Finite element methods for elliptic PDEs on surfaces Finite element methods for elliptic PDEs on...

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Transcript of Finite element methods for elliptic PDEs on surfaces Finite element methods for elliptic PDEs on...

  • Finite element methods for elliptic PDEs on surfaces

    Klaus Deckelnick, Otto–von–Guericke–Universität Magdeburg

    3rd Workshop Analysis, Geometry and Probability

    Universität Ulm

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Motivation

    Two-phase flow with insoluble surfactant

    ρut + ρ(u · ∇)u −∇ · T (u, p) = ρf

    ∇ · u = 0

    } in Ω±(t)

    [ u ]

    = 0[ T (u, p)ν

    ] = σ(c)Hν −∇Γ(σ(c))

    v · ν = u · ν

    ∂•t c −∇Γ · (D∇Γc) + c∇Γ · u = 0

     on Γ(t)

    James & Lowengrub (2004), Ganesan & Tobiska (2009), Barrett, Garcke & Nürnberg (2015)

    Klaus Deckelnick FEM for elliptic surface PDEs

  • A model problem

    given smooth, compact hypersurface Γ ⊂ Rn+1, ∂Γ = ∅, f : Γ→ R;

    find u : Γ→ R such that

    −∆Γu + u = f on Γ. (1)

    Aim Development and analysis of numerical methods for (1)

    Difficulties Simultaneous approximation of the PDE and the geometry

    G. Dziuk, C.M. Elliott: Finite element methods for surface PDEs, Acta Numerica 22, 289-396 (2013)

    Klaus Deckelnick FEM for elliptic surface PDEs

  • A model problem

    given smooth, compact hypersurface Γ ⊂ Rn+1, ∂Γ = ∅, f : Γ→ R;

    find u : Γ→ R such that

    −∆Γu + u = f on Γ. (1)

    Aim Development and analysis of numerical methods for (1)

    Difficulties Simultaneous approximation of the PDE and the geometry

    G. Dziuk, C.M. Elliott: Finite element methods for surface PDEs, Acta Numerica 22, 289-396 (2013)

    Klaus Deckelnick FEM for elliptic surface PDEs

  • A model problem

    given smooth, compact hypersurface Γ ⊂ Rn+1, ∂Γ = ∅, f : Γ→ R;

    find u : Γ→ R such that

    −∆Γu + u = f on Γ. (1)

    Aim Development and analysis of numerical methods for (1)

    Difficulties Simultaneous approximation of the PDE and the geometry

    G. Dziuk, C.M. Elliott: Finite element methods for surface PDEs, Acta Numerica 22, 289-396 (2013)

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Basics on hypersurfaces

    Local description of Γ

    U∩Γ =

    { F (Ω), F : Ω→ Rn+1, rankDF (ω) = n,

    {x ∈ U |φ(x) = 0}, φ : U → R, ∇φ(x) 6= 0.

    Tangent space

    TxΓ =

     span{ ∂F ∂ω1

    (ω), . . . , ∂F∂ωn (ω)}, x = F (ω);( span{∇φ(x)}

    )⊥ , φ(x) = 0.

     Unit normal ν ∈ C 0(Γ,Rn+1), ν(x) ⊥ TxΓ, |ν(x)| = 1.

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Basics on hypersurfaces

    Local description of Γ

    U∩Γ =

    { F (Ω), F : Ω→ Rn+1, rankDF (ω) = n,

    {x ∈ U |φ(x) = 0}, φ : U → R, ∇φ(x) 6= 0.

    Tangent space

    TxΓ =

     span{ ∂F ∂ω1

    (ω), . . . , ∂F∂ωn (ω)}, x = F (ω);( span{∇φ(x)}

    )⊥ , φ(x) = 0.

    

    Unit normal ν ∈ C 0(Γ,Rn+1), ν(x) ⊥ TxΓ, |ν(x)| = 1.

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Basics on hypersurfaces

    Local description of Γ

    U∩Γ =

    { F (Ω), F : Ω→ Rn+1, rankDF (ω) = n,

    {x ∈ U |φ(x) = 0}, φ : U → R, ∇φ(x) 6= 0.

    Tangent space

    TxΓ =

     span{ ∂F ∂ω1

    (ω), . . . , ∂F∂ωn (ω)}, x = F (ω);( span{∇φ(x)}

    )⊥ , φ(x) = 0.

     Unit normal ν ∈ C 0(Γ,Rn+1), ν(x) ⊥ TxΓ, |ν(x)| = 1.

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Differentiation on hypersurfaces

    Definition A function f : Γ→ R is called differentiable on Γ if f ◦ F is differentiable for every local parametrisation F of Γ.

    Tangential gradient

    ∇Γf (x) =

    { ∑n i ,j=1 g

    ij(ω)∂j ( f ◦ F

    ) (ω)∂iF (ω), x = F (ω)

    (In+1 − ν(x)⊗ ν(x))∇f e(x), f e = f on Γ.

    }

    Laplace–Beltrami operator: ∆Γf = divΓ∇Γf

    Mean curvature: H = −divΓν

    Example: Γ = ∂BR(0) ⊂ Rn+1, ν(x) = xR ,H = − n R , x ∈ Γ

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Differentiation on hypersurfaces

    Definition A function f : Γ→ R is called differentiable on Γ if f ◦ F is differentiable for every local parametrisation F of Γ.

    Tangential gradient

    ∇Γf (x) =

    { ∑n i ,j=1 g

    ij(ω)∂j ( f ◦ F

    ) (ω)∂iF (ω), x = F (ω)

    (In+1 − ν(x)⊗ ν(x))∇f e(x), f e = f on Γ.

    }

    Laplace–Beltrami operator: ∆Γf = divΓ∇Γf

    Mean curvature: H = −divΓν

    Example: Γ = ∂BR(0) ⊂ Rn+1, ν(x) = xR ,H = − n R , x ∈ Γ

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Differentiation on hypersurfaces

    Definition A function f : Γ→ R is called differentiable on Γ if f ◦ F is differentiable for every local parametrisation F of Γ.

    Tangential gradient

    ∇Γf (x) =

    { ∑n i ,j=1 g

    ij(ω)∂j ( f ◦ F

    ) (ω)∂iF (ω), x = F (ω)

    (In+1 − ν(x)⊗ ν(x))∇f e(x), f e = f on Γ.

    }

    Laplace–Beltrami operator: ∆Γf = divΓ∇Γf

    Mean curvature: H = −divΓν

    Example: Γ = ∂BR(0) ⊂ Rn+1, ν(x) = xR ,H = − n R , x ∈ Γ

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Integration by parts ∫ Γ ∇Γf dσ =

    ∫ Γ f H ν dσ.

    Function spaces

    C 1(Γ) := {f : Γ→ R | f is continuously differentiable on Γ};

    H1(Γ) := Completion of C 1(Γ) under the norm

    ‖f ‖H1(Γ) = (∫

    Γ |f |2dσ +

    ∫ Γ |∇Γf |2dσ

    )1/2 .

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Integration by parts ∫ Γ ∇Γf dσ =

    ∫ Γ f H ν dσ.

    Function spaces

    C 1(Γ) := {f : Γ→ R | f is continuously differentiable on Γ};

    H1(Γ) := Completion of C 1(Γ) under the norm

    ‖f ‖H1(Γ) = (∫

    Γ |f |2dσ +

    ∫ Γ |∇Γf |2dσ

    )1/2 .

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Weak solutions Suppose that u : Γ→ R satisfies −∆Γu + u = f on Γ.

    Multiply by v and integrate over Γ:

    − ∫

    Γ ∆Γu v dσ = −

    ∫ Γ ∇Γ ·

    ( v∇Γu

    ) dσ +

    ∫ Γ ∇Γu · ∇Γvdσ

    = − ∫

    Γ Hv ∇Γu · ν︸ ︷︷ ︸

    =0

    dσ +

    ∫ Γ ∇Γu · ∇Γvdσ.

    Definition A function u ∈ H1(Γ) is called a weak solution of

    −∆Γu + u = f on Γ if∫ Γ ∇Γu · ∇Γv dσ +

    ∫ Γ u v dσ︸ ︷︷ ︸

    =a(u,v)

    =

    ∫ Γ f v dσ︸ ︷︷ ︸

    =l(v)

    ∀v ∈ H1(Γ).

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Weak solutions Suppose that u : Γ→ R satisfies −∆Γu + u = f on Γ.

    Multiply by v and integrate over Γ:

    − ∫

    Γ ∆Γu v dσ = −

    ∫ Γ ∇Γ ·

    ( v∇Γu

    ) dσ +

    ∫ Γ ∇Γu · ∇Γvdσ

    = − ∫

    Γ Hv ∇Γu · ν︸ ︷︷ ︸

    =0

    dσ +

    ∫ Γ ∇Γu · ∇Γvdσ.

    Definition A function u ∈ H1(Γ) is called a weak solution of

    −∆Γu + u = f on Γ if∫ Γ ∇Γu · ∇Γv dσ +

    ∫ Γ u v dσ︸ ︷︷ ︸

    =a(u,v)

    =

    ∫ Γ f v dσ︸ ︷︷ ︸

    =l(v)

    ∀v ∈ H1(Γ).

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Theorem For every f ∈ L2(Γ) the PDE

    −∆Γu + u = f on Γ

    has a unique weak solution u ∈ H1(Γ). Furthermore, u ∈ H2(Γ) and there exists c > 0 such that

    ‖u‖H2(Γ) ≤ c‖f ‖L2(Γ).

    Idea of proof:

    I Existence and uniqueness: Lax–Milgram theorem

    I Regularity: ũ := u ◦ F (F : Ω→ Rn+1 local parametrisation) is a weak solution of

    − n∑

    i ,j=1

    ∂j ( g ij √ g∂i ũ

    ) + √ gũ =

    √ g f ◦ F in Ω.

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Theorem For every f ∈ L2(Γ) the PDE

    −∆Γu + u = f on Γ

    has a unique weak solution u ∈ H1(Γ). Furthermore, u ∈ H2(Γ) and there exists c > 0 such that

    ‖u‖H2(Γ) ≤ c‖f ‖L2(Γ).

    Idea of proof:

    I Existence and uniqueness: Lax–Milgram theorem

    I Regularity: ũ := u ◦ F (F : Ω→ Rn+1 local parametrisation) is a weak solution of

    − n∑

    i ,j=1

    ∂j ( g ij √ g∂i ũ

    ) + √ gũ =

    √ g f ◦ F in Ω.

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Oriented distance function

    Suppose that Γ = ∂Ω ∈ C 2 for some bounded domain Ω ⊂ Rn+1. Let

    d(x) :=

     infy∈Γ |x − y | x ∈ Rn+1 \ Ω̄

    0 x ∈ Γ − infy∈Γ |x − y | x ∈ Ω.

    Lemma

    (a) There exists δ > 0 such that d ∈ C 2(Γδ), where Γδ = {x ∈ Rn+1 | |d(x)| < δ};

    (b) (Fermi coordinates) For every x ∈ Γδ there exists a unique p(x) ∈ Γ such that

    x = p(x) + d(x)ν(p(x)).

    see: D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of 2nd Order, Springer

    Klaus Deckelnick FEM for elliptic surface PDEs

  • Oriented distance function