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

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Finite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto–von–Guericke–Universit¨ at Magdeburg 3rd Workshop Analysis, Geometry and Probability Universit¨ at Ulm Klaus Deckelnick FEM for elliptic surface PDEs

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

Page 1: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Finite element methods for elliptic PDEs onsurfaces

Klaus Deckelnick, Otto–von–Guericke–Universitat Magdeburg

3rd Workshop Analysis, Geometry and Probability

Universitat Ulm

Klaus Deckelnick FEM for elliptic surface PDEs

Page 2: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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 & Nurnberg (2015)

Klaus Deckelnick FEM for elliptic surface PDEs

Page 3: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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 thegeometry

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

Klaus Deckelnick FEM for elliptic surface PDEs

Page 4: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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 thegeometry

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

Klaus Deckelnick FEM for elliptic surface PDEs

Page 5: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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 thegeometry

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

Klaus Deckelnick FEM for elliptic surface PDEs

Page 6: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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

Page 7: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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

Page 8: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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

Page 9: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Differentiation on hypersurfaces

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

Tangential gradient

∇Γf (x) =

∑ni ,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

Page 10: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Differentiation on hypersurfaces

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

Tangential gradient

∇Γf (x) =

∑ni ,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

Page 11: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Differentiation on hypersurfaces

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

Tangential gradient

∇Γf (x) =

∑ni ,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

Page 12: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Integration by parts ∫Γ∇Γf dσ =

∫Γf 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

Page 13: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Integration by parts ∫Γ∇Γf dσ =

∫Γf 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

Page 14: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Weak solutionsSuppose 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

= −∫

ΓHv ∇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 +

∫Γu v dσ︸ ︷︷ ︸

=a(u,v)

=

∫Γf v dσ︸ ︷︷ ︸

=l(v)

∀v ∈ H1(Γ).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 15: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Weak solutionsSuppose 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

= −∫

ΓHv ∇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 +

∫Γu v dσ︸ ︷︷ ︸

=a(u,v)

=

∫Γf v dσ︸ ︷︷ ︸

=l(v)

∀v ∈ H1(Γ).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 16: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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

−n∑

i ,j=1

∂j(g ij√g∂i u

)+√gu =

√g f F in Ω.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 17: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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

−n∑

i ,j=1

∂j(g ij√g∂i u

)+√gu =

√g f F in Ω.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 18: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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 \ Ω

0 x ∈ Γ− infy∈Γ |x − y | x ∈ Ω.

Lemma

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

(b) (Fermi coordinates) For every x ∈ Γδ there exists a uniquep(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

Page 19: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

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 \ Ω

0 x ∈ Γ− infy∈Γ |x − y | x ∈ Ω.

Lemma

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

(b) (Fermi coordinates) For every x ∈ Γδ there exists a uniquep(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

Page 20: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Approach I: FEM on triangulated surface

Idea Approximate Γ ⊂ Rn+1 by Γh =⋃

T∈Th

T , 0 < h ≤ h0, where

I Th consists of n–simplices with vertices a1, . . . , aN ∈ Γ;

I Th is admissible and regular; h := maxT∈Th diamT ;

I The mapping p : Γh → Γ is bijective.

Let Sh := vh ∈ C 0(Γh) | vh|T ∈ P1(T ),T ∈ Th.

Every uh ∈ Sh can be uniquely written as

uh(x) =N∑j=1

ujφj(x), x ∈ Γh,

where φi ∈ Sh, 1 ≤ j ≤ N satisfies φj(aj) = 1, φj(ak) = 0, k 6= j .

Klaus Deckelnick FEM for elliptic surface PDEs

Page 21: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Approach I: FEM on triangulated surface

Idea Approximate Γ ⊂ Rn+1 by Γh =⋃

T∈Th

T , 0 < h ≤ h0, where

I Th consists of n–simplices with vertices a1, . . . , aN ∈ Γ;

I Th is admissible and regular; h := maxT∈Th diamT ;

I The mapping p : Γh → Γ is bijective.

Let Sh := vh ∈ C 0(Γh) | vh|T ∈ P1(T ),T ∈ Th.

Every uh ∈ Sh can be uniquely written as

uh(x) =N∑j=1

ujφj(x), x ∈ Γh,

where φi ∈ Sh, 1 ≤ j ≤ N satisfies φj(aj) = 1, φj(ak) = 0, k 6= j .

Klaus Deckelnick FEM for elliptic surface PDEs

Page 22: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Figure : Triangulation of the sphere: after 6 refinement steps thetriangulation consists of 512 triangles and 258 vertices.

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

Klaus Deckelnick FEM for elliptic surface PDEs

Page 23: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Surface FEM (Dziuk, 1988)

Find uh =∑N

j=1 ujφj(x) ∈ Sh such that∫Γh

(∇Γh

uh · ∇Γhvh + uh vh

)dσh =

∫Γh

fh vh dσh ∀vh ∈ Sh

⇐⇒N∑j=1

uj

∫Γh

(∇Γh

φj · ∇Γhφi + φj φi

)dσh︸ ︷︷ ︸

=:Aij

=

∫Γh

fhφi dσh︸ ︷︷ ︸=:Fi

1 ≤ i ≤ N.

Theorem The discrete problem has a unique solution uh ∈ Sh and

‖u − ulh‖L2(Γ) + h‖∇Γ(u − ulh)‖L2(Γ) ≤ ch2‖u‖H2(Γ),

provided ‖f − f lh‖L2(Γ) ≤ ch2. Here, ulh(y) := uh(p−1(y)), y ∈ Γ.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 24: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Surface FEM (Dziuk, 1988)

Find uh =∑N

j=1 ujφj(x) ∈ Sh such that∫Γh

(∇Γh

uh · ∇Γhvh + uh vh

)dσh =

∫Γh

fh vh dσh ∀vh ∈ Sh

⇐⇒N∑j=1

uj

∫Γh

(∇Γh

φj · ∇Γhφi + φj φi

)dσh︸ ︷︷ ︸

=:Aij

=

∫Γh

fhφi dσh︸ ︷︷ ︸=:Fi

1 ≤ i ≤ N.

Theorem The discrete problem has a unique solution uh ∈ Sh and

‖u − ulh‖L2(Γ) + h‖∇Γ(u − ulh)‖L2(Γ) ≤ ch2‖u‖H2(Γ),

provided ‖f − f lh‖L2(Γ) ≤ ch2. Here, ulh(y) := uh(p−1(y)), y ∈ Γ.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 25: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Surface FEM (Dziuk, 1988)

Find uh =∑N

j=1 ujφj(x) ∈ Sh such that∫Γh

(∇Γh

uh · ∇Γhvh + uh vh

)dσh =

∫Γh

fh vh dσh ∀vh ∈ Sh

⇐⇒N∑j=1

uj

∫Γh

(∇Γh

φj · ∇Γhφi + φj φi

)dσh︸ ︷︷ ︸

=:Aij

=

∫Γh

fhφi dσh︸ ︷︷ ︸=:Fi

1 ≤ i ≤ N.

Theorem The discrete problem has a unique solution uh ∈ Sh and

‖u − ulh‖L2(Γ) + h‖∇Γ(u − ulh)‖L2(Γ) ≤ ch2‖u‖H2(Γ),

provided ‖f − f lh‖L2(Γ) ≤ ch2. Here, ulh(y) := uh(p−1(y)), y ∈ Γ.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 26: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Approach II: FEM on bulk triangulation

Idea

I Extend the surface PDE to a neighbourhood U of Γ

I Solve the extended PDE using a FEM method on U

For φ : U → R with ∇φ(x) 6= 0, x ∈ U we let

Γr := x ∈ U |φ(x) = r and suppose that Γ = Γ0.

Observe that for f : U → R

∇Γr f|Γr= [Pφ∇f ]|Γr

, where Pφ := In+1 −∇φ|∇φ|

⊗ ∇φ|∇φ|

,

∆Γr f|Γr=

[1

|∇φ|∇ ·(Pφ∇f |∇φ|

)]|Γr

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 27: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Approach II: FEM on bulk triangulation

Idea

I Extend the surface PDE to a neighbourhood U of Γ

I Solve the extended PDE using a FEM method on U

For φ : U → R with ∇φ(x) 6= 0, x ∈ U we let

Γr := x ∈ U |φ(x) = r and suppose that Γ = Γ0.

Observe that for f : U → R

∇Γr f|Γr= [Pφ∇f ]|Γr

, where Pφ := In+1 −∇φ|∇φ|

⊗ ∇φ|∇φ|

,

∆Γr f|Γr=

[1

|∇φ|∇ ·(Pφ∇f |∇φ|

)]|Γr

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 28: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Approach II: FEM on bulk triangulation

Idea

I Extend the surface PDE to a neighbourhood U of Γ

I Solve the extended PDE using a FEM method on U

For φ : U → R with ∇φ(x) 6= 0, x ∈ U we let

Γr := x ∈ U |φ(x) = r and suppose that Γ = Γ0.

Observe that for f : U → R

∇Γr f|Γr= [Pφ∇f ]|Γr

, where Pφ := In+1 −∇φ|∇φ|

⊗ ∇φ|∇φ|

,

∆Γr f|Γr=

[1

|∇φ|∇ ·(Pφ∇f |∇φ|

)]|Γr

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 29: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Variant 1: Burger (2009)

− 1

|∇φ|∇ ·(Pφ∇u|∇φ|

)+ u = f in

⋃−δ<r<δ

Γr . (2)

Properties

I u solves (2) =⇒ u|Γrsolves

−∆Γr v + v = f|Γrfor − δ < r < δ

I (2) is only degenerate elliptic because Pφ∇φ = 0

I Existence: Burger (2009)

I Regularity: D., Dziuk, Elliott & Heine (2010).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 30: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Variant 1: Burger (2009)

− 1

|∇φ|∇ ·(Pφ∇u|∇φ|

)+ u = f in

⋃−δ<r<δ

Γr . (2)

Properties

I u solves (2) =⇒ u|Γrsolves

−∆Γr v + v = f|Γrfor − δ < r < δ

I (2) is only degenerate elliptic because Pφ∇φ = 0

I Existence: Burger (2009)

I Regularity: D., Dziuk, Elliott & Heine (2010).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 31: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Narrow band around Γ

Let (Th)0<h≤h0 be a family of triangulations of U and set

Γh := x ∈ U ; Ihφ(x) = 0,

Dh := x ∈ U ; |Ihφ(x)| < h

T Γh := T ∈ Th ; |T ∩ Dh| > 0.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 32: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Narrow band around Γ

Let (Th)0<h≤h0 be a family of triangulations of U and set

Γh := x ∈ U ; Ihφ(x) = 0,

Dh := x ∈ U ; |Ihφ(x)| < h

T Γh := T ∈ Th ; |T ∩ Dh| > 0.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 33: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Figure : Narrow bands around a torus

(T. Ranner: Computational surface partial differential equations, PhD Thesis, University of Warwick (2013))

Klaus Deckelnick FEM for elliptic surface PDEs

Page 34: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Let Vh := spanϕj ; aj ∈ T ∈ T Γh

Find uh ∈ Vh such that for all vh ∈ Vh∫Dh

(Ph∇uh · ∇vh + uh vh

)|∇Ihφ| dx =

∫Dh

f vh|∇Ihφ|dx

where

Ph = In+1 −∇Ihφ|∇Ihφ|

⊗ ∇Ihφ|∇Ihφ|

.

Theorem (D., Dziuk, Elliott & Heine, 2010)

Suppose that the solution u of (2) belongs to W 2,∞(U) and thatf ∈W 1,∞(U). Then( 1

2h

∫Dh

(|Ph∇(u − uh)|2 + |u − uh|2

)dx) 1

2 ≤ ch

Klaus Deckelnick FEM for elliptic surface PDEs

Page 35: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Let Vh := spanϕj ; aj ∈ T ∈ T Γh

Find uh ∈ Vh such that for all vh ∈ Vh∫Dh

(Ph∇uh · ∇vh + uh vh

)|∇Ihφ| dx =

∫Dh

f vh|∇Ihφ|dx

where

Ph = In+1 −∇Ihφ|∇Ihφ|

⊗ ∇Ihφ|∇Ihφ|

.

Theorem (D., Dziuk, Elliott & Heine, 2010)

Suppose that the solution u of (2) belongs to W 2,∞(U) and thatf ∈W 1,∞(U). Then( 1

2h

∫Dh

(|Ph∇(u − uh)|2 + |u − uh|2

)dx) 1

2 ≤ ch

Klaus Deckelnick FEM for elliptic surface PDEs

Page 36: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Γ = x = (x1, x2, x3) ∈ R3 |3∑

i=1

[(x2i +x2

i+1−4)2 +(x2i+2−1)2] = 3

Figure : Computed solution for

f (x) = 100∑4

j=1 exp(−|x − x (j)|2), x (1), . . . , x (4) given (left)

f (x) = 10000 sin(5(x1 + x2 + x3) + 2.5) (right)

Klaus Deckelnick FEM for elliptic surface PDEs

Page 37: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Variant 2: D., Elliott & Ranner (2014)

For a given u : Γ→ R we define an extension ue : U → R by

ue(x) := u(p(x)), where x = p(x) + d(x)ν(p(x)).

Properties of ue :

a) ∇ue · ν = 0 =⇒ P∇ue = ∇ue ;

b) If −∆Γu + u = f on Γ, then ue satisfies

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p)

in U.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 38: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Variant 2: D., Elliott & Ranner (2014)

For a given u : Γ→ R we define an extension ue : U → R by

ue(x) := u(p(x)), where x = p(x) + d(x)ν(p(x)).

Properties of ue :

a) ∇ue · ν = 0 =⇒ P∇ue = ∇ue ;

b) If −∆Γu + u = f on Γ, then ue satisfies

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p)

in U.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 39: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Let Vh := spanϕj ; aj ∈ T ∈ T Γh

Find uh ∈ Vh such that for all vh ∈ Vh∫Dh

(∇uh · ∇vh + uh vh

)|∇Ihd |dx =

∫Dh

f e vh|∇Ihd |dx .

Theorem The discrete problem has a unique solution uh ∈ Vh and(1

2h

∫Dh

|∇(ue − uh)|2|∇Ihd |dx) 1

2

≤ ch‖f ‖L2(Γ)

‖ue − uh‖L2(Γh) ≤ ch2‖f ‖L2(Γ).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 40: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Let Vh := spanϕj ; aj ∈ T ∈ T Γh

Find uh ∈ Vh such that for all vh ∈ Vh∫Dh

(∇uh · ∇vh + uh vh

)|∇Ihd |dx =

∫Dh

f e vh|∇Ihd |dx .

Theorem The discrete problem has a unique solution uh ∈ Vh and(1

2h

∫Dh

|∇(ue − uh)|2|∇Ihd |dx) 1

2

≤ ch‖f ‖L2(Γ)

‖ue − uh‖L2(Γh) ≤ ch2‖f ‖L2(Γ).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 41: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Sketch of the proof

Abstract error bound

1

2h

∫Dh

(∇uh · ∇vh + uhvh)|∇Ihd |dx︸ ︷︷ ︸=:ah(uh,vh)

=1

2h

∫Dh

f evh |∇Ihd |dx︸ ︷︷ ︸=:lh(vh)

.

Strang’s Second Lemma:

Let ‖v‖h :=√

ah(v , v). Then:

‖ue − uh‖h ≤ 2 infvh∈Vh

‖ue − vh‖h + supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 42: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Sketch of the proof

Abstract error bound

1

2h

∫Dh

(∇uh · ∇vh + uhvh)|∇Ihd |dx︸ ︷︷ ︸=:ah(uh,vh)

=1

2h

∫Dh

f evh |∇Ihd |dx︸ ︷︷ ︸=:lh(vh)

.

Strang’s Second Lemma:

Let ‖v‖h :=√

ah(v , v). Then:

‖ue − uh‖h ≤ 2 infvh∈Vh

‖ue − vh‖h + supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 43: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Interpolation error

infvh∈Vh

‖ue − vh‖h ≤ ‖ue − Ihue‖h ≤ ch‖u‖H2(Γ).

Consistency error

Fh(x) := x + (Ihd(x)− d(x))ν(p(x)).

Properties

a) Fh is a bijection from |Ihd | < h = Dh onto Dh = |d | < h;

b) |Fh(x)− x | ≤ ch2, |DFh(x)− In+1| ≤ ch;

c) |detDFh(x)− |∇Ihd(x)| | ≤ ch2.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 44: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Interpolation error

infvh∈Vh

‖ue − vh‖h ≤ ‖ue − Ihue‖h ≤ ch‖u‖H2(Γ).

Consistency error

Fh(x) := x + (Ihd(x)− d(x))ν(p(x)).

Properties

a) Fh is a bijection from |Ihd | < h = Dh onto Dh = |d | < h;

b) |Fh(x)− x | ≤ ch2, |DFh(x)− In+1| ≤ ch;

c) |detDFh(x)− |∇Ihd(x)| | ≤ ch2.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 45: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Recall that

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p).

We obtain for arbitrary vh ∈ Vh

1

2h

∫Dh

(−∆ue + ue)vh F−1h dx =

1

2h

∫Dh

(f e + dg)vh F−1h dx .

1

2h

∫Dh

(−∆ue + ue)vh F−1h =

1

2h

∫Dh

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)=

1

2h

∫Dh

(∇ue Fh · DF−th ∇vh + ue Fh vh

)detDFhdx

=1

2h

∫Dh

(∇ue · ∇vh + ue vh

)|∇Ihd |dx︸ ︷︷ ︸

=ah(ue ,vh)

+O(h‖vh‖h).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 46: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Recall that

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p).

We obtain for arbitrary vh ∈ Vh

1

2h

∫Dh

(−∆ue + ue)vh F−1h dx =

1

2h

∫Dh

(f e + dg)vh F−1h dx .

1

2h

∫Dh

(−∆ue + ue)vh F−1h =

1

2h

∫Dh

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)=

1

2h

∫Dh

(∇ue Fh · DF−th ∇vh + ue Fh vh

)detDFhdx

=1

2h

∫Dh

(∇ue · ∇vh + ue vh

)|∇Ihd |dx︸ ︷︷ ︸

=ah(ue ,vh)

+O(h‖vh‖h).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 47: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Recall that

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p).

We obtain for arbitrary vh ∈ Vh

1

2h

∫Dh

(−∆ue + ue)vh F−1h dx =

1

2h

∫Dh

(f e + dg)vh F−1h dx .

1

2h

∫Dh

(−∆ue + ue)vh F−1h =

1

2h

∫Dh

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)

=1

2h

∫Dh

(∇ue Fh · DF−th ∇vh + ue Fh vh

)detDFhdx

=1

2h

∫Dh

(∇ue · ∇vh + ue vh

)|∇Ihd |dx︸ ︷︷ ︸

=ah(ue ,vh)

+O(h‖vh‖h).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 48: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Recall that

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p).

We obtain for arbitrary vh ∈ Vh

1

2h

∫Dh

(−∆ue + ue)vh F−1h dx =

1

2h

∫Dh

(f e + dg)vh F−1h dx .

1

2h

∫Dh

(−∆ue + ue)vh F−1h =

1

2h

∫Dh

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)=

1

2h

∫Dh

(∇ue Fh · DF−th ∇vh + ue Fh vh

)detDFhdx

=1

2h

∫Dh

(∇ue · ∇vh + ue vh

)|∇Ihd |dx︸ ︷︷ ︸

=ah(ue ,vh)

+O(h‖vh‖h).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 49: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Recall that

−∆ue + ue = f e + d(x) g(x , (∇Γu) p, (D2

Γu) p).

We obtain for arbitrary vh ∈ Vh

1

2h

∫Dh

(−∆ue + ue)vh F−1h dx =

1

2h

∫Dh

(f e + dg)vh F−1h dx .

1

2h

∫Dh

(−∆ue + ue)vh F−1h =

1

2h

∫Dh

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)=

1

2h

∫Dh

(∇ue Fh · DF−th ∇vh + ue Fh vh

)detDFhdx

=1

2h

∫Dh

(∇ue · ∇vh + ue vh

)|∇Ihd |dx︸ ︷︷ ︸

=ah(ue ,vh)

+O(h‖vh‖h).

Klaus Deckelnick FEM for elliptic surface PDEs

Page 50: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Similarly

• 1

2h

∫Dh

f e vh F−1h dx =

1

2h

∫Dh

f e vh |∇Ihd |dx︸ ︷︷ ︸=lh(vh)

+O(h2‖vh‖h);

• | 1

2h

∫Dh

d(x)g(x , (∇Γu) p, (D2Γu) p)vh F−1

h | ≤ ch‖vh‖h.

In conclusion

‖ue − uh‖h ≤ 2 infvh∈Vh

‖ue − vh‖h︸ ︷︷ ︸≤ch

+ supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h︸ ︷︷ ︸≤ch

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 51: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Similarly

• 1

2h

∫Dh

f e vh F−1h dx =

1

2h

∫Dh

f e vh |∇Ihd |dx︸ ︷︷ ︸=lh(vh)

+O(h2‖vh‖h);

• | 1

2h

∫Dh

d(x)g(x , (∇Γu) p, (D2Γu) p)vh F−1

h | ≤ ch‖vh‖h.

In conclusion

‖ue − uh‖h ≤ 2 infvh∈Vh

‖ue − vh‖h︸ ︷︷ ︸≤ch

+ supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h︸ ︷︷ ︸≤ch

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 52: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Similarly

• 1

2h

∫Dh

f e vh F−1h dx =

1

2h

∫Dh

f e vh |∇Ihd |dx︸ ︷︷ ︸=lh(vh)

+O(h2‖vh‖h);

• | 1

2h

∫Dh

d(x)g(x , (∇Γu) p, (D2Γu) p)vh F−1

h | ≤ ch‖vh‖h.

In conclusion

‖ue − uh‖h ≤ 2 infvh∈Vh

‖ue − vh‖h︸ ︷︷ ︸≤ch

+ supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h︸ ︷︷ ︸≤ch

.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 53: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Sharp interface methods

Γh = x ∈ U ; Ihd(x) = 0

T Γh = T ∈ Th ; |T ∩ Γh| > 0

V Γh = spanϕj ; xj ∈ T ∈ T Γ

h .

Variant I (Olshanskii, Reusken & Grande, 2009):∫Γh

(Ph∇uh · ∇vh + uhvh

)dσh =

∫Γh

f evhdσh.

Variant II (D., Elliott, Ranner, 2014):∫Γh

(∇uh · ∇vh + uhvh

)dσh =

∫Γh

f evhdσh.

Higher order elements: Reusken, 2014.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 54: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Sharp interface methods

Γh = x ∈ U ; Ihd(x) = 0

T Γh = T ∈ Th ; |T ∩ Γh| > 0

V Γh = spanϕj ; xj ∈ T ∈ T Γ

h .

Variant I (Olshanskii, Reusken & Grande, 2009):∫Γh

(Ph∇uh · ∇vh + uhvh

)dσh =

∫Γh

f evhdσh.

Variant II (D., Elliott, Ranner, 2014):∫Γh

(∇uh · ∇vh + uhvh

)dσh =

∫Γh

f evhdσh.

Higher order elements: Reusken, 2014.

Klaus Deckelnick FEM for elliptic surface PDEs

Page 55: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Comparison

Surface FEM

I construction of triangulation can be difficult, after that easyto implement

I efficient with respect to degrees of freedom

I coupling with bulk equations may be difficult

Narrow band bulk FEM

I no surface mesh required

I evaluation of narrow band integrals not straightforward

I possibly bad conditioning

I coupling with bulk equations can be done on the same mesh

Klaus Deckelnick FEM for elliptic surface PDEs

Page 56: Finite element methods for elliptic PDEs on surfacesFinite element methods for elliptic PDEs on surfaces Klaus Deckelnick, Otto{von{Guericke{Universit at Magdeburg 3rd Workshop Analysis,

Comparison

Surface FEM

I construction of triangulation can be difficult, after that easyto implement

I efficient with respect to degrees of freedom

I coupling with bulk equations may be difficult

Narrow band bulk FEM

I no surface mesh required

I evaluation of narrow band integrals not straightforward

I possibly bad conditioning

I coupling with bulk equations can be done on the same mesh

Klaus Deckelnick FEM for elliptic surface PDEs