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
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)
[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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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