Mimetic Finite Difference methods

Transcription

1 Mimetic Finite Difference methods An introduction Andrea Cangiani Università di Roma La Sapienza Seminario di Modellistica Differenziale Numerica 2 dicembre 2008 Andrea Cangiani (IAC CNR) mimetic finite difference methods 1 / 30

2 Some cronology... RECENTLY ( ) Shashkov-Steinberg, JCP 1995 (support-operator methods) (Hyman)-Shashkov-Steinberg, JCP 1996-(7) (rough diffusion) Hyman-Shashkov, SINUM 1999 (Maxwell) Campbell-Shashkov, JCP 2001 (gas dynamics) MORE RECENTLY ( ) Kuznetsov-Lipnikov-Shashkov, Comp. Geos (polygons) Brezzi-Lipnikov-Shashkov, SINUM 2005 (error analysis) Brezzi-Lipnikov-Simoncini, M 3 AS 2005 (a new family of MFD) Brezzi-Lipnikov-Shashkov, M 3 AS 2006 (curved faces) Andrea Cangiani (IAC CNR) mimetic finite difference methods 2 / 30

3 Some cronology (continued) MORE RECENTLY ( ) continued Beirao Da Veiga, NM 2007 (residual error estimator) C.-Manzini, CMAME 2008 (post-processing) C.-Manzini-Russo, SINUM accepted (convection-reaction-diffusion) IN PROGRESS Convection-dominated diffusion Higher-order MFD Nodal MFD Mimetic curl operator Mimetic discretization of Stokes Andrea Cangiani (IAC CNR) mimetic finite difference methods 3 / 30

4 Features RECENTLY Generalisation of finite differences to hexahedral meshes. Discrete differential operators defined so as to mimic the properties of the underlying continuum operators (e.g. vector calculus identities, conservation laws, solution symmetries) Applied to wide range of problems. MORE RECENTLY Generalisation of (low order) Mixed Finite Elements/finite volumes to general polyhedral meshes. Constructs a family of methods. Discrete differential operators defined so as to mimic the properties of the underlying continuum operators. May gain from extra freedom given by method construction Currently limited to linear diffusion problems Andrea Cangiani (IAC CNR) mimetic finite difference methods 4 / 30

5 Compatible Spatial Discretizations We define compatible spatial discretizations as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. (IMA Hot Topics Workshop Compatible Spatial Discretizations for Partial Differential Equations, May 11-15, 2004) Advantages: Conserve crucial features of physical, geometrical, and mathematical model: Conservation laws Symmetry Positivity and monotonicity Duality properties of differential operators Provide reliability and accuracy Andrea Cangiani (IAC CNR) mimetic finite difference methods 5 / 30

6 Compatible Spatial Discretizations The following can be cast as compatible spatial discretizations: mixed finite element methods, mimetic finite differences, support operator methods, control volume methods, discrete differential forms, Whitney forms, conservative differencing, discrete Hodge operators, discrete Helmholtz decomposition, finite integration techniques, staggered grid and dual grid methods, etc. Support operator method: Most PDEs are written in terms of invariant differential operators (div, grad, and curl) Define discrete analogues of these invariant operators that satisfy exacly the discrete analogs of the identities satisfied by the continuum operators (e.g. Gauss, Stokes, Hodge orthogonal decomposition). Andrea Cangiani (IAC CNR) mimetic finite difference methods 6 / 30

7 Linear diffusion in mixed form Consider the linear diffusion equation bvp div(kgrad p) = b in Ω IR 2,3 p = 0 on Ω with K strongly elliptic full symmetric tensor. Mixed formulation: F = K grad p div F = b (Constitutive Equation) (Conservation Equation) We also introduce the flux operator Gp = Kgrad p. Andrea Cangiani (IAC CNR) mimetic finite difference methods 7 / 30

8 Linear diffusion in mixed form The operators div and grad satisfy the integral identity p divf dv + F gradp dv = 0 which we rewrite: p divf dv Ω Introducing the inner-products: Ω the identity (1) becomes: Ω Ω (p, q) W = Ω p q dv (F, G) V = Ω K 1 F G dv (divf, p) W (F, Gp) V = 0 K 1 F Gp dv = 0 (1) expressing the fact that divergence and flux are adjoint: G = grad Andrea Cangiani (IAC CNR) mimetic finite difference methods 8 / 30

9 Linear diffusion in mixed form Consider the linear diffusion equation bvp div(kgrad p) = b in Ω IR 2,3 p = 0 on Ω with K strongly elliptic full symmetric tensor. Mixed formulation: F = K grad p div F = b (Constitutive Equation) (Conservation Equation) Mixed variational formulation: find (p, F ) W V s.t. (K 1 F, G) (p, divg) = 0 G V (divf, q) = (b, q) q W W = L 2 (Ω), V = H(div; Ω) Andrea Cangiani (IAC CNR) mimetic finite difference methods 9 / 30

10 Mixed Finite Element RT0-P0 Discretisation Ω E Andrea Cangiani (IAC CNR) mimetic finite difference methods 10 / 30

11 Mixed Finite Element RT0-P0 Discretisation Let Ω h be a triangularization of Ω IR 2. For E Ω h, let P 0 (E) = Polynomials of degree zero on E RT 0 (E) = {a + bx : a (P 0 (E)) 2, b P 0 (E)} from which we define the conforming FE spaces: W h = {q h : q h E P 0 (E), E Ω h } V h = {G h : G h E RT 0 (E) and G h n cont. on Ω h } MFE RT0-P0 discretization: find (p h, F h ) W h V h s.t. (K 1 F h, G h ) (p h, divg h ) = 0 G h V h (divf h, q h ) = (b, q h ) q h W h Andrea Cangiani (IAC CNR) mimetic finite difference methods 11 / 30

12 RT0-P0 nodal variables Finite Element Nodal variables Lifting (E, P 0, N 0 ) N 0 (q h ) = q E L E 0 q E P 0 q E := 1 E E q LE 0 q E q E (E, RT 0, N RT ) N RT (G h ) = (GE e )e = G E L E RT G E RT 0 GE e := 1 e e G ne E L E RT G E e n e E = GE e Global FE space Global nodal spaces Global lifting W h Q h := {q = (q E ) E Ωh } L 0 : Q h W h { } G = (GE ) E Ωh : V h X h := GE e + GE e L RT : X h V h. + = 0 e Ω h Where E and E + are the two elements sharing the edge. Andrea Cangiani (IAC CNR) mimetic finite difference methods 12 / 30

13 Discrete divergence Let L RT G = G h. Then, E div G h E = E Gauss divg h = divl E RT G = divg h E = 1 E E G n E = e GE e e E e GE e =: div hg E e E We may now define the discrete divergence operator satisfying divl RT G = div h G. div h : X h Q h G div h G = (div h G E ) E (p h, div G h ) = E E p E div h G E = [p, div h G] Qh Andrea Cangiani (IAC CNR) mimetic finite difference methods 13 / 30

14 Scalar product in X h (K 1 F h, G h ) = Ω = E K 1 L RT F L RT G K 1 L E RT F E L E RT G E E = E [F E, G E ] E =: [F, G] Xh MFE in terms of the spaces Q h and X h : find (p, F) Q h X h : [F, G] Xh [p, div h G] Qh = 0 G X h [div h F, q] Qh = [b, q] Qh q Q h. with b = b I, I =interpolation operator defined for q V by (q I ) E := 1 q E Ω h E E Andrea Cangiani (IAC CNR) mimetic finite difference methods 14 / 30

15 Mimetic finite difference discretization Ω h partition of Ω into polygonal (polyhedral) elements. E E q e E G E Ω Why polyhedral meshes? They naturally arise in the treatment of complex solution domains and heterogeneous materials (e.g. reservoir models) They facilitate adaptive mesh refinement/de-refinement Andrea Cangiani (IAC CNR) mimetic finite difference methods 15 / 30

16 Discrete MFD pressure and flux spaces Define the discrete pressure space Q h and flux space X h. q Q h q = (q E ) E Ωh ( piecewise constant) ) e E G X h G = ( GE e ( constant normal component) E Ω h e E E + FE e + FE e + = 0 We have the interpolation operators: For q W, (q I ) E = 1 E E q E Ω h, ( I) e For G V, G E = 1 e e G ne E E Ω h e E And the discrete divergence operator: div h : X h Q h G div h G = (div h G E ) E div h G E = 1 E e GE e e E Andrea Cangiani (IAC CNR) mimetic finite difference methods 16 / 30

17 Discretisation of the conservation equation Let b = b I. The mimetic discrete conservation equation reads: i.e. Equivalently, 1 E div h F = b e F e E = (b) E E (2) e E [div h F, q] Qh = [b, q] Qh q Q h where the scalar product for the pressure space Q h is given by [p, q] Qh = E Ω h E p E q E This MFD discrete conservation equation (2) coincides with the MFE RT0-P0 and also with many finite volume discretisation of the conservation equation. Andrea Cangiani (IAC CNR) mimetic finite difference methods 17 / 30

18 Finite volume discretisation of the constitutive equation Directly discretize the constitutive equation defining the flux: F = K grad p (with K = k I) B B K L n AB K L A A n AB F k p K p L K L F k ( pk p L n AB + p ) B p A n KL K L γ AB B A γ KL Andrea Cangiani (IAC CNR) mimetic finite difference methods 18 / 30

19 Mimetic discretisation of the constitutive equation Introduce a scalar product for the flux space X h [F, G] Xh = [F, G] E. ([, ] E to be defined!). E Ω h Moreover, define a discrete flux operator G h : Q h X h as the adjoint of div h : [G, G h q] Xh = [q, div h G] Qh, q Q h G X h. This definition naturally establishes a discrete Green formula with respect to the discrete scalar products. We say that the discrete operators mimic the continuous ones. Andrea Cangiani (IAC CNR) mimetic finite difference methods 19 / 30

20 Discretisation of the constitutive equation (cont.) The mimetic discretisation of the constitutive equation reads: F = G h p or, equivalently, [F, G] Xh = [G h p, G] Xh =[p, div h G] Qh G X h The (family of) Mimetic Finite Difference (MFD) schemes: [F, G] Xh [p, div h G] Qh = 0 G X h [div h F, q] Qh = [b, q] Qh q Q h or, equivalently, [ A B t B 0 ] [ F p ] = [ 0 b ] Andrea Cangiani (IAC CNR) mimetic finite difference methods 20 / 30

21 MFE and MFD As we already saw, if Ω h is made of triangles (tetrahedrons), The MFD conservation equation discretisation is equivalent to RT0-P0 If we define the X h -scalar product to be [F, G] Xh = K 1 L RT F L RT G the constitutive equation disretisation is equivalent....and the two methods coincide! Which properties define L RT? Locally, on each E Ω h, 1 L E RT G e n e E = G e E e E, 2 divl E RT G = div hg E 3 L E RT G I = G if G is constant Ω Andrea Cangiani (IAC CNR) mimetic finite difference methods 21 / 30

22 The flux scalar product: P 0 -compatible liftings IDEA: any reasonable lifting can be used to define a scalar product yielding a reasonable MFD formulation. Set X E = X h E. We define P 0 -compatible lifting a linear map L E : X E L 2 (E) such that for all G X E, 1. L E G e n e E = G E e, e E, 2. divl E G = div h G E, and, for all constant vector C, 3. L E C I = C. The following defines a MFD scalar product for the flux space: [F, G] E = K 1 L E F L E G ds. Ω Andrea Cangiani (IAC CNR) mimetic finite difference methods 22 / 30

23 The flux scalar product: P 0 -compatible liftings Assume that K is constant on E. A crucial property. For any linear function q 1 and G X E [(K q 1 ) I, G] E = K 1 L E ( (K q 1 ) I ) L E G ds E 3 = q 1 L E G ds E Green = q 1 divl E G ds + q 1 (L E G n ext ) dl E E 2,1 = q 1 div h G ds + q 1 dl E e E G e E e (local consistency) the scalar product is independent from the lifting if one of the factors is the interpolant of a constant vector field. the scalar product is exact on the interpolant of constant vector fields: [(e i ) I, (e j ) I ] E = (K 1 ) i,j E Andrea Cangiani (IAC CNR) mimetic finite difference methods 23 / 30

24 A general locally consistent scalar product IDEA: Use local consistency to construct a scalar product without a lifting! Let k E be the number of edges (faces) of E. Set: [F, G] E = k E s,t=1 M s,t E F es E G et E Imposing local consistency and symmetry we get a set of positive semi-definite matrices Imposing stability: exists s, s > 0 independent of h s.t. k E s (G e i E )2 E [G, G] E s i=1 k E i=1 (G e i E )2 E G X E. we restrict to a set of acceptable (symmetric & pos. def.) matrices defining a family of reasonable (e.g. exact on constants) scalar products. Andrea Cangiani (IAC CNR) mimetic finite difference methods 24 / 30

25 A family of consistent and stable MFD scalar products [Brezzi-Lipnikov-Simoncini, Math. Models Methods Appl. Sci. (2005)] Imposing: local consistency ME 0 symmetric & semidef. stability M E = ME 0 +CUC t symmetric & pos. def., where C is a given (computable) full-rank k E (k E d) matrix, U is any symmetric & pos. def. (k E d) (k E d) matrix Proposition: Under a reasonable assumption on the minimum eigenvalue of U, the matrix M E is induced by a lifting. In practice (so far) we use: U = u ( E Trace(K 1 )) II ke d, u IR and the proposition requires that u is sufficiently large. Andrea Cangiani (IAC CNR) mimetic finite difference methods 25 / 30

26 MFD error in function of u pressure. -error pressure L2-error linear pressure L2-error flux. -error Quadrilaterals 20x20 BLS Example Andrea Cangiani (IAC CNR) mimetic finite difference methods 26 / 30

27 Back to Mixed finite elements C. & Manzini, CMAME, 2008 Example. If Ω h is made of triangles, M E = M 0 E + u E [ ei e j ] i,j. with M 0 E symmetric and positive semidefinite. x e x B u E = ke ( ) T i=1 xei x B K 1 ( ) x ei x B 12 E RT 0 P0 Example Rectangles central finite differences Andrea Cangiani (IAC CNR) mimetic finite difference methods 27 / 30

28 Assumptions on the partition The number of edges per face and the number of faces per element is unif. bounded. The length l of each edge is such that l Ch (shape regularity). Every element is uniformly strictly starshaped (if Ω IR 3 ) Every face is uniformly strictly starshaped Andrea Cangiani (IAC CNR) mimetic finite difference methods 28 / 30

29 Error estimates [Brezzi-Lipnikov-Shashkov, SIAM J. Num. Anal. (2005)] If Ω is a convex polyhedron (polygon) with Lipschitz continuous boundary, Ω h is a partition satisfying the previous slide assumptions, the scalar product satisfies local consistency and stability. p H 2 (Ω) Then, p I p Qh + F I F Xh C h, p L 0 p L 2 C h where 2 Q h = [, ] Qh and 2 X h = [, ] Xh. If, moreover, The scalar product is induced by (some) lifting. Then, p I p Qh C h 2 (pressure super convergence) Andrea Cangiani (IAC CNR) mimetic finite difference methods 29 / 30