Mixed and Mixed Hybrid Finite Element Methods: Theory ...

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

F.A. Radu Max-Planck Institute for Mathematics in the Sciences Leipzig, Germany mailto:[email protected]

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.

F.A. Radu

WS 06/07, IMPRS, Leipzig

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.

F.A. Radu


Introduction

Knowledge requirements • Numerical mathematics, → A. Quarteroni et al., Numerical Mathematics, 2000. • Numerical methods for partial differential equations, → P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differential equations, 2003. • Functional Analysis, → B. Rynne and M. Youngson, Linear functional analysis, 2000.

F.A. Radu


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.

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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

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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,

q = −D∇c

Fick’s law of diffusion.

(1) (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. F.A. Radu


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 L2 (Ω)d , so no continuity.

F.A. Radu


Introduction

Why MFEM → Error estimates (Poisson equation, enough regularity for the domain, homogeneuos boundary conditions, regular triangulation, smooth data;): FE : ku − uh k0 + k∇u − ∇uh k0 ≤ Ch MFEM(RT0 ) :

ku − uh k0 + kq − qh kdiv ≤ Ch

MFEM(BDM1) :

ku − uh k0 + kq − qh kdiv ≤ Ch kq − qh k0 ≤ Ch2

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Richy versus Feflow

Richy (left pressure, right flux)

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Richy versus Feflow

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

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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.

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Applications: I. Reactive transport in porous medium

Reactive transport in porous medium

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To describe • Water flow, including the unsaturated zone near the subsurface. • Advective and dispersive transport of multiple contaminants. • Non-equilibrium and equilibrium sorption. • Biodegradation.

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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

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 IRd (d = 1, 2 or 3)

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in J × Ω


The equation results from • mass conservation ∂t Θ(ψ) + ∇ · q = 0 • Darcy’s law q = −K(ψ)∇(ψ + z)

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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.

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• the soil-water retention Θ(ψ), • the unsaturated hydraulic conductivity K(Θ), Gardner

Θexp (ψ) = Θr + (Θs − Θr )eαψ Kexp (ψ) = Ks eαψ

Haverkamp

ΘHav (ψ) = Θr + KHav (ψ) =

van GenuchtenMualem

(Θs −Θr ) 1+(αψ)n

Ks 1+(βψ)p

ΘvG (ψ) = Θr + (Θs − Θr )Φ(ψ) Φ(ψ) =

1 , (1+(αψ)n )m

m=1−

1 n

p 1 KvG (ψ) = Ks Φ(ψ)(1 − (1 − Φ(ψ) m )m )2

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General model with multicomponent organic transport and biodegradation N mobile species, M immobile species ∂t (Θci ) + ρb ∂t si − ∇ · (Di ∇ci − qci ) = −Ri , ∂t si = ki (φ(ci ) − si ) or

si = φ(ci ),

“ ” ci Ri , ∂t ci + kd i ci = 1 − γi ci max

i ∈ 1, ..., N

i ∈ N + 1, ..., N + M.

ci concentration of the species, si concentration of the absorbed species , Di diffusion coefficient, ρb bulk density, Ri degradation rate, φ sorption isotherm, kd i death rate, ci max a maximal realistic concentration, γi ∈ {0, 1}.

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Boundary Conditions ci = gDi on J × ΓDi , −Di ∇ci · n = gN i on J × ΓN i , −Di ∇ci · n + ci q · n = gF i on J × ΓF i , | {z } qi ·n

Remark. ΓDi , ΓN i , ΓF i are species depending.

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Benzene Biodegradation

•Water Flow :

F3 days rain, 4 days dry

??? (0,3)

Γ1 ..........

(2,3)

Ω1

Fvan Genuchten-Mualem Model •Biodegradation : F2 mobile species, 1 biomass

Ω2

Fno sorption FMonod Model

(0,0)

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(2,0)



∂t (ΘcD ) − ∇ · (DD ∇cD − qcD ) = −R , ∂t (ΘcA ) − ∇ · (DA ∇cA − qcA ) = −αA/D R ,

cX

∂t cX + kd cX = Y 1 − γX cX

max

R

with donator/contaminant cD , acceptor cA , biomass cX .

Reactive term:

 R = µmax cX

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cD KMD + cD

  

 cA KMA


c2A + cA + KIA

 . 


Benzene concentration at T = 30, 60, 90, 120, 150, 160 days

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Oxygen concentration at T = 30, 60, 90, 120, 150, 160 days

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Biomass concentration at T = 30, 60, 90, 120, 150, 160 days

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Real case study: Xylene Degradation •Water Flow :

F stationary flow F variable permeability

•Biodegradation : F 2 mobile species, 1 biomass F without sorption F Monod Model

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Xylene degradation: variable permeability

Domain

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Pressure Profile


Flux


Concentration [mg/l] profiles at T = 1 [year] (without additional delivery of contaminant)

Xylene

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Oxygen


Biomass


Concentration [mg/l] profiles at T = 3 [years] (without additional delivery of contaminant)

Xylene

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Oxygen


Biomass


Concentration [mg/l] profiles at T = 5 [years] (without additional delivery of contaminant)

Xylene

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Oxygen


Biomass

Applications: II. Drug release from collagen matrices

Modelling drug release from collagen matrices • Department of Pharmacy, Pharmaceutical Technology and Biopharmacy, University of Munich, Germany

F.A. Radu



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. F.A. Radu



To describe

PHASE I: • Swelling (very short, 20 - 30 min), → free boundary problem, front-tracking method: Bonnerot and Jamet.

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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.

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Enzymatic degradation

The general behaviour of an enzymatically catalyzed degradation process can be summarized by the equations:

k1

E + S → ES k

ES →2 P + E k1 = the constant rate of formation of the enzyme-substrate complex. k2 = the catalysis rate.

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Mathematical model: Enzymatic degradation

∂t CE − ∇ · (DE (CK )∇CE ) + kakt CE

∂t CES

∂t CK ∂t CP − ∇ · (DP ∇CP )

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k1 (CE )α γ = − C + k C K 2 ES , k1 α 1 + M axSorp (CE )

=

k1 (CE )α γ C − k C K 2 ES , k1 α 1 + M axSorp (CE )

= −k1 (CE )α CK , γ = k2 CES .


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.

F.A. Radu


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 CA − ∇ · (DA (CK )∇CA ) = −∂t (CAi ) , where CA , CAi denote the concentrations of free and immobilized drug. F.A. Radu


Drug release

We assume CAi = f (CA , CK ) • the simplest approach: CAi = σCK ( Tzafriri 2000) 2 • CAi = σCK √ • CAi = σ CK

•• 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 0 CAi , 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.

F.A. Radu


Numerical simulations: Algorithm n CK

?

enzyme transport

sorption

polymer degradation

coupled solver (Newton)

• 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. 27-40, 1997.

F.A. Radu


drug release solver

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).

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Numerical results: Phase II (2D)

Concentration [µmol/cm3 ] profiles of collagen and drug at T = 30, 60, 120 [min].

F.A. Radu


References Numerical results: Phase II (2D)

References [1] R. A. A DAMS, Sobolev Spaces, Academic Press, New York, 1975. [2] T. A RBOGAST, M. F. W HEELER AND N. Y. Z HANG, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal. 33, pp. 1669–1687, 1996. [3] J. B ARANGER , J-F. M AITRE AND F. O UDIN, Connection between finite volume and mixed finite element methods, RAIRO Model. Math. Anal. Numer., Vol. 30 No. 4, pp. 445–465, 1996. [4] P. B ASTIAN , K. B IRKEN , K. J OHANSSEN , S. L ANG , N. N EUSS , H. R ENTZ -R EICHERT AND C. W IENERS, UG–a flexible toolbox for solving partial differential equations, Comput. Visualiz. Sci. 1, pp. 27–40, 1997.

F.A. Radu


References Numerical results: Phase II (2D) [5] F. B REZZI AND M. F ORTIN, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, 1991. [6] Z. C HEN, Finite Element Methods and Their Applications, Springer Verlag, 2005. [7] P. K NABNER AND L. A NGERMANN, Numerical methods for elliptic and parabolic partial differential equations, Springer Verlag, 2003. [8] S. M ICHELETTI , R. S ACCO, F. S ALERI, On Some Mixed Finite Element Methods with Numerical Integration, SIAM J. Sci. Comput. 23, No.1, 245–270, 2001. [9] A. Q UARTERONI AND A. VALLI, Numerical approximations of partial differential equations, Springer-Verlag, 1994. [10] A. Q UARTERONI , R. S ACCO AND F. S ALERI, Numerical mathematics, Springer-Verlag, New York, 2000. [11] F. R ADU, I. S. P OP AND P. K NABNER, Order of convergence estimates

F.A. Radu


References Numerical results: Phase II (2D) for an Euler implicit, mixed finite element discretization of Richards’ equation, SIAM J. Numer. Anal. 42, No. 4, pp. 1452–1478, 2004. [12] B. RYNNE AND M. YOUNGSON, Linear functional analysis, Springer-Verlag, 2000. [13] C. W OODWARD 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. 701–724, 2000.

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Software • Richy1D. • UG. • Matlab or octave.

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Homepage http://personal-homepages. mis.mpg.de/fradu/mfem course.html mailto:[email protected]

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Lecture developing

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Lecture developing 0 Preliminary topics.

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1 Theory of MFEM exemplified on the Poisson equation.

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1 Theory of MFEM exemplified on the Poisson equation. 1.1 Conforming variational formulation and equivalence with a minimization problem for the Poisson equation.

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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.

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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.

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1.4 Existence and uniqueness for the continuous mixed problem.

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1.4 Existence and uniqueness for the continuous mixed problem. 1.5 The discrete mixed variational problem. Existence and Uniqueness. The inf-sup condition.

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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.

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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.

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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.

F.A. Radu



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. 1.8.1 More complicated equations (parabolic problems).

F.A. Radu



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. 1.8.1 More complicated equations (parabolic problems). 1.8.2 Error estimates through duality techniques.

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2 The discrete problem.

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2 The discrete problem. 2.1 Mixed FE spaces.

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2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators.

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2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators. 2.2.1 Construction.

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2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators. 2.2.1 Construction. 2.2.2 Technical lemmas.

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2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators. 2.2.1 Construction. 2.2.2 Technical lemmas. 2.2 MHFEM.

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2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators. 2.2.1 Construction. 2.2.2 Technical lemmas. 2.2 MHFEM. 2.3 Implementation.

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2 The discrete problem. 2.1 Mixed FE spaces. 2.2 Interpolation (projection) operators. 2.2.1 Construction. 2.2.2 Technical lemmas. 2.2 MHFEM. 2.3 Implementation. 2.3.1 MFEM and multigrid.

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3 Applications of MFEM and MHFEM.

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3 Applications of MFEM and MHFEM. 3.1 Reactive flow in porous media.

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3 Applications of MFEM and MHFEM. 3.1 Reactive flow in porous media. 3.2 Controlled drug release.

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4 Connection between MFEM and other numerical schemes.

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4 Connection between MFEM and other numerical schemes. 4.1 Cell centered FV.

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4 Connection between MFEM and other numerical schemes. 4.1 Cell centered FV. 4.2 Multi point flux approximation (MPFA) method.

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5 MFEM and adaptivity.

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5 MFEM and adaptivity.

Hopefully we will enjoy it !!!

F.A. Radu


Mixed and Mixed Hybrid Finite Element Methods: Theory ...

Mixed and Mixed Hybrid Finite Element Methods: Theory ...

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