite element method M S C f o E T U Academy of Sciences I T Czech Republic T S N I Raviart-Thomas ( RT 0 ) finite elements � � v ∈ L 2 (Ω); ∇ · v ∈ L 2 (Ω) and v · n = 0 on ∂ Ω E V ⊂ H (Ω; div) = Q ⊂ L 2 (Ω) Mixed formulation Find a pair { u , p } ∈ V × Q that satisfies � � � � Ω k − 1 u · v dx − Ω p ∇ · v dx = − ∂ Ω N p N v · n ds − Ω v z dx , ∀ v ∈ V � � − Ω q ∇ · u dx = − Ω fq dx , ∀ q ∈ Q Jakub ˇ S´ ıstek BDDC for flows in porous media 5 / 31

A H T E M A I T Mixed finite element method M S C f o E T U Academy of Sciences I T Czech Republic T S N I Raviart-Thomas ( RT 0 ) finite elements � � v ∈ L 2 (Ω); ∇ · v ∈ L 2 (Ω) and v · n = 0 on ∂ Ω E V ⊂ H (Ω; div) = Q ⊂ L 2 (Ω) Mixed formulation Find a pair { u , p } ∈ V × Q that satisfies � � � � Ω k − 1 u · v dx − Ω p ∇ · v dx = − ∂ Ω N p N v · n ds − Ω v z dx , ∀ v ∈ V � � − Ω q ∇ · u dx = − Ω fq dx , ∀ q ∈ Q Jakub ˇ S´ ıstek BDDC for flows in porous media 5 / 31

A T H E M A I T Mixed–hybrid finite element method M C S o f E T U Academy of Sciences T I Czech Republic T S N I Space of Lagrange multipliers � � V i = v ∈ H ( T i ; div) : v ∈ RT 0 ( T i ) V − 1 = V 1 × · · · × V N E � � λ ∈ L 2 ( F ) : λ = v · n | F , v ∈ V Λ = F . . . set of all faces of the elements in triangulation T Mixed–hybrid formulation Find a triple { u , p , λ } ∈ V − 1 × Q × Λ that satisfies �� � � � � N E T i k − 1 u · v dx − T i p ∇ · v dx + ∂ T i \ ∂ Ω λ ( v · n ) | ∂ T i ds = i =1 i � ∂ Ω N p N v · n ds − � N E � − ∀ v ∈ V T i v z dx , i =1 �� � � − � N E T i q ∇ · u dx = − Ω fq dx , ∀ q ∈ Q �� � i =1 � N E ∂ T i \ ∂ Ω µ ( u · n ) | ∂ T i ds = 0 , ∀ µ ∈ Λ i =1 Jakub ˇ S´ ıstek BDDC for flows in porous media 6 / 31

A T H E M A I T Mixed–hybrid finite element method M C S o f E T U Academy of Sciences T I Czech Republic T S N I Space of Lagrange multipliers � � V i = v ∈ H ( T i ; div) : v ∈ RT 0 ( T i ) V − 1 = V 1 × · · · × V N E � � λ ∈ L 2 ( F ) : λ = v · n | F , v ∈ V Λ = F . . . set of all faces of the elements in triangulation T Mixed–hybrid formulation Find a triple { u , p , λ } ∈ V − 1 × Q × Λ that satisfies �� � � � � N E T i k − 1 u · v dx − T i p ∇ · v dx + ∂ T i \ ∂ Ω λ ( v · n ) | ∂ T i ds = i =1 i � ∂ Ω N p N v · n ds − � N E � − ∀ v ∈ V T i v z dx , i =1 �� � � − � N E T i q ∇ · u dx = − Ω fq dx , ∀ q ∈ Q �� � i =1 � N E ∂ T i \ ∂ Ω µ ( u · n ) | ∂ T i ds = 0 , ∀ µ ∈ Λ i =1 Jakub ˇ S´ ıstek BDDC for flows in porous media 6 / 31

A H T E M A I T System of linear algebraic equations M S C f o E T U Academy of Sciences I T Czech Republic T S N I Saddle-point system       B T B T A u g F     =   B 0 0 p f (1) B F 0 0 λ 0 A . . . symmetric positive definite (s.p.d.), block-diagonal matrix with respect to elements � � B B = . . . full row rank if ∂ Ω N � = ∅ B F analysis e.g. in [Brezzi, Fortin (1991)], [Maryˇ ska, Rozloˇ zn´ ık, T˚ uma (2000)], [Tu (2007)], . . . problem (1) has a unique solution Jakub ˇ S´ ıstek BDDC for flows in porous media 7 / 31

A T H E M A T I Modelling of cracks M C S o f T E U Academy of Sciences T I Czech Republic T S N I Combined meshes T 123 = T 1 ∪ T 2 ∪ T 3 T i d − 1 ⊂ F d d = 2 , 3 . . . spatial dimension System with fluxes u d k − 1 + ∇ p d = −∇ z d δ d u d . . . flux — volume per second per unit δ d . . . conversion to velocity in dimension d ( δ 3 = 1, δ 2 is thickness of a fracture, δ 1 cross-section of a channel) Jakub ˇ S´ ıstek BDDC for flows in porous media 8 / 31

A T H E M A T I Modelling of cracks M C S o f T E U Academy of Sciences T I Czech Republic T S N I Combined meshes T 123 = T 1 ∪ T 2 ∪ T 3 T i d − 1 ⊂ F d d = 2 , 3 . . . spatial dimension System with fluxes u d k − 1 + ∇ p d = −∇ z d δ d u d . . . flux — volume per second per unit δ d . . . conversion to velocity in dimension d ( δ 3 = 1, δ 2 is thickness of a fracture, δ 1 cross-section of a channel) Jakub ˇ S´ ıstek BDDC for flows in porous media 8 / 31

A T H E M A T I Coupling of mesh dimensions M C S o f T E U Academy of Sciences I T Czech Republic S T N I Introduce Robin (a.k.a. Newton) boundary conditions 3D–2D 3 · n + + u − f 2 = δ 2 ˜ f 2 + u + 3 · n − 3 · n + = σ + u + 3 ( p + 3 − p 2 ) 3 · n − = σ − u − 3 ( p − 3 − p 2 ) σ + / − > 0 . . . transition coefficients on sides of a 2D element 3 2D–1D � f 1 = δ 1 ˜ u k 2 · n k f 1 + k 2 · n k = σ k u k 2 ( p k 2 − p 1 ) σ k 2 > 0 . . . transition coefficient from k -th 2D element to 1D channel Jakub ˇ S´ ıstek BDDC for flows in porous media 9 / 31

A T H E M A T I Coupling of mesh dimensions M C S o f T E U Academy of Sciences I T Czech Republic S T N I Introduce Robin (a.k.a. Newton) boundary conditions 3D–2D 3 · n + + u − f 2 = δ 2 ˜ f 2 + u + 3 · n − 3 · n + = σ + u + 3 ( p + 3 − p 2 ) 3 · n − = σ − u − 3 ( p − 3 − p 2 ) σ + / − > 0 . . . transition coefficients on sides of a 2D element 3 2D–1D � f 1 = δ 1 ˜ u k 2 · n k f 1 + k 2 · n k = σ k u k 2 ( p k 2 − p 1 ) σ k 2 > 0 . . . transition coefficient from k -th 2D element to 1D channel Jakub ˇ S´ ıstek BDDC for flows in porous media 9 / 31

A T H E M A T I System of linear algebraic equations M C S o f E T U Academy of Sciences T I Czech Republic T S N I Saddle-point system with couplings       B T B T A g u F  − C T    =   B − C p f (2) F − � − C F λ 0 B F C A . . . symmetric positive definite (s.p.d.), block-diagonal matrix with respect to elements � C � C T F C = . . . symmetric positive semi-definite � C F C � B � B = . . . generally no longer full row rank B F Jakub ˇ S´ ıstek BDDC for flows in porous media 10 / 31

A T H E M A T I System of linear algebraic equations M C S o f E T U Academy of Sciences T I Czech Republic T S N I Theorem (Solvability of the saddle-point system) Let natural boundary conditions be prescribed at a certain part of the boundary, i.e. ∂ Ω N , d � = ∅ for at least one d ∈ { 1 , 2 , 3 } . Then the discrete mixed-hybrid problem (2) has a unique solution. details in [ˇ S´ ıstek, Bˇ rezina, Soused´ ık (2015)] Jakub ˇ S´ ıstek BDDC for flows in porous media 11 / 31