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
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