Design Principles of the Mimetic Finite Difference Schemes - PowerPoint PPT Presentation

Design Principles of the Mimetic Finite Difference Schemes Konstantin Lipnikov Los Alamos National Laboratory, Theoretical Division Applied Mathematics and Plasma Physics Group October 2015, Georgia Tech, GA Co-authors: L.Beirao da Veiga, F.Brezzi, V.Gyrya, G.Manzini, D.Moulton, V.Simoncini, M.Shashkov, D.Svyatskiy Funding: DOE Office of Science, ASCR Program Acknowledgements: R.Garimella, MSTK, (software.lanl.gov/MeshTools/trac) Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Objective The mimetic finite difference method preserves or mimics critical mathematical and physical properties of systems of PDEs such as conservation laws, exact identities, solution symmetries, secondary equations, maximum principles, etc. These properties are important for multiphysics simulations. The task of building mimetic schemes becomes more difficult on unstructured polygonal and polyhedral meshes. Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Outline 1 Discrete vector and tensor calculus Coordinate invariant definition of primary mimetic operator Duality & derived mimetic operators Properties of mimetic operators 2 Mimetic inner products Consistency condition Stability condition Numerical example 3 Flexibility of mimetic discretization framework Nonlinear parabolic problem M-adaptation Selection of DOFs (meshes with curved faces; Stokes) Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Mesh notation n – n ode, discrete space N h e – e dge, length | e | , tangent τ e , discrete space E h f – f ace, area | f | , normal n f , discrete space F h c – c ell, volume | c | , discrete space C h Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Engineering mesh Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete vector and tensor calculus Coordinate invariant definition of primary mimetic operators 1 Duality & derived mimetic operators Properties of mimetic operators 1 K.L., M.Manzini, M.Shashkov, JCP 2014 Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Coordinate invariant definition of primary operators Primary mimetic operators appear naturally from the Stokes theorem in one, two and three dimensions. � � � e = p n 2 − p n 1 ∂p GRAD h p h d x = p ( x n 2 ) − p ( x n 1 ) | e | ∂ τ e e Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Coordinate invariant definition of primary operators Primary mimetic operators appear naturally from the Stokes theorem in one, two and three dimensions. � � � e = p n 2 − p n 1 ∂p GRAD h p h d x = p ( x n 2 ) − p ( x n 1 ) | e | ∂ τ e e � � � � � f = 1 CURL h u h α f,e | e | u e ( curl u ) · n f d x = u · τ d x | f | f ∂f e ∈ ∂f Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Coordinate invariant definition of primary operators Primary mimetic operators appear naturally from the Stokes theorem in one, two and three dimensions. � � � e = p n 2 − p n 1 ∂p GRAD h p h d x = p ( x n 2 ) − p ( x n 1 ) | e | ∂ τ e e � � � � � f = 1 CURL h u h α f,e | e | u e ( curl u ) · n f d x = u · τ d x | f | f ∂f e ∈ ∂f � � � � � c = 1 u · n d x DIV h u h α c,f | f | u f div u d x = | c | c ∂c f ∈ ∂c where α = ± 1 and degrees of freedom are � � u e = 1 u f = 1 p n = p ( x n ) , u · τ e d x, u · n f d x | e | | f | e f Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (1/3) The integration by part formula is � � ∀ u ∈ H div (Ω) , q ∈ H 1 (div u ) q d x = − u · ∇ q d x 0 (Ω) Ω Ω In other words, ∇ = − div ∗ with respect to L 2 products. Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (1/3) The integration by part formula is � � ∀ u ∈ H div (Ω) , q ∈ H 1 (div u ) q d x = − u · ∇ q d x 0 (Ω) Ω Ω In other words, ∇ = − div ∗ with respect to L 2 products. We define � GRAD h = −DIV ∗ h with respect to inner products � � � � u h , � DIV h u h , q h C h = − GRAD h q h ∀ u h ∈ F h , q h ∈ C h F h The primary and derived mimetic operators (rectangular matrices) are not discretized independently of one another. Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (2/3) An inner product is defined by an SPD matrix M Q : [ u h , v h ] Q h = ( u h ) T M Q v h , ∀ u h , v h ∈ Q h . Using this in the discrete duality formula, we have F ( DIV h ) T M C � GRAD h = − M − 1 Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Duality & derived mimetic operators (3/3) Similarly to the derived gradient operator, we have E ( CURL h ) T M F � CURL h = M − 1 and N ( GRAD h ) T M E DIV h = − M − 1 � Derived mimetic operators are fully characterized by the inner products and primary mimetic operators. Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete Laplacians (1/2) The first discrete Laplacian is ∆ h = DIV h � GRAD h : C h → C h Using the definition of the derived gradient operator: F ( DIV h ) T M C ∆ h = −DIV h M − 1 Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete Laplacians (1/2) The first discrete Laplacian is ∆ h = DIV h � GRAD h : C h → C h Using the definition of the derived gradient operator: F ( DIV h ) T M C ∆ h = −DIV h M − 1 Hence, we have symmetry and definiteness: F ( DIV h ) T M C p h = [∆ h p h , q h ] C h [∆ h q h , p h ] C h = − q T h M C DIV h M − 1 and � � 2 � ( DIV h ) T M C q h � � M − 1 / 2 [∆ h q h , q h ] C h = − ≤ 0 � F Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Discrete Laplacians (2/2) The second discrete Laplacian is ∆ h = � DIV h GRAD h : N h → N h Using the definition of the derived divergence operator: N ( GRAD h ) T M E GRAD h ∆ h = − M − 1 Hence, we have symmetry and definiteness: h ( GRAD h ) T M E GRAD h p h = [∆ h p h , q h ] N h [∆ h q h , p h ] N h = − q T and � � 2 � � � M 1 / 2 [∆ h q h , q h ] N h = − GRAD h q h � ≤ 0 E Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Exact identities By construction, we have the exact identities: DIV h CURL h = 0 and CURL h GRAD h = 0 . Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Exact identities By construction, we have the exact identities: DIV h CURL h = 0 and CURL h GRAD h = 0 . The derived operators satisfy similar identities: � �� � N ( GRAD h ) T M E E ( CURL h ) T M F DIV h � � M − 1 M − 1 CURL h = − N ( CURL h GRAD h ) T M F = 0 − M − 1 = and � �� � E ( CURL h ) T M F F ( DIV h ) T M C CURL h � � M − 1 M − 1 GRAD h = − E ( DIV h CURL h ) T M C = 0 . − M − 1 = Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Helmholtz decomposition theorems Theorem I Let domain Ω and mesh Ω h be simply-connected. Then, for any v h ∈ F h there exists a unique q h ∈ C h and a unique u h ∈ E h with � DIV h u h = 0 such that v h = � GRAD h q h + CURL h u h Theorem II Let domain Ω and mesh Ω h be simply-connected. Then, for any v h ∈ E h there exist a discrete field q h ∈ N h , which is defined up to a constant field, and a unique discrete field u h ∈ F h with DIV h u h = 0 such that v h = GRAD h q h + � CURL h u h Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Related methods Incomplete list of various compatible discretization methods and frameworks includes Cell method Compatible discrete operators Co-volume method Summation by parts Hybrid FV, mixed FV, discrete duality FV Mixed FE, weak Galerkin, VEM, Kuznetsov-Repin Exterior calculus Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Special properties the mimetic framework There is a lot of freedom in construction of primary and derived operators. This is especially improtant for PDEs with non-constant coefficients. Using the weighed L 2 product, � � k − 1 u · ( k ∇ ) q d x, (div u ) q d x = − Ω Ω we construct primary DIV h that approximates div( · ) and derived � GRAD h that approximates k ∇ ( · ) . Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes

Special properties the mimetic framework There is a lot of freedom in construction of primary and derived operators. This is especially improtant for PDEs with non-constant coefficients. Using the weighed L 2 product, � � k − 1 u · ( k ∇ ) q d x, (div u ) q d x = − Ω Ω we construct primary DIV h that approximates div( · ) and derived � GRAD h that approximates k ∇ ( · ) . Using � � (div ( k u )) q d x = − k u · ∇ q d x, Ω Ω we construct primary DIV h that approximates div( k · ) and derived � GRAD h that approximates ∇ ( · ) . Konstantin Lipnikov Design Principles of the Mimetic Finite Difference Schemes