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