Adaptive Finite Element Methods Lecture 5: The Laplace ... · PDF fileOutline Motivation AFEM...

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

Parametric Surfaces

Optimal 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:

Seek u ∈ H10 (γ) :

∇γu · ∇γv =∫

f v, ∀v ∈ H10 (γ).∫

∇γu·∇γv =∫

bK ∇(uFk)T G−1K ∇(vFK)

√det(GK), GK = DFT

Biomembranes: Modeling (w. A. Bonito and M.S. Pauletti)

• Bending (Willmore) energy: J(Γ) = 12

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

Biomembrane: Geometric vs Fluid Red Blood Cell

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

|divΓ ν − δ divΓ n|2dσ +λ

|∇Γn|2dσ

µ(|n|2 − 1)dσ +12ε

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)

Coupling of Director Fields and Flexible Surfaces: Simulations

Cone-like structure near positive degree-one defects pointing outwards ⇒stomatocyte shape

Outline

Parametric Surfaces

Optimal 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).

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).

Asymptotics: Role of ω• γ is the graph of class C1,α given by

z(x, y) =(0.75− x2 − y2

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.

Case α = 3/5

10−3

10−2

10−1

η+λ/

ω = 0.1ω = 1ω = 10

N−0.5

10−3

10−2

10−1

ω = 0.1ω = 1ω = 10

N−0.5

ηk + λk/ω (left) and ηk + λk (right) versus DOF for ω = 0.1, 1, 10.

10−4

10−3

10−2

10−1

N−0.5

η+λ/ωηλ/ω

10−4

10−3

10−2

10−1

N−0.5

η+λ/ωηλ/ω

10−4

10−3

10−2

10−1

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).

Case α = 2/5

10−3

10−2

10−1

η+λ/

ω = 0.1ω = 1ω = 10

N−0.4

10−2

10−1

ω = 0.1ω = 1ω = 10

N−0.4

ηk + λk/ω (left) and ηk + λk (right) versus DOF for ω = 0.1, 1, 10.

10−4

10−3

10−2

10−1

N−0.5

η+λ/ωηλ/ωN−0.4

10−4

10−3

10−2

10−1

N−0.5

10−4

10−3

10−2

10−1

N−0.5

ηk, λk/ω and ηk + λk/ω for ω = 0.1 (left) ω = 1 (middle) and ω = 10 (right).

Outline

Parametric Surfaces

Optimal 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

• 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 .

• Γ0 =⋃I

i=1 Γi0, P i

0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi

0(Γi0) ∈ C1(Γi

Γi0 = F i

0(Ω);

• Γ0 =⋃I

i=1 Γi0, P i

0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi

0(Γi0) ∈ C1(Γi

Γi0 = F i

0(Ω);

• Γ0 =⋃I

i=1 Γi0, P i

0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi

0(Γi0) ∈ C1(Γi

Γi0 = F i

0(Ω);

• Γ0 =⋃I

i=1 Γi0, P i

0 : Γi0 → Rd+1, 1 ≤ i ≤ I; Γi

0(Γi0) ∈ C1(Γi

Γi0 = F i

0(Ω);

Parametric Domain Ω, Macro Element Γi0, and Surface 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.

Interpolation of Parametric Surfaces

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.

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

• 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 .

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

• Mesh T :=T = FT (T ) | T ∈ T (Ω)

V(T ) :=

λΓ(T );

Outline

Parametric Surfaces

Optimal 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 :=√

• Discrete quantities: GΓ := TTΓ TΓ, QΓ :=

√detGΓ.

• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT

]∈ R(d+1)×(d+1);

g =(gγ,ij

)1≤i,j≤d

:=(∂iX T ∂jX

)1≤i,j≤d

= TT T;

• ∇v = ∇γv T;

∇γv = ∇γv T D =[∇v, 0

]D = ∇v D,

√detGΓ.

• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT

]∈ R(d+1)×(d+1);

g =(gγ,ij

)1≤i,j≤d

:=(∂iX T ∂jX

)1≤i,j≤d

= TT T;

• ∇v = ∇γv T;

∇γv = ∇γv T D =[∇v, 0

]D = ∇v D,

√detGΓ.

• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT

]∈ R(d+1)×(d+1);

g =(gγ,ij

)1≤i,j≤d

:=(∂iX T ∂jX

)1≤i,j≤d

= TT T;

• ∇v = ∇γv T;

∇γv = ∇γv T D =[∇v, 0

]D = ∇v D,

√detGΓ.

• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT

]∈ R(d+1)×(d+1);

g =(gγ,ij

)1≤i,j≤d

:=(∂iX T ∂jX

)1≤i,j≤d

= TT T;

• ∇v = ∇γv T;

∇γv = ∇γv T D =[∇v, 0

]D = ∇v D,

√detGΓ.

• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT

]∈ R(d+1)×(d+1);

g =(gγ,ij

)1≤i,j≤d

:=(∂iX T ∂jX

)1≤i,j≤d

= TT T;

• ∇v = ∇γv T;

∇γv = ∇γv T D =[∇v, 0

]D = ∇v D,

√detGΓ.

• T := [∂1X , . . . , ∂dX ] ∈ R(d+1)×d, T :=[T,νT

]∈ R(d+1)×(d+1);

g =(gγ,ij

)1≤i,j≤d

:=(∂iX T ∂jX

)1≤i,j≤d

= TT T;

• ∇v = ∇γv T;

∇γv = ∇γv T D =[∇v, 0

]D = ∇v D,

√detGΓ.

Variational Formulation and Galerkin Method

• Weak formulation on a closed surface γ:

〈−∆γv, ϕ〉 =I∑

∫γi

∇γv∇Tγ ϕ = 〈f, ϕ〉, ∀ ϕ ∈ H1

#(γ);

• Weak formulation on parametric domain Ω:∫γi

∇γv∇Tγ ϕ =

∇vDDT ∇ϕT q;

• Strong form of ∆γ on parametric domain Ω:

∆γv =1qdiv

(q∇vg−1

• Discrete problem (U Galerkin solution):

U ∈ V(T ) :∫

∇ΓU∇TΓ V =

FΓ V ∀ V ∈ V(T );

• Elementwise integration by parts formula: for all T ∈ T there holds∫T

∇ΓU∇TΓV =

−∆ΓU V +∫

∇ΓU nTT V ∀V ∈ V(T ).

Outline

Parametric Surfaces

Optimal Cardinality

Consistency Error

• Error equation: For all v, w ∈ H1(γ) there holds∫Γ

∇Γv∇TΓ w −

∇γv∇Tγ w =

∇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 .

Consistency Error

∇Γv∇TΓ w −

∇γv∇Tγ w =

∇γvEΓ∇Tγ w,

EΓ :=1qT(QΓG−1

Γ − qg−1)TT ; (1)

6Λ0L3 , then

‖q −QΓ‖L∞(γ) + ‖g −GΓ‖L∞(γ) . λΓ; (2)

‖EΓ‖L∞( bT ) . λΓ(T ) ∀ T ∈ T .

Consistency Error

∇Γv∇TΓ w −

∇γv∇Tγ w =

∇γvEΓ∇Tγ w,

EΓ :=1qT(QΓG−1

Γ − qg−1)TT ; (1)

6Λ0L3 , then

‖q −QΓ‖L∞(γ) + ‖g −GΓ‖L∞(γ) . λΓ; (2)

‖EΓ‖L∞( bT ) . λΓ(T ) ∀ T ∈ T .

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

FΓ(v − V )−∑S∈S

JS(U)(v − V ),

I2 :=∫

∇ΓU · ∇Γv −∫

∇γU · ∇γv =∫

∇γUEΓ∇Tγ v,

I3 :=∫

fv −∫

FΓv = 0

provided FΓ = qQΓ

Bounds for the Energy Error

• PDE error indicator: for any V ∈ V(T )

ηT (V, T )2 := h2T ‖FΓ‖2L2(T ) +

∑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.

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

• 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

• Quasi-monotonicity of data oscillation: there exists C5 ≥ 1 such that

oscT∗(f) ≤ C5 oscT (f) ∀!T∗ ≥ T .

Outline

Parametric Surfaces

Optimal 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.

λΓ+ ≤ τ

#M+ ≤ Cτ−1/t,

λΓ+ ≤ τ

#M+ ≤ Cτ−1/t,

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)

• Equivalence of estimator and total error: if ω ≤√

C22Λ2

0Λ1then

λΓ ≤ Λ0λΓ+ ≤ ωΛ0ε ≤√

2Λ1ηT (U);

• Complexity of REFINE: #Tk −#T0 ≤ C6

∑k−1j=0 #Mj ∀ k ≥ 1.

Outline

Parametric Surfaces

Optimal 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 −

j − Λ2λ2i ≤ e2

j+1 ≤ e2j −

j + Λ2λ2i .

This is because

e2j = e2

j+1 + E2j + 2

∇γ(u− Uj+1)∇Tγ (Uj+1 − Uj)︸︷︷︸

.‖f‖L2(γ)λjEj

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

√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

)∀ 0 ≤ j < J.

Moreover, the number of inner iterates J of ADAPT PDE is uniformlybounded.

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)

j − ξηj(Mj)2)

+ (1 + δ−1)(Λ3E

2j + Λ2λ

)to get

e2j+1 + βη2

j+1 ≤ e2j +

(− 1

2+ β(1 + δ−1)Λ3

(1 + β(1 + δ−1)

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

This implies

e2j+1 +βη2

j+1 ≤ e2j +Λ2

(1+β(1+ δ−1)

j +β(1+ δ)(η2

j − ξηj(Mj)2).

e2j+1 ≤ e2

j −12E2

j + Λ2λ2j

η2j+1 ≤ (1 + δ)

j − ξηj(Mj)2)

+ (1 + δ−1)(Λ3E

2j + Λ2λ

)to get

e2j+1 + βη2

j+1 ≤ e2j +

(− 1

2+ β(1 + δ−1)Λ3

(1 + β(1 + δ−1)

j + β(1 + δ)(η2

j − ξηj(Mj)2).

β(1 + δ−1)Λ3 =12

⇒ β(1 + δ) =δ

This implies

e2j+1 +βη2

j+1 ≤ e2j +Λ2

(1+β(1+ δ−1)

j +β(1+ δ)(η2

j − ξηj(Mj)2).

e2j+1 ≤ e2

j −12E2

j + Λ2λ2j

η2j+1 ≤ (1 + δ)

j − ξηj(Mj)2)

+ (1 + δ−1)(Λ3E

2j + Λ2λ

)to get

e2j+1 + βη2

j+1 ≤ e2j +

(− 1

2+ β(1 + δ−1)Λ3

(1 + β(1 + δ−1)

j + β(1 + δ)(η2

j − ξηj(Mj)2).

β(1 + δ−1)Λ3 =12

⇒ β(1 + δ) =δ

This implies

e2j+1 +βη2

j+1 ≤ e2j +Λ2

(1+β(1+ δ−1)

j +β(1+ δ)(η2

j − ξηj(Mj)2).

e2j+1 ≤ e2

j −12E2

j + Λ2λ2j

η2j+1 ≤ (1 + δ)

j − ξηj(Mj)2)

+ (1 + δ−1)(Λ3E

2j + Λ2λ

)to get

e2j+1 + βη2

j+1 ≤ e2j +

(− 1

2+ β(1 + δ−1)Λ3

(1 + β(1 + δ−1)

j + β(1 + δ)(η2

j − ξηj(Mj)2).

β(1 + δ−1)Λ3 =12

⇒ β(1 + δ) =δ

This implies

e2j+1 +βη2

j+1 ≤ e2j +Λ2

(1+β(1+ δ−1)

j +β(1+ δ)(η2

j − ξηj(Mj)2).

Use Dorfler marking ηj(Mj) ≥ θηj to deduce

η2j − ξηj(Mj)2 ≤

(1− ξθ2

Recall λ0 ≤ ωε ≤ ωηj , whence quasi-monotonicity λj ≤ Λ0ωηj implies

e2j+1 + βη2

j+1 ≤ e2j − β(1 + δ)

+ β((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

Employ upper bound

e2j ≤ C1

(ηj + Λ1λ

)≤ C1

(1 + ω2Λ1Λ2

j = C3η2j

to deduce

e2j+1+βη2

j+1 ≤(1− δ

4Λ3C3

)︸︷︷︸

=α1(δ)

((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

)︸︷︷︸

=α2(δ)

Choose δ = ξθ2

4−2ξθ2 and β = ξθ2

2Λ3(4−ξθ2) to obtain α1, α2 < 1.

(1− ξθ2

e2j+1 + βη2

j+1 ≤ e2j − β(1 + δ)

+ β((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

Employ upper bound

e2j ≤ C1

(ηj + Λ1λ

)≤ C1

(1 + ω2Λ1Λ2

j = C3η2j

to deduce

e2j+1+βη2

j+1 ≤(1− δ

4Λ3C3

)︸︷︷︸

=α1(δ)

((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

)︸︷︷︸

=α2(δ)

Choose δ = ξθ2

(1− ξθ2

e2j+1 + βη2

j+1 ≤ e2j − β(1 + δ)

+ β((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

Employ upper bound

e2j ≤ C1

(ηj + Λ1λ

)≤ C1

(1 + ω2Λ1Λ2

j = C3η2j

to deduce

e2j+1+βη2

j+1 ≤(1− δ

4Λ3C3

)︸︷︷︸

=α1(δ)

((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

)︸︷︷︸

=α2(δ)

Choose δ = ξθ2

(1− ξθ2

e2j+1 + βη2

j+1 ≤ e2j − β(1 + δ)

+ β((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

Employ upper bound

e2j ≤ C1

(ηj + Λ1λ

)≤ C1

(1 + ω2Λ1Λ2

j = C3η2j

to deduce

e2j+1+βη2

j+1 ≤(1− δ

4Λ3C3

)︸︷︷︸

=α1(δ)

((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

)︸︷︷︸

=α2(δ)

Choose δ = ξθ2

(1− ξθ2

e2j+1 + βη2

j+1 ≤ e2j − β(1 + δ)

+ β((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

Employ upper bound

e2j ≤ C1

(ηj + Λ1λ

)≤ C1

(1 + ω2Λ1Λ2

j = C3η2j

to deduce

e2j+1+βη2

j+1 ≤(1− δ

4Λ3C3

)︸︷︷︸

=α1(δ)

((1 + δ)

(1− ξθ2

)+ Λ2

)Λ20ω

)︸︷︷︸

=α2(δ)

Choose δ = ξθ2

Outline

Parametric Surfaces

Optimal 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−

• Bt: Class for γ

|γ|Bt := supN≥1

(N t inf

T ∈TN

)< ∞.

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 |

As+ω−r|γ|

)(#Tk−#T0

with r = mins, t.

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)).

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

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)

. Moreover, γ ∈ Bt and

|γ|Bt . |γ|W 1+tdp (Γ0)

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