Mixed and Mixed Hybrid Finite Element Methods: Theory, Implementation and Applications

Transcription

1 Mixed and Mixed Hybrid Finite Element Methods: Theory, Implementation and Applications Max-Planck Institute for Mathematics in the Sciences Leipzig, Germany

2 Introduction Objective Offer the skills to be able to set, analyse and implement a mixed or mixed hybrid finite element method for second order partial differential equations.

3 Introduction Lecture organization Two hours weekly. Desirable: Some of you to take on from me some tasks in the frame of the lecture, study and present them in the form of a paper. Exercises: I will always let you some exercises, it would be very good if you will solve them and come and discuss the solution (if necessary) with me.

4 Introduction Knowledge requirements Numerical mathematics, A. Quarteroni et al., Numerical Mathematics, Numerical methods for partial differential equations, P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differential equations, Functional Analysis, B. Rynne and M. Youngson, Linear functional analysis, 2000.

5 Introduction What are Mixed Finite Element Methods? They are FE methods founded on a variational principle expressing an equilibrium (saddle point) condition and not a minimization principle (conforming FE). MFEM approximate both a scalar variable (e.g. pressure) and a vector variable (its gradient, the flux) simultaneously; from here comes also the name mixed. They are nonconforming methods in the sense that the primal variable is not necessary continuous, as in the case of conforming FE.

6 Introduction History It was first introduced by engineers in the 1960 s to solve problems in solid continua: Fraeijs de Veubeke, 1965 Hellan, 1967 Hermann, 1967

7 Introduction Why MFEM Local conservation. Very important when the equation to be discretized corresponds to a conservation law (usually mass). Example: t c + q = f mass conservation, (1) q = D c Fick s law of diffusion. (2) MFEM: The equation (1) holds not only global, but also locally on each simplex. The same property we have by the finite volume (FV), not on each simplex but on each control volumina.

8 Introduction Why MFEM An intrinsic and accurate approximation of the flux. Transport equation (convection-diffusion equation): t (Θc) (D c) + (qc) = R The flux q is much important than the pressure, a good approximation of it is of great interest. MFEM: the normal component of the discrete flux is continuous over edges, because the flux is in H(div; Ω). By FE, the flux is only in L 2 (Ω) d, so no continuity.

9 Introduction Why MFEM Error estimates (Poisson equation, enough regularity for the domain, homogeneuos boundary conditions, regular triangulation, smooth data;): FE : u u h 0 + u u h 0 Ch MFEM(RT 0 ) : u u h 0 + q q h div Ch MFEM(BDM1) : u u h 0 + q q h div Ch q q h 0 Ch 2

10 Richy versus Feflow Richy (left pressure, right flux)

11 Richy versus Feflow Feflow (left pressure, right flux) Simulations performed by Ch. Kohlepp, Univ. Erlangen-Nuernberg.

12 Richy versus Feflow Why MFEM Applicable for equations with jumps in the coefficients and irregular geometries. heterogeneous soil or materials. anisotrop soil or materials. sophisticated domains.

13 Applications: I. Reactive transport in porous medium Reactive transport in porous medium

14 Applications: I. Reactive transport in porous medium To describe Water flow, including the unsaturated zone near the subsurface. Advective and dispersive transport of multiple contaminants. Non-equilibrium and equilibrium sorption. Biodegradation.

15 Applications: I. Reactive transport in porous medium An appropiate model for the water flow in porous media is Richards equation (here in the pressure formulation): t Θ(ψ) K(ψ) (ψ + z) = 0 in J Ω Water content: θ(ψ) [0, 1] Pressure head: ψ Unsaturated hydraulic conductivity: K(ψ) Height against the gravitational direction: z Time interval: J = (0,T) Domain: Ω in IR d (d = 1, 2 or 3)

16 Applications: I. Reactive transport in porous medium The equation results from mass conservation t Θ(ψ) + q = 0 Darcy s law q = K(ψ) (ψ + z)

17 Applications: I. Reactive transport in porous medium The equation results from mass conservation t Θ(ψ) + q = 0 Darcy s law q = K(ψ) (ψ + z) Nonlinearities Θ, K: strictly monotone increasing for ψ 0, constant for ψ 0 (saturated region) = elliptic - parabolic equation.

18 Applications: I. Reactive transport in porous medium the soil-water retention Θ(ψ), the unsaturated hydraulic conductivity K(Θ), Gardner Θ exp (ψ) = Θ r + (Θ s Θ r )e αψ K exp (ψ) = K s e αψ Haverkamp Θ Hav (ψ) = Θ r + (Θ s Θ r ) 1+(αψ) n K Hav (ψ) = K s 1+(βψ) p van Genuchten- Θ vg (ψ) = Θ r + (Θ s Θ r )Φ(ψ) Mualem Φ(ψ) = 1 (1+(αψ) n ) m, m = 1 1 n K vg (ψ) = K s p Φ(ψ)(1 (1 Φ(ψ) 1 m ) m ) 2

19 Applications: I. Reactive transport in porous medium General model with multicomponent organic transport and biodegradation N mobile species, M immobile species t (Θc i ) + ρ b t s i (D i c i qc i ) = R i, t s i = k i (φ(c i ) s i ) or s i = φ(c i ), i 1,..., N t c i + k d ic i = c 1 γ i i c i R i, i N + 1,..., N + M. max c i concentration of the species, s i concentration of the absorbed species, D i diffusion coefficient, ρ b bulk density, R i degradation rate, φ sorption isotherm, k d i death rate, c i max a maximal realistic concentration, γ i {0, 1}.

20 Applications: I. Reactive transport in porous medium Boundary Conditions c i = g Di on J Γ Di, D i c i n = g Ni on J Γ Ni, D i c i n + c i q n }{{} q i n = g F i on J Γ F i, Remark. Γ Di, Γ Ni, Γ F i are species depending.

21 Applications: I. Reactive transport in porous medium Benzene Biodegradation Γ 1... (0,3) (2,3) Ω 1 Ω 2 Water Flow : Biodegradation : 3 days rain, 4 days dry van Genuchten-Mualem Model 2 mobile species, 1 biomass no sorption Monod Model (0,0) (2,0)

22 Applications: I. Reactive transport in porous medium t (Θc D ) (D D c D qc D ) = R, t (Θc A ) (D A c A qc A ) = α A/D R, t c X + k d c X = Y ( 1 γ X c X c X max with donator/contaminant c D, acceptor c A, biomass c X. ) R Reactive term: R = µ max c X ( ) c D K MD + c D c A K MA + c A + c2 A K IA.

23 Applications: I. Reactive transport in porous medium Benzene concentration at T = 30, 60, 90, 120, 150, 160 days

24 Applications: I. Reactive transport in porous medium Oxygen concentration at T = 30, 60, 90, 120, 150, 160 days

25 Applications: I. Reactive transport in porous medium Biomass concentration at T = 30, 60, 90, 120, 150, 160 days

26 Applications: I. Reactive transport in porous medium Real case study: Xylene Degradation Water Flow : stationary flow variable permeability Biodegradation : 2 mobile species, 1 biomass without sorption Monod Model

27 Applications: I. Reactive transport in porous medium Xylene degradation: variable permeability Domain Pressure Profile Flux

28 Applications: I. Reactive transport in porous medium Concentration [mg/l] profiles at T = 1 [year] (without additional delivery of contaminant) Xylene Oxygen Biomass

29 Applications: I. Reactive transport in porous medium Concentration [mg/l] profiles at T = 3 [years] (without additional delivery of contaminant) Xylene Oxygen Biomass

30 Applications: I. Reactive transport in porous medium Concentration [mg/l] profiles at T = 5 [years] (without additional delivery of contaminant) Xylene Oxygen Biomass

31 Applications: II. Drug release from collagen matrices Modelling drug release from collagen matrices Department of Pharmacy, Pharmaceutical Technology and Biopharmacy, University of Munich, Germany

32 Applications: II. Drug release from collagen matrices Motivation Matrix systems of insoluble collagen are a promising and advantageous drug delivery system for prolonged protein release over several days. Applications Collagen implants have been evaluated for tumor treatment, bone, and nerve regeneration as well as therapy of infections. Goal Optimizing the controlled release from degradable collagen matrices.

33 Applications: II. Drug release from collagen matrices To describe PHASE I: Swelling (very short, min), free boundary problem, front-tracking method: Bonnerot and Jamet.

34 Applications: II. Drug release from collagen matrices PHASE II: diffusion of the enzyme in the matrix, adsorption of the enzyme from the fluid to the collagen fibers, enzymatic degradation of the polymer, enzyme activity (death), drug release.

35 Enzymatic degradation The general behaviour of an enzymatically catalyzed degradation process can be summarized by the equations: E + S k 1 ES k 2 ES P + E k 1 = the constant rate of formation of the enzyme-substrate complex. k 2 = the catalysis rate.

36 Mathematical model: Enzymatic degradation t C E (D E (C K ) C E ) + k akt C E = 1 + k 1 (C E ) α k 1 MaxSorp (C E) C α K + k 2 C γ ES, t C ES = 1 + k 1 (C E ) α k 1 MaxSorp (C E) C α K k 2 C γ ES, t C K = k 1 (C E ) α C K, t C P (D P C P ) = k 2 C γ ES.

37 Drug release The model for the enzymatic degradation is validated by comparison of the experimentally obtained data with the numerically simulated data for the collagen.

38 Drug release The model for the enzymatic degradation is validated by comparison of the experimentally obtained data with the numerically simulated data for the collagen. Next step: Drug Release The release of the active agent is governed by a diffusion equation with a source term due to liberation of the immobilized active agent by matrix degradation: t C A (D A (C K ) C A ) = t (C Ai ), where C A, C Ai denote the concentrations of free and immobilized drug.

39 Drug release We assume C Ai = f(c A, C K ) the simplest approach: C Ai = σc K ( Tzafriri 2000) C Ai C Ai = σc 2 K = σ C K σ can be obtained experimentally (if one considers release from a collagen matrices, without enzyme (i.e. no degradation), the concentration of the collagen remaining in the matrix gives us CAi 0, and therefore also σ.) the form of the function f is determinated by fitting a set of release data and then validated for an other one.

40 Numerical simulations: Algorithm C n K enzyme polymer sorption transport degradation coupled solver (Newton) drug release solver the algorithm was implemented in ug. P. Bastian et al., UG-a flexible toolbox for solving partial differential equation, Comput. Visualization in Science 1, pp , 1997.

41 Numerical results: Phase II (2D) Enzymatic degradation of collagen (left) and drug release from an insoluble collagen matrix: comparison of numerical and experimental results (points).

42 Numerical results: Phase II (2D) Concentration [µmol/cm 3 ] profiles of collagen and drug at T = 30, 60, 120 [min].

43 Numerical results: Phase II (2D) References References [1] R. A. ADAMS, Sobolev Spaces, Academic Press, New York, [2] T. ARBOGAST, M. F. WHEELER AND N. Y. ZHANG, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal. 33, pp , [3] J. BARANGER, J-F. MAITRE AND F. OUDIN, Connection between finite volume and mixed finite element methods, RAIRO Model. Math. Anal. Numer., Vol. 30 No. 4, pp , [4] P. BASTIAN, K. BIRKEN, K. JOHANSSEN, S. LANG, N. NEUSS, H. RENTZ-REICHERT AND C. WIENERS, UG a flexible toolbox for solving partial differential equations, Comput. Visualiz. Sci. 1, pp , 1997.

44 Numerical results: Phase II (2D) References [5] F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, [6] Z. CHEN, Finite Element Methods and Their Applications, Springer Verlag, [7] P. KNABNER AND L. ANGERMANN, Numerical methods for elliptic and parabolic partial differential equations, Springer Verlag, [8] S. MICHELETTI, R. SACCO, F. SALERI, On Some Mixed Finite Element Methods with Numerical Integration, SIAM J. Sci. Comput. 23, No.1, , [9] A. QUARTERONI AND A. VALLI, Numerical approximations of partial differential equations, Springer-Verlag, [10] A. QUARTERONI, R. SACCO AND F. SALERI, Numerical mathematics, Springer-Verlag, New York, [11] F. RADU, I. S. POP AND P. KNABNER, Order of convergence estimates

45 Numerical results: Phase II (2D) References for an Euler implicit, mixed finite element discretization of Richards equation, SIAM J. Numer. Anal. 42, No. 4, pp , [12] B. RYNNE AND M. YOUNGSON, Linear functional analysis, Springer-Verlag, [13] C. WOODWARD AND C. DAWSON, Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media, SIAM J. Numer. Anal. 37, pp , 2000.

46 Numerical results: Phase II (2D) References Software Richy1D. UG. Matlab or octave.

47 Numerical results: Phase II (2D) References Homepage mis.mpg.de/fradu/mfem course.html

48 Numerical results: Phase II (2D) References Lecture developing

49 Numerical results: Phase II (2D) References Lecture developing 0 Preliminary topics.

50 Numerical results: Phase II (2D) References Lecture developing 0 Preliminary topics. 1 Theory of MFEM exemplified on the Poisson equation.

51 Numerical results: Phase II (2D) References Lecture developing 0 Preliminary topics. 1 Theory of MFEM exemplified on the Poisson equation. 1.1 Conforming variational formulation and equivalence with a minimization problem for the Poisson equation.

52 Numerical results: Phase II (2D) References Lecture developing 0 Preliminary topics. 1 Theory of MFEM exemplified on the Poisson equation. 1.1 Conforming variational formulation and equivalence with a minimization problem for the Poisson equation. 1.2 Mixed variational formulation and equivalence with the conforming method.

53 Numerical results: Phase II (2D) References Lecture developing 0 Preliminary topics. 1 Theory of MFEM exemplified on the Poisson equation. 1.1 Conforming variational formulation and equivalence with a minimization problem for the Poisson equation. 1.2 Mixed variational formulation and equivalence with the conforming method. 1.3 Abstract formulation of the continuous mixed problem. Equivalence with a saddle point problem.

54 Numerical results: Phase II (2D) References problem. 1.4 Existence and uniqueness for the continuous mixed

55 Numerical results: Phase II (2D) References 1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition.

56 Numerical results: Phase II (2D) References 1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition. 1.6 Error estimates.

57 Numerical results: Phase II (2D) References 1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition. 1.6 Error estimates. 1.7 Criteria for checking the inf-sup.

58 Numerical results: Phase II (2D) References 1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition. 1.6 Error estimates. 1.7 Criteria for checking the inf-sup. 1.8 Extensions of the theory.

59 Numerical results: Phase II (2D) References 1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition. 1.6 Error estimates. 1.7 Criteria for checking the inf-sup. 1.8 Extensions of the theory More complicated equations (parabolic problems).

60 Numerical results: Phase II (2D) References 1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition. 1.6 Error estimates. 1.7 Criteria for checking the inf-sup. 1.8 Extensions of the theory More complicated equations (parabolic problems) Error estimates through duality techniques.

61 Numerical results: Phase II (2D) References 2 The discrete problem.

62 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces.

63 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators.

64 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction.

65 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas.

66 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas. 2.2 MHFEM.

67 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas. 2.2 MHFEM. 2.3 Implementation.

68 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas. 2.2 MHFEM. 2.3 Implementation MFEM and multigrid.

69 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas. 2.2 MHFEM. 2.3 Implementation MFEM and multigrid. 3 Applications of MFEM and MHFEM.

70 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas. 2.2 MHFEM. 2.3 Implementation MFEM and multigrid. 3 Applications of MFEM and MHFEM. 3.1 Reactive flow in porous media.

71 Numerical results: Phase II (2D) References 2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators Construction Technical lemmas. 2.2 MHFEM. 2.3 Implementation MFEM and multigrid. 3 Applications of MFEM and MHFEM. 3.1 Reactive flow in porous media. 3.2 Controlled drug release.

72 Numerical results: Phase II (2D) References

73 Numerical results: Phase II (2D) References 4 Connection between MFEM and other numerical schemes.

74 Numerical results: Phase II (2D) References 4 Connection between MFEM and other numerical schemes. 4.1 Cell centered FV.

75 Numerical results: Phase II (2D) References 4 Connection between MFEM and other numerical schemes. 4.1 Cell centered FV. 4.2 Multi point flux approximation (MPFA) method.

76 Numerical results: Phase II (2D) References 4 Connection between MFEM and other numerical schemes. 4.1 Cell centered FV. 4.2 Multi point flux approximation (MPFA) method. 5 MFEM and adaptivity.

77 Numerical results: Phase II (2D) References 4 Connection between MFEM and other numerical schemes. 4.1 Cell centered FV. 4.2 Multi point flux approximation (MPFA) method. 5 MFEM and adaptivity. Hopefully we will enjoy it!!!