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

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

Motivation

Two-phase flow with insoluble surfactant

ρut + ρ(u · ∇)u −∇ · T (u, p) = ρf

∇ · u = 0

in Ω±(t)

= 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)

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)

A model problem

−∆Γu + u = f on Γ. (1)

A model problem

−∆Γu + u = f on Γ. (1)

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.

U∩Γ =

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

Tangent space

TxΓ =

span ∂F∂ω1(ω), . . . , ∂F∂ωn

)⊥, φ(x) = 0.

U∩Γ =

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

Tangent space

TxΓ =

span ∂F∂ω1(ω), . . . , ∂F∂ωn

)⊥, φ(x) = 0.

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

Tangential gradient

∇Γf (x) =

∑ni ,j=1 g

ij(ω)∂j(f F

)(ω)∂iF (ω), x = F (ω)

R , x ∈ Γ

Tangential gradient

∇Γf (x) =

∑ni ,j=1 g

ij(ω)∂j(f F

)(ω)∂iF (ω), x = F (ω)

R , x ∈ Γ

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

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

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︸︷︷︸

∫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︸︷︷︸

∀v ∈ H1(Γ).

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︸︷︷︸

∫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︸︷︷︸

∀v ∈ H1(Γ).

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.

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.

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.

(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

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.

(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

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 .

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 .

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))

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

∫Γh

(∇Γh

φj · ∇Γhφi + φj φi

)dσh︸︷︷︸

∫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.

Find uh =∑N

(∇Γh

)dσh =

∫Γh

⇐⇒N∑j=1

∫Γh

(∇Γh

)dσh︸︷︷︸

∫Γh

1 ≤ i ≤ N.

Find uh =∑N

(∇Γh

)dσh =

∫Γh

⇐⇒N∑j=1

∫Γh

(∇Γh

)dσh︸︷︷︸

∫Γh

1 ≤ i ≤ N.

Approach II: FEM on bulk triangulation

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=

|∇φ|∇ ·(Pφ∇f |∇φ|

)]|Γr

⊗ ∇φ|∇φ|

∆Γr f|Γr=

|∇φ|∇ ·(Pφ∇f |∇φ|

)]|Γr

⊗ ∇φ|∇φ|

∆Γr f|Γr=

|∇φ|∇ ·(Pφ∇f |∇φ|

)]|Γr

Variant 1: Burger (2009)

|∇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).

Variant 1: Burger (2009)

|∇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).

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.

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.

Figure : Narrow bands around a torus

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

Let Vh := spanϕj ; aj ∈ T ∈ T Γh

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

(Ph∇uh · ∇vh + uh vh

)|∇Ihφ| dx =

f vh|∇Ihφ|dx

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

(|Ph∇(u − uh)|2 + |u − uh|2

)dx) 1

2 ≤ ch

(Ph∇uh · ∇vh + uh vh

)|∇Ihφ| dx =

f vh|∇Ihφ|dx

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

(|Ph∇(u − uh)|2 + |u − uh|2

)dx) 1

2 ≤ ch

Γ = x = (x1, x2, x3) ∈ R3 |3∑

[(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)

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)

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)

(∇uh · ∇vh + uh vh

)|∇Ihd |dx =

f e vh|∇Ihd |dx .

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

|∇(ue − uh)|2|∇Ihd |dx) 1

≤ ch‖f ‖L2(Γ)

‖ue − uh‖L2(Γh) ≤ ch2‖f ‖L2(Γ).

(∇uh · ∇vh + uh vh

)|∇Ihd |dx =

f e vh|∇Ihd |dx .

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

|∇(ue − uh)|2|∇Ihd |dx) 1

≤ ch‖f ‖L2(Γ)

‖ue − uh‖L2(Γh) ≤ ch2‖f ‖L2(Γ).

Sketch of the proof

Abstract error bound

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

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

Sketch of the proof

Abstract error bound

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

f evh |∇Ihd |dx︸︷︷︸=:lh(vh)

Strang’s Second Lemma:

Let ‖v‖h :=√

ah(v , v). Then:

‖ue − vh‖h + supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h

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.

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.

Recall that

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

Γu) p).

We obtain for arbitrary vh ∈ Vh

(−∆ue + ue)vh F−1h dx =

(f e + dg)vh F−1h dx .

(−∆ue + ue)vh F−1h =

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

(∇ue Fh · DF−th ∇vh + ue Fh vh

)detDFhdx

(∇ue · ∇vh + ue vh

)|∇Ihd |dx︸︷︷︸

=ah(ue ,vh)

+O(h‖vh‖h).

Recall that

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

Γu) p).

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)detDFhdx

)|∇Ihd |dx︸︷︷︸

=ah(ue ,vh)

+O(h‖vh‖h).

Recall that

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

Γu) p).

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)detDFhdx

)|∇Ihd |dx︸︷︷︸

=ah(ue ,vh)

+O(h‖vh‖h).

Recall that

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

Γu) p).

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)detDFhdx

)|∇Ihd |dx︸︷︷︸

=ah(ue ,vh)

+O(h‖vh‖h).

Recall that

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

Γu) p).

(∇ue · ∇(vh F−1

h ) + ue vh F−1h

)detDFhdx

)|∇Ihd |dx︸︷︷︸

=ah(ue ,vh)

+O(h‖vh‖h).

Similarly

f e vh F−1h dx =

f e vh |∇Ihd |dx︸︷︷︸=lh(vh)

+O(h2‖vh‖h);

• | 1

d(x)g(x , (∇Γu) p, (D2Γu) p)vh F−1

h | ≤ ch‖vh‖h.

In conclusion

‖ue − vh‖h︸︷︷︸≤ch

+ supvh∈Vh

|ah(ue , vh)− lh(vh)|‖vh‖h︸︷︷︸≤ch

Similarly

f e vh F−1h dx =

+O(h2‖vh‖h);

• | 1

In conclusion

+ supvh∈Vh

Similarly

f e vh F−1h dx =

+O(h2‖vh‖h);

• | 1

In conclusion

+ supvh∈Vh

Sharp interface methods

Γh = x ∈ U ; Ihd(x) = 0

T Γh = T ∈ Th ; |T ∩ Γh| > 0

V Γh = spanϕj ; xj ∈ T ∈ T Γ

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.

Sharp interface methods

Γh = x ∈ U ; Ihd(x) = 0

T Γh = T ∈ Th ; |T ∩ Γh| > 0

V Γh = spanϕj ; xj ∈ T ∈ T Γ

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.

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

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

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