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  • Coupled Surface and Saturated/Unsaturated

    Ground Water Flow in Heterogeneous Media

    Heiko Berninger∗, Ralf Kornhuber, and Oliver Sander

    Université de Genève∗, Freie Universität Berlin and Matheon

    Multiscale Simulation & Analysis in Energy and the Environment,

    Radon Special Semester 2011

  • Outline

    Richards equation with homogeneous equations of state: Berninger, Kh. & Sander 10

    • solver friendly finite element discretization: linear efficiency and robustness

    Heterogeneous equations of state: Thesis of Berninger 07, Berninger, Kh. & Sander 07,09,...

    • nonlinear domain decomposition: linear efficiency and robustness

    Coupled Richards and shallow water equations: Ern et al. 06, Sochala et al. 09, Dawson 08, Berninger et al. 11

    • continuous of mass flow and discontinuous pressure (clogging)

    • mass conserving discretization (discontinuous Galerkin, ...) Dedner et al. 09, ...

    • Steklov–Poincaré formulation and substructuring

    • numerical experiments

    All computations made with Dune Bastian, Gräser, Sander, ...

  • Outline

    Richards equation with homogeneous equations of state: Berninger, Kh. & Sander 10

    • solver friendly finite element discretization: linear efficiency and robustness

    Heterogeneous equations of state: Thesis of Berninger 07, Berninger, Kh. & Sander 07,09,...

    • nonlinear domain decomposition: linear efficiency and robustness

    Coupled Richards and shallow water equations: Ern et al. 06, Sochala et al. 09, Dawson 08, Berninger et al. 11

    • continuous of mass flow and discontinuous pressure (clogging)

    • mass conserving discretization (discontinuous Galerkin, ...) Dedner et al. 09

    • Steklov–Poincaré formulation and substructuring

    • numerical experiments

    All computations made with Dune Bastian, Gräser, Sander, ...

  • Runoff Generation for Lowland Areas

    γE

    γSP

    saturated

    γD

    h

    γSP

    γE

    vadose

    mathematical challenges:

    • saturated/unsaturated ground water flow: non-smooth degenerate pdes l Signorini-type bc (seepage face)

    • coupling subsurface and surface water: heterogeneous domain decomposition

    • uncertain parameters (permeability, ...): stochastic pdes Forster & Kh. 10, Forster 11

  • Saturated/Unsaturated Groundwater Flow: Richards Equation

    ∂t θ(p) + div v(x, p) = 0 , v(x, p) = −K(x) kr(θ(p))∇(p− ̺gz)

    equations of state: (Brooks & Corey, Burdine)

    θ(p) =

    {

    θm + (θM − θm) (

    p pb

    )−ε

    (p ≤ pb)

    θM (p ≥ pb) kr(θ) =

    (

    θ − θm θM − θm

    )3+2ε

    −5 −4 −3 −2 −1 0 1 2 3 4 5 0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    saturation vs. pressure: p 7→ θ(p)

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    relative permeability vs. saturation: θ 7→ kr(θ)

  • Saturated/Unsaturated Groundwater Flow: Richards Equation

    ∂t θ(p) + div v(x, p) = 0 , v(x, p) = −K(x) kr(θ(p))∇(p− ̺gz)

    equations of state: (Brooks & Corey, Burdine)

    θ(p) =

    {

    θm + (θM − θm) (

    p pb

    )−ε

    (p ≤ pb)

    θM (p ≥ pb) kr(θ) =

    (

    θ − θm θM − θm

    )3+2ε

    −5 −4 −3 −2 −1 0 1 2 3 4 5 0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    saturation vs. pressure: p 7→ θ(p)

    pb, ε → 0

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    relative permeability vs. saturation: θ 7→ kr(θ)

  • Homogeneous Equations of State Alt & Luckhaus 83, Otto 97

    Kirchhoff Transformation: κ(p) :=

    ∫ p

    0

    kr(θ(q)) dq =⇒ ∇κ(p) = kr(θ(p))∇p

    −3 −1 0 3 −4/3

    −1

    0

    3

    generalized pressure: u := κ(p)

    −2 −4/3 −1 0 2 0

    0.21

    0.95

    1

    M(u) := θ(κ−1(u))

    separation of ill–conditioning and numerical solution: semilinear variational equation

    u(t) ∈ H10(Ω) :

    M(u)t v dx+

    (

    K∇u−kr(M(u))̺gez )

    ∇v dx = 0 ∀v ∈ H10(Ω)

  • Solver-Friendly Discretization

    lumped implicit/explicit-upwind discretization in time, finite elements Sj ⊂ H 1 0(Ω):

    un+1j ∈ Sj :

    ISj(M(u n+1 j ) v) dx+

    τK∇un+1j ∇v dx = ℓunj (v) ∀v ∈ Sj

    equivalent convex minimization problem

    uj ∈ Sj : J (uj) + φj(uj) ≤ J (v) + φj(v) ∀v ∈ Sj

    quadratic energy J (v) = 12 (τK∇v,∇v) − ℓunj (v)

    convex, l.s.c., proper functional φj(v) = ∑

    Φ(v(p)) hp = ∫

    Ω ISj(Φ(v)) dx

    nonlinear convex function Φ : R → R ∪ {+∞} with ∂Φ = M

  • Algebraic Solution: Monotone Multigrid Kh. 99, 02, Gräser & Kh. 07

    • given iterate uνj

    • fine grid smoothing: − successive 1D minimization of J + φj in direction of nodal basis functions of Sj:

    1 step of nonlinear Gauss–Seidel iteration → smoothed iterate ūνj

    • coarse grid correction:

    − Newton linearization of M(u) at ūνj − constrain corrections to smooth regime of M

    1 step of damped MMG → new iterate uν+1j

    =⇒ (J + φj)(u ν+1 j ) ≤ (J + φj)(u

    ν j )

    0.02 0.04 0.06 0.08 0.1 0.12 0.14 0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    1.4

    1.6

    1.8

    � � �

    � � �

    ūνj (p) u

    M(u)

    Theorem: (for V –cycle) Global convergence and asymptotic multigrid convergence ∀ −pb, ε ≥ 0 (robustness).

  • Algebraic Solution: Monotone Multigrid Kh. 99, 02, Gräser & Kh. 07

    • given iterate uνj

    • fine grid smoothing: − successive 1D minimization of J + φj in direction of nodal basis functions of Sj:

    1 step of nonlinear Gauss–Seidel iteration → smoothed iterate ūνj

    • coarse grid correction:

    − Newton linearization of M(u) at ūνj − constrain corrections to smooth regime of M

    1 step of damped MMG → new iterate uν+1j

    =⇒ (J + φj)(u ν+1 j ) ≤ (J + φj)(u

    ν j )

    0.02 0.04 0.06 0.08 0.1 0.12 0.14 0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    1.4

    1.6

    1.8

    � � �

    � � �

    ūνj (p) u

    M(u)

    Signorini condition

    0

    Theorem: (for V –cycle) Global convergence and asymptotic multigrid convergence ∀ −pb, ε ≥ 0 (robustness).

  • Solver–Friendly Discretization

    • Kirchhoff transformation

    • discretization: discrete minimization problem for uj

    • algebraic solution by monotone multigrid (descent method!)

    • inverse discrete Kirchhoff transformation

  • Solver–Friendly Discretization

    • Kirchhoff transformation: u = κ(p)

    • discretization: discrete minimization problem for uj

    • algebraic solution by monotone multigrid (descent method!)

    • inverse discrete Kirchhoff transformation

  • Solver–Friendly Discretization

    • Kirchhoff transformation: u = κ(p)

    • discretization: discrete minimization problem for uj

    • algebraic solution by monotone multigrid (descent method!)

    • inverse discrete Kirchhoff transformation

  • Solver–Friendly Discretization

    • Kirchhoff transformation: u = κ(p)

    • discretization: discrete minimization problem for uj

    • algebraic solution by monotone multigrid

    • inverse discrete Kirchhoff transformation

  • Solver–Friendly Discretization

    • Kirchhoff transformation: u = κ(p)

    • discretization: discrete minimization problem for uj

    • algebraic solution by monotone multigrid

    • discrete inverse Kirchhoff transformation: pj = Ij(κ −1(uj))

  • Reinterpretation and Convergence Analysis Berninger, Kh. & Sander 10

    reinterpretation in terms of physical variables:

    inexact finite element discretization with special quadrature points

    convergence properties:

    generalized variables: uj → u and M(uj) → M(u) in H 1(Ω)

    physical variables pj → p and θj(pj) = Ij (M(uj)) → θ(p) in L 2(Ω)

  • Reinterpretation and Convergence Analysis Berninger, Kh. & Sander 10

    reinterpretation in terms of physical variables:

    inexact finite element discretization with special quadrature points

    convergence properties:

    generalized variables: uj → u and M(uj) → M(u) in H 1(Ω)

    physical variables pj → p and θj(pj) = Ij (M(uj)) → θ(p) in L 2(Ω)

  • Experimental Order of L2-Convergence

    model problem: time discretized Richards equation without g