Adaptive Finite Element MethodsLecture 5: The Laplace-Beltrami Operator
Ricardo H. Nochetto
Department of Mathematics andInstitute for Physical Science and Technology
University of Maryland, USAwww2.math.umd.edu/˜rhn
Joint work with
Andrea Bonito, Texas A&M University, USA
J. Manuel Cascon, University of Salamanca, Spain
Khamron Mekchay, Chulalongkorn University, Thailand
Pedro Morin, Universidad Nacional del Litoral, Argentina
7th Zurich Summer School, August 2012A Posteriori Error Control and Adaptivity
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
The Laplace-Beltrami Problem
−∆γu = f on γ, u = 0 on ∂γ.
• γ is a parametric Lipschitz surface, piecewise C1, with Lipschitzboundary ∂γ or without boundary (closed surface).
• f ∈ L2(γ) (see Lecture 3 for H−1(γ) data).
• Weak Formulation:
Fk
K
Seek u ∈ H10 (γ) :
∫γ
∇γu · ∇γv =∫
γ
f v, ∀v ∈ H10 (γ).∫
K
∇γu·∇γv =∫
bK ∇(uFk)T G−1K ∇(vFK)
√det(GK), GK = DFT
KDFK
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Biomembranes: Modeling (w. A. Bonito and M.S. Pauletti)
• Bending (Willmore) energy: J(Γ) = 12
∫Γ
H2, H mean curvature
• Geometric Gradient Flow (with area and volume constraint):
v = −δΓJ = −(∆ΓH +
12H3 − 2κH
)ν −
(λHν + pν
)where ∆Γ is the Laplace-Beltrami operator on Γ.
• Fluid-Membrane Interaction (with area constraint):
ρDtv − div (−pI + µD(v)︸ ︷︷ ︸Σ
) = b in Ωt,
div v = 0 in Ωt,
[Σ]ν = δΓJ on Γt
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Biomembrane: Geometric vs Fluid Red Blood Cell
play
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Director Fields on Flexible Surfaces (w. S. Bartels, G. Dolzmann)
• Coupling of mean curvature Hu = divΓ ν with a director field n via
J(Γ, n) =12
∫Γ
|divΓ ν − δ divΓ n|2dσ +λ
2
∫Γ
|∇Γn|2dσ
+12
∫Γ
µ(|n|2 − 1)dσ +12ε
∫Γ
f(n · ν)dσ
• µ the Lagrange multiplier for the rigid constraint |n| = 1
• f(x) = (x2 − ξ20)2 with ξ0 ∈ [0, 1] penalizes the deviation of the angle
between ν and n from arccos ξ0
• Spontaneous curvature H0 = δ divΓ n induced by director field n
• Relaxation dynamics (L2- gradient flow): V normal velocity of Γ
V = −δΓJ(Γ, n), ∂tn = −δnJ(Γ, n)
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Coupling of Director Fields and Flexible Surfaces: Simulations
Cone-like structure near positive degree-one defects pointing outwards ⇒stomatocyte shape
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
AFEM: Comparison with the Standard Approach
AFEM: Given an initial surface-mesh pair (Γ0, T0), and parametersε0 > 0, 0 < ρ < 1, and ω > 0, set k = 0 and iterate
[T +k ,Γ+
k ] = ADAPT SURFACE (Tk, ωεk)[Tk+1,Γk+1] = ADAPT PDE (T +
k , εk)εk+1 = ρεk; k = k + 1.
where ADAPT SURFACE deals with surface Γ via the geometricindicator λΓ(T )T∈T , and ADAPT PDE is the usual loop
SOLVE → ESTIMATE → MARK → REFINE
• SOLVE: requires an optimal iterative multilevel solver (Bonito-Pasciak,Kornhuber-Yserentant);
• ESTIMATE: provides error indicators ηT (U, T )T∈T for the PDE;
• MARK: uses Dorfler marking to select a minimal set M such thatηT (U,M) ≤ θηT (U, T );
• REFINE: refines M and creates a new mesh T∗ ≥ T which could beeither conforming (bisection) or have hanging-nodes (quadrilaterals).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
AFEM: Comparison with the Standard Approach
AFEM: Given an initial surface-mesh pair (Γ0, T0), and parametersε0 > 0, 0 < ρ < 1, and ω > 0, set k = 0 and iterate
[T +k ,Γ+
k ] = ADAPT SURFACE (Tk, ωεk)[Tk+1,Γk+1] = ADAPT PDE (T +
k , εk)εk+1 = ρεk; k = k + 1.
where ADAPT SURFACE deals with surface Γ via the geometricindicator λΓ(T )T∈T , and ADAPT PDE is the usual loop
SOLVE → ESTIMATE → MARK → REFINE
• SOLVE: requires an optimal iterative multilevel solver (Bonito-Pasciak,Kornhuber-Yserentant);
• ESTIMATE: provides error indicators ηT (U, T )T∈T for the PDE;
• MARK: uses Dorfler marking to select a minimal set M such thatηT (U,M) ≤ θηT (U, T );
• REFINE: refines M and creates a new mesh T∗ ≥ T which could beeither conforming (bisection) or have hanging-nodes (quadrilaterals).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Asymptotics: Role of ω• γ is the graph of class C1,α given by
z(x, y) =(0.75− x2 − y2
)1+α
+,
over the flat domain Ω = (0, 1)2, and consider α = 3/5, α = 2/5.• We say that z ∈ Bt if z can be approximated in W 1
∞ with N degreesof freedom and accuracy N−t:
α = 3/5 ⇒ z ∈ B 12
α = 2/5 ⇒ z ∈ Bt ∀t < 2/5.
• (u, f) ∈ A 12.
C1.6-surface, with ω = 1: Meshes after 10, 20 and 30 refinements have beenperformed. They are composed of 192, 1216 and 5564 elements, respectively.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Case α = 3/5
101
102
103
104
105
106
107
10−3
10−2
10−1
100
101
102
η+λ/
ω
ω = 0.1ω = 1ω = 10
N−0.5
101
102
103
104
105
106
107
10−3
10−2
10−1
100
101
102
η+λ
ω = 0.1ω = 1ω = 10
N−0.5
ηk + λk/ω (left) and ηk + λk (right) versus DOF for ω = 0.1, 1, 10.
101
102
103
104
105
106
107
10−4
10−3
10−2
10−1
100
101
N−0.5
η+λ/ωηλ/ω
101
102
103
104
105
106
107
10−4
10−3
10−2
10−1
100
101
N−0.5
η+λ/ωηλ/ω
101
102
103
104
105
106
107
10−4
10−3
10−2
10−1
100
N−0.5
η+λ/ωηλ/ω
ηk, λk/ω and ηk + λk/ω for ω = 0.1 (left) ω = 1 (middle) and ω = 10 (right).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Case α = 2/5
101
102
103
104
105
106
107
108
10−3
10−2
10−1
100
101
102
η+λ/
ω
ω = 0.1ω = 1ω = 10
N−0.4
101
102
103
104
105
106
107
108
10−2
10−1
100
101
102
η+λ
ω = 0.1ω = 1ω = 10
N−0.4
ηk + λk/ω (left) and ηk + λk (right) versus DOF for ω = 0.1, 1, 10.
101
102
103
104
105
106
107
108
10−4
10−3
10−2
10−1
100
101
102
N−0.5
η+λ/ωηλ/ωN−0.4
101
102
103
104
105
106
107
10−4
10−3
10−2
10−1
100
101
N−0.5
η+λ/ωηλ/ωN−0.4
101
102
103
104
105
106
107
10−4
10−3
10−2
10−1
100
N−0.5
η+λ/ωηλ/ωN−0.4
ηk, λk/ω and ηk + λk/ω for ω = 0.1 (left) ω = 1 (middle) and ω = 10 (right).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Representation of Parametric Surfaces
• Surface γ is described as the deformation of a d dimensionalpolyhedral surface Γ0 by a globally Lipschitz homeomorphismP0 : Γ0 → γ ⊂ Rd+1;
• Γ0 =⋃I
i=1 Γi0, P i
0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi
0 is a macro-element (asimplex for simplicity) and induces γi := P i
0(Γi0) ∈ C1(Γi
0);
• Parametric domain Ω ⊂ Rd, and affine map F i0 : Rd → Rd+1 such that
Γi0 = F i
0(Ω);
• Let X i := P i0 F i
0 : Ω → γi be a local parametrization of γ which isglobally bi-Lipschitz:
L−1|x− y| ≤ |X i(x)−X i(y)| ≤ L|x− y|, ∀x, y ∈ Ω.
• We will not write the superscript i and refer instead to Γ and X .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Representation of Parametric Surfaces
• Surface γ is described as the deformation of a d dimensionalpolyhedral surface Γ0 by a globally Lipschitz homeomorphismP0 : Γ0 → γ ⊂ Rd+1;
• Γ0 =⋃I
i=1 Γi0, P i
0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi
0 is a macro-element (asimplex for simplicity) and induces γi := P i
0(Γi0) ∈ C1(Γi
0);
• Parametric domain Ω ⊂ Rd, and affine map F i0 : Rd → Rd+1 such that
Γi0 = F i
0(Ω);
• Let X i := P i0 F i
0 : Ω → γi be a local parametrization of γ which isglobally bi-Lipschitz:
L−1|x− y| ≤ |X i(x)−X i(y)| ≤ L|x− y|, ∀x, y ∈ Ω.
• We will not write the superscript i and refer instead to Γ and X .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Representation of Parametric Surfaces
• Surface γ is described as the deformation of a d dimensionalpolyhedral surface Γ0 by a globally Lipschitz homeomorphismP0 : Γ0 → γ ⊂ Rd+1;
• Γ0 =⋃I
i=1 Γi0, P i
0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi
0 is a macro-element (asimplex for simplicity) and induces γi := P i
0(Γi0) ∈ C1(Γi
0);
• Parametric domain Ω ⊂ Rd, and affine map F i0 : Rd → Rd+1 such that
Γi0 = F i
0(Ω);
• Let X i := P i0 F i
0 : Ω → γi be a local parametrization of γ which isglobally bi-Lipschitz:
L−1|x− y| ≤ |X i(x)−X i(y)| ≤ L|x− y|, ∀x, y ∈ Ω.
• We will not write the superscript i and refer instead to Γ and X .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Representation of Parametric Surfaces
• Surface γ is described as the deformation of a d dimensionalpolyhedral surface Γ0 by a globally Lipschitz homeomorphismP0 : Γ0 → γ ⊂ Rd+1;
• Γ0 =⋃I
i=1 Γi0, P i
0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi
0 is a macro-element (asimplex for simplicity) and induces γi := P i
0(Γi0) ∈ C1(Γi
0);
• Parametric domain Ω ⊂ Rd, and affine map F i0 : Rd → Rd+1 such that
Γi0 = F i
0(Ω);
• Let X i := P i0 F i
0 : Ω → γi be a local parametrization of γ which isglobally bi-Lipschitz:
L−1|x− y| ≤ |X i(x)−X i(y)| ≤ L|x− y|, ∀x, y ∈ Ω.
• We will not write the superscript i and refer instead to Γ and X .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Representation of Parametric Surfaces
• Surface γ is described as the deformation of a d dimensionalpolyhedral surface Γ0 by a globally Lipschitz homeomorphismP0 : Γ0 → γ ⊂ Rd+1;
• Γ0 =⋃I
i=1 Γi0, P i
0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi
0 is a macro-element (asimplex for simplicity) and induces γi := P i
0(Γi0) ∈ C1(Γi
0);
• Parametric domain Ω ⊂ Rd, and affine map F i0 : Rd → Rd+1 such that
Γi0 = F i
0(Ω);
• Let X i := P i0 F i
0 : Ω → γi be a local parametrization of γ which isglobally bi-Lipschitz:
L−1|x− y| ≤ |X i(x)−X i(y)| ≤ L|x− y|, ∀x, y ∈ Ω.
• We will not write the superscript i and refer instead to Γ and X .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Parametric Domain Ω, Macro Element Γi0, and Surface γi
ΩF i
0
Γi0
P i0
γi
Representation of each component γi when d = 2 as a parametrization from a flat
triangle Γi0 ⊂ R3 as well as from the master triangle Ω ⊂ R2. The map F i
0 : Ω → Γi0
is affine.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Interpolation of Parametric Surfaces
Ω
T
F0
F
TP0
X
Effect of one bisection of the macro-element F0(Ω) when d = 2 (left). The parametricdomain Ω is split into two triangles in R2 via the affine map F−1
0 (bottom), whereasγ is interpolated by a new piecewise linear surface Γ = F(Ω) (right), with F = IT Xthe piecewise linear interpolant of the parametrization X defined in Ω. The superscripti is omitted for simplicity from now on.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Discrete Surface, Spaces, and Geometric Estimator
• Partitions T (Ω) of Ω are shape regular;
• Let IT : C0(Ω) → V(T (Ω)) be the Lagrange interpolation operator;
• Let FT = IT X be the interpolant of X in V(T (Ω)) and Γ := FT (Ω)
• Mesh T :=T = FT (T ) | T ∈ T (Ω)
and forest T(T0) = T ;
• Finite element space
V(T ) :=
V ∈ C0(Γ)∣∣ V is pw linear and
∫Γ
V = 0
;
• Geometric element indicator and geometric estimator
λΓ(T ) := ‖∇(X − FT )‖L∞( bT ), λΓ := maxT∈T
λΓ(T );
• Quasi-monotonicity: λΓ∗ ≤ Λ0λΓ with Λ0 ≥ 1 for all T∗ ≥ T .
• The forest T(T0) = T is shape regular provided λΓ0 ≤ 12Λ0L .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Basic Differential Geometry
• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT
]∈ R(d+1)×(d+1);
• The first fundamental form of γ
g =(gγ,ij
)1≤i,j≤d
:=(∂iX T ∂jX
)1≤i,j≤d
= TT T;
• ∇v = ∇γv T;
• D = T−1 ∈ R(d+1)×(d+1) and D ∈ Rd×(d+1) is D without the last row
∇γv = ∇γv T D =[∇v, 0
]D = ∇v D,
• Inverse of g: g−1 = DDT ;
• Elementary area: q :=√
detg
• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=
√detGΓ.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Variational Formulation and Galerkin Method
• Weak formulation on a closed surface γ:
〈−∆γv, ϕ〉 =I∑
i=1
∫γi
∇γv∇Tγ ϕ = 〈f, ϕ〉, ∀ ϕ ∈ H1
#(γ);
• Weak formulation on parametric domain Ω:∫γi
∇γv∇Tγ ϕ =
∫Ω
∇vDDT ∇ϕT q;
• Strong form of ∆γ on parametric domain Ω:
∆γv =1qdiv
(q∇vg−1
);
• Discrete problem (U Galerkin solution):
U ∈ V(T ) :∫
Γ
∇ΓU∇TΓ V =
∫Γ
FΓ V ∀ V ∈ V(T );
• Elementwise integration by parts formula: for all T ∈ T there holds∫T
∇ΓU∇TΓV =
∫T
−∆ΓU V +∫
∂T
∇ΓU nTT V ∀V ∈ V(T ).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Consistency Error
• Error equation: For all v, w ∈ H1(γ) there holds∫Γ
∇Γv∇TΓ w −
∫γ
∇γv∇Tγ w =
∫γ
∇γvEΓ∇Tγ w,
where EΓ ∈ R(d+1)×(d+1) stands for the following error matrix
EΓ :=1qT(QΓG−1
Γ − qg−1)TT ; (1)
• Properties of of GΓ and QΓ: if the initial mesh T0 satisfiesλΓ0 ≤ 1
6Λ0L3 , then
‖q −QΓ‖L∞(γ) + ‖g −GΓ‖L∞(γ) . λΓ; (2)
• Estimate of EΓ: if λΓ0 ≤ 16Λ0L3 , then
‖EΓ‖L∞( bT ) . λΓ(T ) ∀ T ∈ T .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Consistency Error
• Error equation: For all v, w ∈ H1(γ) there holds∫Γ
∇Γv∇TΓ w −
∫γ
∇γv∇Tγ w =
∫γ
∇γvEΓ∇Tγ w,
where EΓ ∈ R(d+1)×(d+1) stands for the following error matrix
EΓ :=1qT(QΓG−1
Γ − qg−1)TT ; (1)
• Properties of of GΓ and QΓ: if the initial mesh T0 satisfiesλΓ0 ≤ 1
6Λ0L3 , then
‖q −QΓ‖L∞(γ) + ‖g −GΓ‖L∞(γ) . λΓ; (2)
• Estimate of EΓ: if λΓ0 ≤ 16Λ0L3 , then
‖EΓ‖L∞( bT ) . λΓ(T ) ∀ T ∈ T .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Consistency Error
• Error equation: For all v, w ∈ H1(γ) there holds∫Γ
∇Γv∇TΓ w −
∫γ
∇γv∇Tγ w =
∫γ
∇γvEΓ∇Tγ w,
where EΓ ∈ R(d+1)×(d+1) stands for the following error matrix
EΓ :=1qT(QΓG−1
Γ − qg−1)TT ; (1)
• Properties of of GΓ and QΓ: if the initial mesh T0 satisfiesλΓ0 ≤ 1
6Λ0L3 , then
‖q −QΓ‖L∞(γ) + ‖g −GΓ‖L∞(γ) . λΓ; (2)
• Estimate of EΓ: if λΓ0 ≤ 16Λ0L3 , then
‖EΓ‖L∞( bT ) . λΓ(T ) ∀ T ∈ T .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Geometric Error and Estimator
• Element and jump residuals:
RT (V ) := FΓ|T + ∆ΓV |T = FΓ|T ∀T ∈ T ,
JS(V ) := ∇ΓV +|S · n+S +∇ΓV −|S · n−S ∀S ∈ S;
• Error-residual equation:∫
γ∇γ(u− U) · ∇γv = I1 + I2 + I3 with
I1 :=∑T∈T
∫T
FΓ(v − V )−∑S∈S
∫S
JS(U)(v − V ),
I2 :=∫
Γ
∇ΓU · ∇Γv −∫
γ
∇γU · ∇γv =∫
γ
∇γUEΓ∇Tγ v,
I3 :=∫
γ
fv −∫
Γ
FΓv = 0
provided FΓ = qQΓ
f .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Bounds for the Energy Error
• PDE error indicator: for any V ∈ V(T )
ηT (V, T )2 := h2T ‖FΓ‖2L2(T ) +
12
∑S⊂∂T
hT ‖JS(V )‖2L2(S) ∀T ∈ T ;
• Data oscillation: if FΓ stands for the meanvalue of FΓ on T ∈ T , then
oscT (f, T ) := hT ‖FΓ − FΓ‖L2(T ) ∀T ∈ T ; (3)
• A posteriori upper and lower bounds:
‖∇γ(u− U)‖2L2(γ) ≤ C1ηT (U)2 + Λ1λ2Γ,
C2ηT (U)2 ≤ ‖∇γ(u− U)‖2L2(γ) + oscT (f)2 + Λ1λ2Γ.
• Localized upper bound:
‖∇γ(U∗ − U)‖2L2(γ) ≤ C1ηT (U,R)2 + Λ1λΓ(R)2.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Properties of the PDE and Data Oscillation
• Dominance:oscT (f, T ) ≤ ηT (U, T ) ∀T ∈ T ;
• Total error: ET (U, f) :=(‖∇γ(u− U)‖2L2(γ) + oscT (f)2
) 12;
• PDE estimator vs total error: if λ2Γ ≤
C22Λ1
ηT (U)2, then
C4ηT (U) ≤ ET (U, f) ≤ C3ηT (U);
• Reduction of residual estimator: for T∗ ≥ T and ξ = 1− 2−b/d with bbeing the number of bisections (or partitions) per step
ηT∗(U∗)2 ≤ (1 + δ)
(ηT (U)2 − ξηT (U,M)2
)+ (1 + δ−1)
(Λ3‖∇γ(U∗ − U)‖2L2(γ) + Λ2λ
2Γ
);
• Quasi-monotonicity of data oscillation: there exists C5 ≥ 1 such that
oscT∗(f) ≤ C5 oscT (f) ∀!T∗ ≥ T .
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Module ADAPT SURFACE
• Greedy algorithm:
[T +,Γ+] = ADAPT SURFACE(T ,Γ, τ)while M := T ∈ T |λT (T ) > τ 6= ∅
T := REFINE(T ,M)Γ := FT (Ω)
end whilereturn(T ,Γ)
where REFINE(T ,M) refines all elements in the marked set M andkeeps conformity. Upon termination
λΓ+ ≤ τ
• ADAPT SURFACE is t-optimal: there exists a constant C such thatthe set M+ of all the elements marked for refinement in a call toADAPT SURFACE(T ,Γ, τ) satisfies
#M+ ≤ Cτ−1/t,
• Class Bt : |γ|Bt= supN≥1 N t infT ∈TN
λΓ < ∞. If γ ∈ W 1+tdp (Γ0) for
some tp > 1, then γ ∈ Bt and ADAPT SURFACE is t-optimal.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Module ADAPT SURFACE
• Greedy algorithm:
[T +,Γ+] = ADAPT SURFACE(T ,Γ, τ)while M := T ∈ T |λT (T ) > τ 6= ∅
T := REFINE(T ,M)Γ := FT (Ω)
end whilereturn(T ,Γ)
where REFINE(T ,M) refines all elements in the marked set M andkeeps conformity. Upon termination
λΓ+ ≤ τ
• ADAPT SURFACE is t-optimal: there exists a constant C such thatthe set M+ of all the elements marked for refinement in a call toADAPT SURFACE(T ,Γ, τ) satisfies
#M+ ≤ Cτ−1/t,
• Class Bt : |γ|Bt= supN≥1 N t infT ∈TN
λΓ < ∞. If γ ∈ W 1+tdp (Γ0) for
some tp > 1, then γ ∈ Bt and ADAPT SURFACE is t-optimal.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Module ADAPT SURFACE
• Greedy algorithm:
[T +,Γ+] = ADAPT SURFACE(T ,Γ, τ)while M := T ∈ T |λT (T ) > τ 6= ∅
T := REFINE(T ,M)Γ := FT (Ω)
end whilereturn(T ,Γ)
where REFINE(T ,M) refines all elements in the marked set M andkeeps conformity. Upon termination
λΓ+ ≤ τ
• ADAPT SURFACE is t-optimal: there exists a constant C such thatthe set M+ of all the elements marked for refinement in a call toADAPT SURFACE(T ,Γ, τ) satisfies
#M+ ≤ Cτ−1/t,
• Class Bt : |γ|Bt= supN≥1 N t infT ∈TN
λΓ < ∞. If γ ∈ W 1+tdp (Γ0) for
some tp > 1, then γ ∈ Bt and ADAPT SURFACE is t-optimal.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Module ADAPT PDE
• The module ADAPT PDE is the standard adaptive sequence:
[T ,Γ] = ADAPT PDE(T , ε)U = SOLVE(T )ηT (U, T )T∈T = ESTIMATE(T , U)while ηT (U) > ε
M := MARK(T , ηT (U, T )T∈T )T := REFINE(T ,M)Γ := FT (Ω)U = SOLVE(T )ηT (U, T )T∈T = ESTIMATE(T , U)
end whilereturn(T ,Γ)
• Equivalence of estimator and total error: if ω ≤√
C22Λ2
0Λ1then
λΓ ≤ Λ0λΓ+ ≤ ωΛ0ε ≤√
C2
2Λ1ηT (U);
• Complexity of REFINE: #Tk −#T0 ≤ C6
∑k−1j=0 #Mj ∀ k ≥ 1.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Quasi-Orthogonality
• Notation:
ej := ‖∇γ(u− Uj)‖L2(γ), Ej := ‖∇γ(Uj+1 − Uj)‖L2(γ),
ηj := ηTj(Uj), ηj(Mj) := ηTj
(Uj ,Mj), λj := λΓj;
• Quasi-orthogonality: instead of Phytagoras we now have for i = j, j + 1
e2j −
32E2
j − Λ2λ2i ≤ e2
j+1 ≤ e2j −
12E2
j + Λ2λ2i .
This is because
e2j = e2
j+1 + E2j + 2
∫γ
∇γ(u− Uj+1)∇Tγ (Uj+1 − Uj)︸ ︷︷ ︸
.‖f‖L2(γ)λjEj
.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Conditional Contraction Property
Theorem. Let Tj ,Γj , UjJj≥0 be a sequence of meshes, piecewise affine
surfaces and discrete solutions generated by ADAPT PDE (T 0, ε) withinAFEM with tolerance ε, i.e. λ0 ≤ ωε. Assume that the AFEM parameterω satisfies
ω ≤ ω2 :=ξθ2
Λ0
√32Λ2(2Λ3 + 1)
,
where ξ = 1− 2−b/d. Then there exist constants 0 < α < 1 and β > 0such that
e2j+1 + βη2
j+1 ≤ α2(e2j + βη2
j
)∀ 0 ≤ j < J.
Moreover, the number of inner iterates J of ADAPT PDE is uniformlybounded.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 1
Combine, quasi-orthogonality of energy error
e2j+1 ≤ e2
j −12E2
j + Λ2λ2j
with reduction of residual error estimator:
η2j+1 ≤ (1 + δ)
(η2
j − ξηj(Mj)2)
+ (1 + δ−1)(Λ3E
2j + Λ2λ
2j
)to get
e2j+1 + βη2
j+1 ≤ e2j +
(− 1
2+ β(1 + δ−1)Λ3
)E2
j
+ Λ2
(1 + β(1 + δ−1)
)λ2
j + β(1 + δ)(η2
j − ξηj(Mj)2).
Choose β, depending on δ, so that
β(1 + δ−1)Λ3 =12
⇒ β(1 + δ) =δ
2Λ3.
This implies
e2j+1 +βη2
j+1 ≤ e2j +Λ2
(1+β(1+ δ−1)
)λ2
j +β(1+ δ)(η2
j − ξηj(Mj)2).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 1
Combine, quasi-orthogonality of energy error
e2j+1 ≤ e2
j −12E2
j + Λ2λ2j
with reduction of residual error estimator:
η2j+1 ≤ (1 + δ)
(η2
j − ξηj(Mj)2)
+ (1 + δ−1)(Λ3E
2j + Λ2λ
2j
)to get
e2j+1 + βη2
j+1 ≤ e2j +
(− 1
2+ β(1 + δ−1)Λ3
)E2
j
+ Λ2
(1 + β(1 + δ−1)
)λ2
j + β(1 + δ)(η2
j − ξηj(Mj)2).
Choose β, depending on δ, so that
β(1 + δ−1)Λ3 =12
⇒ β(1 + δ) =δ
2Λ3.
This implies
e2j+1 +βη2
j+1 ≤ e2j +Λ2
(1+β(1+ δ−1)
)λ2
j +β(1+ δ)(η2
j − ξηj(Mj)2).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 1
Combine, quasi-orthogonality of energy error
e2j+1 ≤ e2
j −12E2
j + Λ2λ2j
with reduction of residual error estimator:
η2j+1 ≤ (1 + δ)
(η2
j − ξηj(Mj)2)
+ (1 + δ−1)(Λ3E
2j + Λ2λ
2j
)to get
e2j+1 + βη2
j+1 ≤ e2j +
(− 1
2+ β(1 + δ−1)Λ3
)E2
j
+ Λ2
(1 + β(1 + δ−1)
)λ2
j + β(1 + δ)(η2
j − ξηj(Mj)2).
Choose β, depending on δ, so that
β(1 + δ−1)Λ3 =12
⇒ β(1 + δ) =δ
2Λ3.
This implies
e2j+1 +βη2
j+1 ≤ e2j +Λ2
(1+β(1+ δ−1)
)λ2
j +β(1+ δ)(η2
j − ξηj(Mj)2).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 1
Combine, quasi-orthogonality of energy error
e2j+1 ≤ e2
j −12E2
j + Λ2λ2j
with reduction of residual error estimator:
η2j+1 ≤ (1 + δ)
(η2
j − ξηj(Mj)2)
+ (1 + δ−1)(Λ3E
2j + Λ2λ
2j
)to get
e2j+1 + βη2
j+1 ≤ e2j +
(− 1
2+ β(1 + δ−1)Λ3
)E2
j
+ Λ2
(1 + β(1 + δ−1)
)λ2
j + β(1 + δ)(η2
j − ξηj(Mj)2).
Choose β, depending on δ, so that
β(1 + δ−1)Λ3 =12
⇒ β(1 + δ) =δ
2Λ3.
This implies
e2j+1 +βη2
j+1 ≤ e2j +Λ2
(1+β(1+ δ−1)
)λ2
j +β(1+ δ)(η2
j − ξηj(Mj)2).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 2
Use Dorfler marking ηj(Mj) ≥ θηj to deduce
η2j − ξηj(Mj)2 ≤
(1− ξθ2
)η2
j .
Recall λ0 ≤ ωε ≤ ωηj , whence quasi-monotonicity λj ≤ Λ0ωηj implies
e2j+1 + βη2
j+1 ≤ e2j − β(1 + δ)
ξθ2
2η2
j
+ β((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)η2
j .
Employ upper bound
e2j ≤ C1
(ηj + Λ1λ
2j
)≤ C1
(1 + ω2Λ1Λ2
0
)η2
j = C3η2j
to deduce
e2j+1+βη2
j+1 ≤(1− δ
ξθ2
4Λ3C3
)︸ ︷︷ ︸
=α1(δ)
e2j+
((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)︸ ︷︷ ︸
=α2(δ)
η2j
Choose δ = ξθ2
4−2ξθ2 and β = ξθ2
2Λ3(4−ξθ2) to obtain α1, α2 < 1.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 2
Use Dorfler marking ηj(Mj) ≥ θηj to deduce
η2j − ξηj(Mj)2 ≤
(1− ξθ2
)η2
j .
Recall λ0 ≤ ωε ≤ ωηj , whence quasi-monotonicity λj ≤ Λ0ωηj implies
e2j+1 + βη2
j+1 ≤ e2j − β(1 + δ)
ξθ2
2η2
j
+ β((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)η2
j .
Employ upper bound
e2j ≤ C1
(ηj + Λ1λ
2j
)≤ C1
(1 + ω2Λ1Λ2
0
)η2
j = C3η2j
to deduce
e2j+1+βη2
j+1 ≤(1− δ
ξθ2
4Λ3C3
)︸ ︷︷ ︸
=α1(δ)
e2j+
((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)︸ ︷︷ ︸
=α2(δ)
η2j
Choose δ = ξθ2
4−2ξθ2 and β = ξθ2
2Λ3(4−ξθ2) to obtain α1, α2 < 1.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 2
Use Dorfler marking ηj(Mj) ≥ θηj to deduce
η2j − ξηj(Mj)2 ≤
(1− ξθ2
)η2
j .
Recall λ0 ≤ ωε ≤ ωηj , whence quasi-monotonicity λj ≤ Λ0ωηj implies
e2j+1 + βη2
j+1 ≤ e2j − β(1 + δ)
ξθ2
2η2
j
+ β((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)η2
j .
Employ upper bound
e2j ≤ C1
(ηj + Λ1λ
2j
)≤ C1
(1 + ω2Λ1Λ2
0
)η2
j = C3η2j
to deduce
e2j+1+βη2
j+1 ≤(1− δ
ξθ2
4Λ3C3
)︸ ︷︷ ︸
=α1(δ)
e2j+
((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)︸ ︷︷ ︸
=α2(δ)
η2j
Choose δ = ξθ2
4−2ξθ2 and β = ξθ2
2Λ3(4−ξθ2) to obtain α1, α2 < 1.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 2
Use Dorfler marking ηj(Mj) ≥ θηj to deduce
η2j − ξηj(Mj)2 ≤
(1− ξθ2
)η2
j .
Recall λ0 ≤ ωε ≤ ωηj , whence quasi-monotonicity λj ≤ Λ0ωηj implies
e2j+1 + βη2
j+1 ≤ e2j − β(1 + δ)
ξθ2
2η2
j
+ β((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)η2
j .
Employ upper bound
e2j ≤ C1
(ηj + Λ1λ
2j
)≤ C1
(1 + ω2Λ1Λ2
0
)η2
j = C3η2j
to deduce
e2j+1+βη2
j+1 ≤(1− δ
ξθ2
4Λ3C3
)︸ ︷︷ ︸
=α1(δ)
e2j+
((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)︸ ︷︷ ︸
=α2(δ)
η2j
Choose δ = ξθ2
4−2ξθ2 and β = ξθ2
2Λ3(4−ξθ2) to obtain α1, α2 < 1.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Proof of Contraction: Step 2
Use Dorfler marking ηj(Mj) ≥ θηj to deduce
η2j − ξηj(Mj)2 ≤
(1− ξθ2
)η2
j .
Recall λ0 ≤ ωε ≤ ωηj , whence quasi-monotonicity λj ≤ Λ0ωηj implies
e2j+1 + βη2
j+1 ≤ e2j − β(1 + δ)
ξθ2
2η2
j
+ β((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)η2
j .
Employ upper bound
e2j ≤ C1
(ηj + Λ1λ
2j
)≤ C1
(1 + ω2Λ1Λ2
0
)η2
j = C3η2j
to deduce
e2j+1+βη2
j+1 ≤(1− δ
ξθ2
4Λ3C3
)︸ ︷︷ ︸
=α1(δ)
e2j+
((1 + δ)
(1− ξθ2
2
)+ Λ2
(1 +
12Λ3
)Λ20ω
2
β
)︸ ︷︷ ︸
=α2(δ)
η2j
Choose δ = ξθ2
4−2ξθ2 and β = ξθ2
2Λ3(4−ξθ2) to obtain α1, α2 < 1.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Outline
Motivation: Geometric PDE
AFEM: The Role of ω
Parametric Surfaces
The Laplace-Beltrami Operator
A Posteriori Error Analysis
AFEM: Design and Properties
Conditional Contraction Property
Optimal Cardinality
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Approximation Classes
• As(γ): Class for (u, f)
|u, f |As := supn≥1
Ns inf
T ∈TN
infV ∈V(T )
(‖∇γ(u− V )‖L2(γ) + oscT (f)
).
Equivalently, given ε there exists a mesh Tε ∈ T(T0) with Tε ≥ T0 anda discrete function Vε ∈ V(Tε) so that
‖∇γ(u− Vε)‖L2(γ) + oscTε(f) ≤ ε, #Tε −#T0 ≤ |u, f |1s
Asε−
1s ;
• Bt: Class for γ
|γ|Bt := supN≥1
(N t inf
T ∈TN
λΓ
)< ∞.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Convergence Rates
Theorem. Let γ ∈ Bt and (u, f) ∈ As(γ) for some 0 < t, s ≤ n/d. Letε0 ≤ ε∗ be the initial tolerance, and the parameters θ, ω satisfy
0 < θ ≤ θ∗, 0 < ω ≤ ω∗.
where θ∗, ω∗ are explicit. Let the procedure MARK select sets withminimal cardinality, and the procedure ADAPT SURFACE be t-optimalon the surface γ. Let Γk, Tk, Ukk≥0 a sequence of approximatesurfaces, meshes and discrete solution generated by the outer loop ofAFEM.Then there exists a constant C, depending on the Lipschitz constant L ofγ, ‖f‖L2(γ), the refinement depth b, the initial triangulation T0, andAFEM parameters (θ, ω, ρ) such that
‖∇γ(u−Uk)‖L2(γ)+oscTk(f)+ω−1λΓk
≤ C(|u, f |
rs
As+ω−r|γ|
rt
Bt
)(#Tk−#T0
)−r
with r = mins, t.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Ingredients of the Proof
• Localized upper bound (to the refined set)
• Minimality of set M in Dorfler marking
• Explicit restriction of Dorfler parameter θ < θ∗ < 1
• Explicit restriction of surface parameter ω ≤ ω∗ < 1
• Conditional contraction property of PDE
• Complexity of REFINE (Binev-Dahmen-DeVore (d = 2), Stevenson(d > 2), for conforming meshes, and Bonito-Nochetto fornon-conforming meshes (d ≥ 2)).
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto
Outline Motivation AFEM I Surfaces Laplace-Beltrami A Posteriori AFEM II Contraction Cardinality
Greedy Algorithm
• Sobolev numbers:
sob(W 1∞) = 1− d
∞= 1 < sob(W 1+td
p ) = 1 + td− d
p⇒ tp > 1.
• Sobolev embedding: W 1∞ ⊂ W 1+td
p .
Theorem. Let γ be piecewise of class W 1+tdp (Γ0), with tp > 1, t ≤ 1/d,
and globally of class W 1∞. Then [T +,Γ+] = ADAPT SURFACE(T ,Γ, τ)
terminates in a finite number of steps and the set M+ of markedelements satisfies
#M+ ≤ C|γ|1/t
W 1+tdp (Γ0)
τ−1/t,
where |γ|W 1+tdp (Γ0)
=(∑I
i=1 |X i|pW 1+tp
p (Ω)
)1/p
. Moreover, γ ∈ Bt and
|γ|Bt . |γ|W 1+tdp (Γ0)
.
Adaptive Finite Element Methods Lecture 5: The Laplace-Beltrami Operator Ricardo H. Nochetto