adaptive finite element methods lecture 5: extensions ii

Adaptive Finite Element MethodsLecture 5: Extensions II

Ricardo H. Nochetto

Department of Mathematics andInstitute for Physical Science and Technology

University of Maryland, USA

www.math.umd.edu/˜rhn

School - Fundamentals and Practice of Finite ElementsRoscoff, France, April 16-20, 2018

Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines

Outline

Interior Penalty Discontinuous Galerkin Methods (DG) (w. A. Bonito)

Hybridizable Discontinuous Galerkin Methods (HDG) (w. B. Cockburn and W.Zhang)

Quasi-Orthogonality Property (C. Carstenson, M. Feischl, M. Page, and D.Praetorius)

Adaptive Hierarchical B-Splines (w. P. Morin and M.S. Pauletti)Hierarchical BasisA Posteriori Error AnalysisContraction PropertyOptimality

Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto


Extension to Discontinuous Galerkin (DG) Methods

Consider model problem −div(A∇u) = f in d dimensions. Given anonconforming mesh T , and a space V(T ) of discontinuous pw polynomials ofdegree ≤ n, let UT ∈ V(T ) satisfy for all V ∈ V(T )

BT (UT , V ) : = 〈A∇UT ,∇V 〉T − 〈A∇UT , [[V ]]〉Σ− 〈A∇V , [[UT ]]〉Σ + δ〈h−1 [[UT ]] , [[V ]]〉Σ = 〈f, V 〉T

I 〈·, ·〉T elementwise L2-scalar product over TI · mean value operator over set of interelement boundaries Σ

I [[·]] jump operator over set of interelement boundaries Σ

I δ > 0 penalization parameter

I Energy space E(T ) with norm |||v|||2T = ‖A1/2∇v‖2T + δ‖h1/2 [[v]] ‖2ΣI Refs: Arnold, Brezzi, Cockburn, Marini, Suli, Ainsworth, Riviere, etc

Karakashian, Pascal, Hoppe, Kanschat, Warburton.



Preliminaries

• Lifting operator: LT : E(T )→ V (T )d is defined by∫Ω

LT (v) ·AW = 〈[[v]] , AW〉Σ ∀W ∈ V (T )d.

• Discrete problem: UT satisfies BT (UT , V ) = 〈f, V 〉T with

BT (v, w) := 〈A∇v,∇w〉T − 〈LT (w),A∇v〉T− 〈LT (v),A∇w〉T + δ〈h−1 [[v]] , [[w]]〉Σ.

• Coercivity and continuity of BT in V (T ) with norm |||·|||T if δ ≥ δ0.

• Partial consistency: BT (u, v) = 〈f, v〉T ∀v ∈ H10 (Ω).

• Galerkin orthogonality: BT (u− UT ), V ) = 0 ∀ V ∈ V (T ) ∩H10 (Ω).

• Minimal regularity: u ∈ H10 (Ω) and A∇u ∈ L2(Ω).



Preliminaries (continued)

• Orthogonal decomposition: V (T ) = V 0(T )⊕ V ⊥(T ) w.r.t BT (·, ·) withV 0(T ) := V (T ) ∩H1

0 (Ω).

• Continuous Galerkin Solution: U0 ∈ V 0(T ) solves

U0 ∈ V 0(T ) : BT (U0, V 0) = 〈f, V 0〉T ∀V 0 ∈ V 0(T ).

• Nonconforming component: |||V ⊥|||T . δ12 ‖h−

12 [[V ]] ‖Σ ∀ V ∈ V (T )

• Estimator: E2T (U, T ) = ‖h(div(A∇U) + f)‖2T + ‖h1/2 [[A∇U ]] ‖2Σ

• First upper bound: |||u− U |||2Ω . E2T (U, T ) + δ‖h−

12 [[U ]] ‖Σ

• Jump control: δ‖h−1/2 [[V ]] ‖Σ ≤ ET (U, T ) ∀ δ ≥ δ1.

• Upper bound: (Karakashian-Pascal’07) |||u− U |||Ω 4 ET (U, T ).



Preliminaries (continued)

• Quasi-localized upper bound: For all δ ≥ δ1

|||U0∗ − U |||2T . E2

T (U,RT→T∗) + δ−1E2T (U, T ).

• Global lower bound: E2T (U, T ) . |||u− U |||2T + osc2

T (U, T ).

• Quasi Pythagoras: Let T∗ ≥ T be consecutive meshes created by REFINEand U∗ ∈ V (T∗), U ∈ V (T ) be the dG solutions. Then

BT∗(u− U∗, u− U∗) ≤ (1 + ε)BT (u− U, u− U)− C3‖∇(U∗ − U)‖2T∗

+C4

εδ

(E2T (U, T ) + E2

T∗(U∗, T∗)).

Theorem (Contraction for dG). For Dorfler marking with θ ∈ (0, 1), thereexist α = α < 1, γ > 0 and δ2 > 0 such that for all δ ≥ δ2

E2k+1 := |||u− Uk+1|||2k+1 + γE2

k+1(Uk+1, Tk+1) ≤ αE2k



Quasi-Optimal Cardinality of dG on Nonconforming Meshes

σN (u;A, f) := infT ∈TN

infV∈V(T )

(|||u− V |||T + oscT (V, T )

)As :=

(u,A, f) : |u,A, f |s := sup

N≥0NsσN (u;A, f) <∞

.

Proposition The approximation classes As and A0s for dG and cG are the same.

Theorem Let d > 1, polynomial degree n ≥ 1, and 0 < θ∗ < 1, δ∗ > 1 beexplicit parameters. Assume

I minimal Dorfler marking with 0 < θ < θ∗;

I suitable initial labeling of T0 for bisection;

I (u,A, f) ∈ As for 0 < s ≤ d/n.

Then DG-AFEM produces a sequence Tk, Uk∞k=0 of nonconformingadmissible meshes and discrete solutions so that for δ ≥ δ∗

|||Uk − u|||Tk + osck(Uk, Tk) 4 |u,A, f |s(#Tk −#T0

)−1/s.



Conclusions for dG

I General data A, f and Ω.

I Minimal regularity: u ∈ H10 (Ω),div(A∇u) ∈ L2(Ω) of u (via the lifting

operator).

I Only ONE bisection (or partition) for T ∈Mk (no interior node property).

I The non-monotone jump term ‖h−1 [[UT ]] ‖Σ is the trouble-maker but it iscontrolled by the estimator. It does not enter in the upper bound.

I The analysis relies on cG and the penalty parameter δ large: theapproximation classes As for dG and A0

s for cG coincide.



Outline







The HDG Method

Write the Laplace equation −∆u = f as a first order system:

q +∇u = 0, divq = f.

Given a sequence of conforming partitions Tk and p ≥ 0,

Vk = v ∈ L2(Ω) : v|T ∈ Pp(T ) ∀ T ∈ Tk,

Wk = w ∈ L2(Ω) : w|T ∈ Pp(T ) ∀ T ∈ Tk,

Mk = m ∈ L2(Ek) : v|e ∈ Pp(e) ∀ e ∈ Ek.

HDG Method: seek (qk, uk, uk) ∈ Vk ×Wk ×Mk such that

(qk,v)Tk − (uk, divv)Tk = −〈uk,v · n〉∂Tk ,−(qk,∇w)Tk + 〈qk · n, w〉∂Tk = (f, w)Tk

qk = qk + τ(uk − uk)n on ∂Tk,

for all (v, w) ∈ V×W; τ > 0 is a stabilization (not penalty) parameter.



Error Estimators

Let auxiliary spaces of degree p− 1

Vk = v ∈ L2(Ω) : v|T ∈ Pp−1(T ) ∀ T ∈ Tk,

Wk = w ∈ L2(Ω) : w|T ∈ Pp−1(T ) ∀ T ∈ Tk,

Estimator for q: Let

ζ2(f,qk, T ) := ζ2curl(qk, T ) + ζ2

div(f,qk, ∂T )

with

ζ2curl(qk, T ) : = h2

T ‖curl qk‖2T + hT ‖ [[qk]]t ‖2∂T ,

ζ2div(f,qk, T ) : = τ2

Th2T ‖qk − PVk

qk‖2T + h2T ‖f − PWk

f‖2T≈ hT ‖(qk − qk) · n‖2∂T + h2

T ‖f − fT ‖2T .

I ζdiv(qk,Ω) measures the lack of H(div; Ω) conformity

I ζ2curl(qk,Ω) measure the deviation of qk from being a gradient.



Main Results

Lack of monotonicity: ‖q− qk‖Ω is not monotone and a lower bound

ζdiv(f,qk, Tk) . ‖q− qk‖Ω

is not valid.

Quasi-error: 2-parameter quantity

Eβ,γ(qk, f, Tk)2 := ‖q− qk‖2Ω + βζ2div(f,qk, Tk) + γζ2

curl(f,qk, Tk).

Theorem (contraction). There exists β, γ > 0 and 0 < α < 1 such that

Eβ,γ(qk+1, f, Tk+1)2 ≤ αEβ,γ(qk, f, Tk)2.

Theorem (rate optimality). If (q, f) ∈ As then

‖q− qk‖Ω . (#Tk −#T0)−s.



Quasi-Orthogonality Property

Let Tk+1 ≥ Tk be two nested meshes. For any ε ∈ (0, 1/2), we have

‖q− qk+1‖2Ω +1

1− εΘk ≤ ‖q− qk‖2Ω,

where

Θk := ‖qk+1 − qk‖2Ω −1

εQk − ε‖q− qk‖2Ω.

The quantity

Qk := infv∈V 0

k

‖qk+1 − qk − v‖2Ω

measure the deviation of qk+1 − qk from divergence-free because V 0k is the

subspace of Vk of divergence-free functions.



Outline







Basic Assumptions for Adaptivity

• (A1) Stability on non-refined elements: Let T∗ ≥ T andV ∈ V (T ), V∗ ∈ V (T∗) satisfy for all T ∈ T ∩ T∗∣∣∣ηT (V, T )− ηT∗(V∗, T )

∣∣∣ ≤ Cstab|||V − V∗|||T• (A2) Reduction property on refined elements: Let 0 < ρred < 1, T∗ ≥ T

and V ∈ V (T ), V∗ ∈ V (T∗) satisfy for all T ∈ T \ T∗

ηT∗(V∗, T ) ≤ ρred ηT (V, T ) + Cred|||V − V∗|||T

• (A3) Quasi-orthogonality property: The output Uj of AFEM satisfies forall j,N ∈ N

j+N∑k=j

|||Uk+1 − Uk|||2Ω − ε|||u− Uk|||2Ω ≤ Cqo ηTj (Uj)2.

• (A4) Discrete reliability: Let T∗ ≥ T and U ∈∈ V (T ), U∗ ∈ V (T∗) be theGalerkin solutions and satisfy in the refined set R = T \ T∗

|||U − U∗|||Ω ≤ Cdref ηT (U,R).



Quasi-Orthogonality Property Revisited

Assume the usual quasi-orthogonality property: there is ε > 0 such that

(1− ε)|||u− Uk+1|||2Ω ≤ (1 + ε)|||u− Uk|||2Ω − C1|||Uk − Uk+1|||2Ω

or equivalently

C1|||Uk − Uk+1|||2Ω − 2ε|||u− Uk|||2Ω ≤ (1− ε)(|||u− Uk|||2Ω − |||u− Uk+1|||2Ω

).

Add over k from j to j +N and use telescopic cancellation to get

j+N∑k=j

C1|||Uk − Uk+1|||2Ω − 2ε|||u− Uk|||2Ω

≤ (1− ε)j+N∑k=j

(|||u− Uk|||2Ω − |||u− Uk+1|||2Ω

)≤ (1− ε)

(|||u− Uj |||2Ω − |||u− Uj+N+1|||2Ω

)≤ C2ηTj (Uj)

2.

This gives the quasi-orthogonality property upon dividing by C1.



Linear Convergence without Contraction Property

• Estimator reduction: (A1) and (A2) imply the existence of 0 < ρ1 < 1 suchthat

ηk+1(Uk+1)2 ≤ ρ1ηk(Uk)2 + C1|||Uk+1 − Uk|||2Ω.

• Uniform summability: Add from j ∈ N to j +N − 1 ∈ N to obtain

j+N∑k=j+1

ηk(Uk)2 ≤j+N∑k=j+1

ρ1 ηk−1(Uk−1)2 + C1|||Uk − Uk−1|||2Ω

Let ρ2 = ρ1 + ν < 1, add and substract νηk−1(Uk−1)2 and use reliability|||u− Uk−1|||2Ω ≤ Cηk−1(Uk−1)2 to obtain

j+N∑k=j+1


ρ2 ηk−1(Uk−1)2 + C1

(|||Uk − Uk−1|||2Ω − νC|||u− Uk−1|||2Ω

).

Apply quasi-orthogonality (A3) with ε = νC to arrive at

j+N∑k=j+1


ρ2 ηk−1(Uk−1)2 + C2ηj(Uj)2.



Linear Convergence without Contraction Property (continued)

• Uniform summability: Reorder terms to get

(1− ρ2)

j+N∑k=j+1

ηk(Uk)2 ≤ (ρ2 + C2)ηj(Uj)2 ⇒

j+N∑k=j+1

ηk(Uk)2 ≤ ρ2 + C2

1− ρ2︸︷︷︸=C3

ηj(Uj)2.

• Linear convergence: Take N →∞, divide by C3 and add∑∞k=j+1 ηk(Uk)2

(1+C−13 )

∞∑k=j+1

ηk(Uk)2 ≤∞∑k=j

ηk(Uk)2 ⇒∞∑

k=j+1

ηk(Uk)2 ≤ C3

1 + C3︸︷︷︸=ρ3

∞∑k=j

ηk(Uk)2

Applying this inequality recursively yields

ηj+i(Uj+i)2 ≤

∞∑k=j+i

ηk(Uk)2 ≤ ρ3

∞∑k=j+i−1

ηk(Uk)2

≤ ρi−13

∞∑k=j+1

ηk(Uk)2 ≤ C3

ρ3ρi3ηj(Uj)

2.

No contraction between consecutive iterates!



Quasi-Optimal Cardinality

• Optimality of Dorfler marking: for all 0 < µ < 1 there exists 0 < θ0 < 1such that for T∗ ≥ T , R = T \ T∗ (refined set) and all 0 < θ ≤ θ0

ηT∗(U∗)2 ≤ µηT (U)2 ⇒ θηT (U)2 ≤ ηT (U,R)2.

• Optimal estimator decay: written in terms of the error estimator

|u|Bs = supN∈N

minT ∈TN

NsηT (U) <∞ ⇒ ηT (U) ≤ |u|BsN−s

for the best possible mesh T with N elements more than T0.

• Quasi-optimality of AFEM: If assumptions (A1)-(A4) hold, then thereexists 0 < θ0 < 1 suffiiciently small such that for all Dorfler parametersθ ≤ θ0 the iterates of AFEM satisfy

|u|Bs . supk∈N

((#Tk −#T0)sηk(Uk)

). |u|Bs .



Applications

• Nonconforming Crouzeix-Raviart elements: Carstensen and Hoppe ’06.

• Mixed FEM: Carstensen and Hoppe ’06; Chen, Holst and Xu ’09.

• Boundary element methods: Feischl, Karkulik, Melenk, Praetorius 13;Tsogtgerel ’13.

• Non-symmetric elliptic PDEs: Mekchay and Nochetto ’05; Cascon andNochetto ’12; Feischl, Fuhrer and Praetorious ’13.

• FEMs for Stokes:

I Uzawa algorithm: Bansch, Morin, Nochetto ’02; Kondratyuk and Stevenson’08.

I Nonconforming FEM: Becker and Mao ’11; Carstensen, Peterseim, and Rabus’13.

I Taylor-Hood FEM: Feischl ’18.

• Eigenvalue problems: Dai, Xu, and Zhou ’08; Garau and Morin ’10;Carstensen and Gedicke ’11.



Outline







Isogeometric Analysis

Motivation: Main Properties

I Ease the path from geometry representation (CAD) to solvers.

I Use B-splines or NURBS, which are the standard in CAD.

I Easy high order and smoothness through tensor products.

Adaptivity

I T-splines (Scott-Li-Sederberg-Hughes, Beirao da Veiga-Buffa-Sangalli-Vazquez)

I LR-splines (Dokken-Lyche-Pettersen, Johannessen-Kvamsdal-Dokke)I Hierarchical B-splines

I Vuong-Giannelli-Juttler-SimeonI Giannelli-Juttler-Speleers (truncated)I Buffa-Gianelli (element-based residual-type a posteriori estimation)I Kuru-Verhoosel-Van der Zee-van Brummelen (goal oriented)I Buffa-Garau (function-based residual-type a posteriori estimation)I Gantner-Haberlik-Praetorius (optimal convergence)



Spline Space: Case d = 1

I Fix a polynomial degree (e.g. p = 3)I Regularity at interior knots (e.g. C2, maximum p− 1)

I Vector space of finite dimension

v ∈ Cp−1 : v|I ∈ Pp for each subinterval II B-Spline: a basis with a minimal support property.

I Partition of unity.I Translations of a fixed master ϕ.



Spline Space: Case d > 1

I B a B-spline basis for the uniform spline space on [0, 1]I B = ϕ(x+ h) : h ∈ Z translations of a fixed ϕI B2 = ϕ(x)ψ(y) with ϕ ∈ B and ψ ∈ BI 2-variate tensor product spline space is span(B2)I B2 = ϕ(x+ h) : h ∈ Z2 translations of a fixed ϕ



Hierarchical Basis: Case d = 1

I B`: B-spline basis of level `

(Adaptive) hierarchical basis (idea)

I use B-splines from different levels



Hierarchical Basis: Case d = 1

wϕ = supp(ϕ)

Children of a B–spline

ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ



Hierarchical Basis: Case d > 1

wϕ = supp(ϕ)

Children of a B–spline

ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ



Key Steps for Adaptivity of B-Splines

• Error estimation: by solving local problems on supports ωϕ of splines ϕ.

• Weighted Poincare inequality: for ζ such that∫ωϕζϕ = 0 we have(∫

ωϕ

ζ2 1

d2ϕ

ϕ

) 12

.

(∫ωϕ

|∇ζ|2ϕ

) 12

where dϕ is the distance to the boundary of ωϕ. Delicate because derivativesof ϕ vanish on ∂ωϕ

• Refinement strategy: eliminate a spline and replace it by its children.

• Redundancy: procedure to guarantee that the collection of hierarchicalsplines forms a basis (linear independence) and, properly scaled, a partitionof unity.

• Contraction property: for estimator plus oscillation.

• Complexity of refinement: estimate of cardinality of hierarchical basis Hk

#Hk −#H0 ≤ Λ0

k−1∑j=0

#Mj

• Rate optimality: definition of approximation class and optimal cardinality.



Hierarchical Basis Replacement Sequence

Definition: A hierarchical basis replacement sequence is a sequence R`∞`=0

such that R` ⊂ B` and satisfies the properties

1. for ` > 0, R` ⊂ ch(R`−1)

2. there exists N such that RN = ∅

Example d = 1:

I H0 = B0

I H1 = H0 \ R0 ∪ ch(R0),

I H2 = H1 \ R1 ∪ ch(R1),...

I H` = HN for all ` ≥ N (RN = ∅) H := HN



Hierarchical Basis Replacement Sequence

Example d = 2: Given a hierarchical basis replacement sequence R`∞`=0, let

I H0 = B0

I H` = H`−1 \ R`−1 ∪ ch(R`−1), for ` > 0.




Admissible Replacement Sequence

Hierarchical B-splines basis H: this is generated by a replacement sequenceR`∞`=0 as follows

I H0 = B0

I H` = H`−1 \ R`−1 ∪ ch(R`−1) for ` > 0


Admissible replacement sequence R`: the following are desirableproperties of the resulting set H

I H is linearly independent, whence H is a (hierarchical) basis of H;

I 1 =∑ϕ∈H cϕϕ, with cϕ > 0 for all ϕ ∈ H;

I H = cϕϕ : ϕ ∈ H is a basis of H as well as a partition of unity.



Admissible Replacement Sequence: Definition

R` is admissible ifϕ ∈ chR`−1 : supp(ϕ) ⊂

⋃ψ∈R`

supp(ψ)⊂R`



Elliptic model problem

Weak formulation of −∆u = f , f ∈ L2(D).

u ∈ H10 (D) :

∫D

∇u · ∇v =

∫D

f v ∀ v ∈ H10 (D).

Galerkin Approximation

H: Hierarchical space (splines of degree p ≥ 2, with maximum regularity)

U ∈ H ∩H10 (D) :

∫D

∇U · ∇V =

∫D

f V ∀V ∈ H ∩H10 (D).

I error e = u− UI residual R = −∆e = f + ∆U ∈ L2(D) because H ⊂ C1(D)



Error Equation

Galerkin orthogonality: (∇e,∇V ) = (∇(u− U),∇V ) = 0 ∀V ∈ H.

Normalized basis: H = ϕ such that∑ϕ∈H ϕ = 1 (partition of unity).

(∇e,∇e) =

(∇e,∇

[e−

∑ϕ∈H

cϕϕ︸︷︷︸∈H

])=

(∇e,∇

[ ∑ϕ∈H

ϕ

︸︷︷︸1

e−∑ϕ∈H

cϕϕ

])

=∑ϕ∈H

(∇e,∇

(ϕ(e− cϕ)

))=∑ϕ∈H

∫∇e · ∇

(ϕ(e− cϕ)

)=∑ϕ∈H

∫(f + ∆U︸︷︷︸

R

)(e− cϕ)ϕ =∑ϕ∈H

(R, e− cϕ)ϕ

=∑ϕ∈H

(R,Πϕ(e− cϕ)

)ϕ

+∑ϕ∈H

(R, (I −Πϕ)(e− cϕ)

)ϕ



Local Problems

• Error equation:

(∇e,∇e) =∑ϕ∈H

(R,Πϕ(e− cϕ)

)ϕ︸︷︷︸

I

+∑ϕ∈H

(R, (I −Πϕ)(e− cϕ)

)ϕ︸︷︷︸

II

• Local problems on suppϕ:

ζϕ ∈ V ϕ :(∇ζϕ,∇v

)ϕ

=(R, v

)ϕ, ∀v ∈ V ϕ,

V ϕ =

v ∈ B`ϕ+1 :

∫vϕ = 0

if ϕ ∈ HI

v ∈ B`ϕ+1 : v|∂D∩∂ωϕ = 0

if ϕ ∈ HB .

• Estimating term I:

I =∑ϕ∈H

(∇ζϕ,∇Πϕ(e− cϕ))ϕ

≤∑ϕ∈H

‖∇ζϕ‖ϕ︸︷︷︸η(ϕ)

‖∇Πϕ(e− cϕ))‖ϕ

.

(∑ϕ∈H

η2(ϕ)

) 12(∑ϕ∈H

‖∇(e− cϕ)‖2ϕ) 1

2

=

(∑ϕ∈H

η2(ϕ)

) 12

‖∇e‖

.



Local Problems

• Error equation:

(∇e,∇e) =∑ϕ∈H

(R,Πϕ(e− cϕ)

)ϕ︸︷︷︸

I

+∑ϕ∈H

(R, (I −Πϕ)(e− cϕ)

)ϕ︸︷︷︸

II



)ϕ

=(R, v

)ϕ, ∀v ∈ V ϕ,

V ϕ =

v ∈ B`ϕ+1 :

∫vϕ = 0

if ϕ ∈ HI

v ∈ B`ϕ+1 : v|∂D∩∂ωϕ = 0

if ϕ ∈ HB .


I =∑ϕ∈H

(∇ζϕ,∇Πϕ(e− cϕ))ϕ ≤∑ϕ∈H

‖∇ζϕ‖ϕ︸︷︷︸η(ϕ)


.

(∑ϕ∈H

η2(ϕ)

) 12(∑ϕ∈H

‖∇(e− cϕ)‖2ϕ) 1

2

=

(∑ϕ∈H

η2(ϕ)

) 12

‖∇e‖

.



Local Problems

• Error equation:

(∇e,∇e) =∑ϕ∈H

(R,Πϕ(e− cϕ)

)ϕ︸︷︷︸

I

+∑ϕ∈H

(R, (I −Πϕ)(e− cϕ)

)ϕ︸︷︷︸

II



)ϕ

=(R, v

)ϕ, ∀v ∈ V ϕ,

V ϕ =

v ∈ B`ϕ+1 :

∫vϕ = 0

if ϕ ∈ HI

v ∈ B`ϕ+1 : v|∂D∩∂ωϕ = 0

if ϕ ∈ HB .


I =∑ϕ∈H

(∇ζϕ,∇Πϕ(e− cϕ))ϕ ≤∑ϕ∈H

‖∇ζϕ‖ϕ︸︷︷︸η(ϕ)


.

(∑ϕ∈H

η2(ϕ)

) 12(∑ϕ∈H

‖∇(e− cϕ)‖2ϕ) 1

2

=

(∑ϕ∈H

η2(ϕ)

) 12

‖∇e‖.



Local Weighted Projector

• Polynomial space: Let Qϕ be the a tensor product space of polynomials:

Qϕ = span

Πdj=1qj(xj) with qj a polynomial of order mj

,

such that the dimension of Qϕ coincides with that of V ϕ.

• Definition of projector: Let Πϕ : L2(ϕ)→ V ϕ be defined by

Πϕ(v) ∈ V ϕ :

∫ωϕ

Πϕ(v)wϕ =

∫ωϕ

vw ϕ ∀w ∈ Qϕ.

The operator Πϕ is well defined!

• Stability properties:

‖Πϕ(v)‖ϕ . ‖v‖ϕ ∀v ∈ L2(ϕ) [L2 stability]

‖∇Πϕ(v)‖ϕ . ‖∇v‖ϕ ∀v ∈ H1(ϕ) [H1 stability]

where

H1(ϕ) =

v ∈ H1(ϕ) :

∫ωϕvϕ = 0

if ϕ ∈ HI

v ∈ H1(ϕ) : v|∂D∩∂ωϕ = 0

if ϕ ∈ HB .



Upper Bound

‖∇e‖2 .(∑ϕ∈H

η2(ϕ)

) 12

‖∇e‖

︸︷︷︸I

+∑ϕ∈H

(R, (I −Πϕ)(e− cϕ)

)ϕ︸︷︷︸

II

• Local weighted L2-projection: Let Qϕ be the space of polynomials fromthe previous definition. Let Mϕ : L2(ϕ)→ Qϕ be defined as follows:

Mϕ(R) ∈ Qϕ :(MϕR, q)ϕ = (R, q)ϕ, ∀q ∈ Qϕ.

• Estimating term II:

II =∑ϕ∈H

(R−Mϕ(R), (I −Πϕ)(e− cϕ)

)ϕ

≤∑ϕ∈H

‖R−Mϕ(R)‖ϕ‖(I −Πϕ)(e− cϕ)‖ϕ

.∑ϕ∈H

‖R−Mϕ(R)‖ϕhϕ‖∇e‖ϕ

≤( ∑ϕ∈H

h2ϕ‖R−Mϕ(R)‖2ϕ

) 12 ‖∇e‖,



Upper Bound

‖∇e‖2 .(∑ϕ∈H

η2(ϕ)

) 12

‖∇e‖+

∑ϕ∈H


12

‖∇e‖

Upper error estimate:

‖∇e‖L2(D) .

(∑ϕ∈H

η2(ϕ)

) 12

︸︷︷︸estimator

+

∑ϕ∈H


12

︸︷︷︸oscillation

.

Local indicator:

η(ϕ) = ‖∇ζ‖ϕ =

(∫|∇ζ|2ϕ

)1/2

where

ζ ∈ V ϕ :

∫∇ζ · ∇v ϕ =

∫Rv ϕ, ∀v ∈ V ϕ,

V ϕ =

v|ωϕ : v ∈ B`ϕ+1 :

∫vϕ = 0

if ϕ ∈ HI

v|ωϕ : v ∈ B`ϕ+1 : v|∂D∩∂ωϕ = 0

if ϕ ∈ HB .Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto


Upper Bound

Upper error estimate:

‖∇e‖L2(D) .

(∑ϕ∈H

η2(ϕ)

) 12

︸︷︷︸estimator

+

∑ϕ∈H


12

︸︷︷︸oscillation

.

Local indicator:

η(ϕ) = ‖∇ζ‖ϕ =

(∫|∇ζ|2ϕ

)1/2

where

ζ ∈ V ϕ :

∫∇ζ · ∇v ϕ =

∫Rv ϕ, ∀v ∈ V ϕ,

V ϕ =

v|ωϕ : v ∈ B`ϕ+1 :

∫vϕ = 0

if ϕ ∈ HI

v|ωϕ : v ∈ B`ϕ+1 : v|∂D∩∂ωϕ = 0

if ϕ ∈ HB .



Lower Bound

• Definition of ζ = ζϕ:

ζ ∈ V ϕ :(∇ζ,∇v

)ϕ

=(R, v

)ϕ∀ v ∈ V ϕ

• Local indicator:

η2(ϕ) =

∫|∇ζ|2 ϕ = (∇ζ,∇ζ)ϕ = (R, ζ)ϕ

=

∫(f + ∆U) ζ ϕ =

∫∇e · ∇(ζϕ)

=

∫∇e · ∇ζ ϕ︸︷︷︸

I

+

∫∇e · ζ∇ϕ︸︷︷︸

II


I ≤ ‖∇e‖ϕ‖∇ζ‖ϕ = ‖∇e‖ϕ η(ϕ).

• Estimating term II:

II ≤(∫|∇e|2ϕ

) 12(∫

ζ2 |∇ϕ|2

ϕ

) 12

.



Improved Poincare Inequality

• Asymptotic behavior of ϕ: ϕ behaves like a power to the distance dϕ tothe boundary of suppϕ except at the corners. The following is true

|∇ϕ|2

ϕ≤ C ϕ

d2ϕ

in ωϕ = suppϕ.

• Intermediate inequality:∫ωϕ

ζ2 |∇ϕ|2

ϕ.∫ωϕ

ζ2 ϕ

d2ϕ

• Improved Poincare inequality: for ζ such that∫ωϕζϕ = 0 we have

(∫ωϕ

ζ2 1

d2ϕ

ϕ

) 12

.

(∫ωϕ

|∇ζ|2ϕ

) 12

I Inspired by a similar inequality [Duran-Lombardi-Prieto’2013];I Ingredients: integral representation of ζ in terms of ∇ζ, precise control of

boundary behavior of ϕ, estimate of Hardy-Littlehood maximal function.



Efficiency: Local and Global Lower Bound

Local lower bound:

η(ϕ) . ‖∇e‖ϕ =

(∫|∇(u− U)|2ϕ

)1/2

∀ϕ ∈ H,

Global lower bound:∑ϕ∈H

η2(ϕ)

1/2

. ‖∇(u− U)‖L2(D)

because H is a partition of unity, whence∑ϕ∈H

η2(ϕ) .∫|∇(u− U)|2

∑ϕ∈H

ϕ

︸︷︷︸=1

=

∫Ω

|∇(u− U)|2.



AHBS - Adaptive Hierarchical B-Spline Algorithm (version 1.0)

I Given R0 = R`0`, set k = 0

Solve: Let Hk = Hk(Rk), and solve for Uk ∈ HkEstimate: Compute ηk(ϕ) and osck(ϕ) for each ϕ ∈ Hk

Mark: Define Mk ⊂ Hk such that

∑ϕ∈Mk

η2k(ϕ) + osc2

k(Uk, ϕ) ≥ θ2

∑ϕ∈Hk

η2k(ϕ) + osc2

k(Uk, ϕ)

,

where θ is a fixed real number in (0, 1)

Enrich: R`k+1 = R`k ∪M`k, ` = 0, 1, . . .

MakeAdmiss: Enlarge R`k+1 so that R`k+1` is admissible

I increment k by 1 and go to step solve



Properties of AHBS

Discrete lower bound: If ϕ ∈Mk, then

η2k(ϕ) . ‖∇(Uk+1 − Uk)‖2ϕ + h2

ϕ‖Rk −Mϕ(Rk)‖2ϕ,

so that

η2k(Mk) =

∑ϕ∈Mk

η2k(ϕ) ≤ cL‖∇(Uk+1 − Uk)‖2 + cL osc2

k(Uk,Mk).

Energy reduction: ‖∇(u−Uk+1)‖2 = ‖∇(u−Uk)‖2−‖∇(Uk+1−Uk)‖2 yields

‖∇(u− Uk+1)‖2 = ‖∇(u− Uk)‖2 − 1

cLη2k(Mk) + osc2

k(Uk,Mk)

≤ ‖∇(u− Uk)‖2 − θ2

cLη2k(Hk) +

(1 +

1− θ2

cL

)osc2

k(Uk,Hk)

≤ ‖∇(u− Uk)‖2 − θ2

cLcU‖∇(u− Uk)‖2 + C osc2

k(Uk,Hk)

=

(1− θ2

cLcU

)‖∇(u− Uk)‖2 + C osc2

k(Uk,Hk).



Properties of Oscillation

Oscillation reduction: If Wk ∈ Hk then

osc2k+1(Wk) ≤ osc2

k(Wk)− 3

4osc2

k(Wk,Mk).

Lipschitz dependence: If Wk+1, Wk+1 ∈ Hk+1, and ϕ ∈ Hk+1 then

| osck+1(Wk+1, ϕ)− osck+1(Wk+1, ϕ)| ≤ C0‖∇(Wk+1 − Wk+1)‖ϕ,with C0 a constant depending on the maximum level gap g ∈ N. This allowsinverse inequalities to control ∆Wk+1 by ∇Wk+1.

Level gap: Coexistence of 4 levels on an interval level gap g = 3



AHBS Algorithm (version 2.0)

I Given R0 = R`0`, set k = 0

Solve: Let Hk = Hk(Rk), and solve for Uk ∈ HkEstimate: Compute ηk(ϕ) and osck(ϕ) for each ϕ ∈ Hk

Mark: Define Mk ⊂ Hk such that

∑ϕ∈Mk

η2k(ϕ) + osc2

k(Uk, ϕ) ≥ θ2

∑ϕ∈Hk

η2k(ϕ) + osc2

k(Uk, ϕ)

,

where θ is a fixed real number in (0, 1)

Refine: Rk+1 = GapControlledRefine(Rk,Mk)

I increment k by 1 and go to step Solve

Remarks:

I Level gap control leads to convergence of AHBS.

I Rk+1 = GapControlledRefine(Rk,Mk) creates an admissiblereplacement sequence Rk+1 with specified but arbitrary finite level gap.

I GapControlledRefine is constructive and simple to implement.



Contraction Property

Theorem (contraction property). There exist two constants γ > 0 and0 < α < 1 such that, for k = 0, 1, 2, . . . ,

‖∇(u−Uk+1)‖2L2(Ω)+γ osc2k+1(Uk+1) ≤ α

(‖∇(u−Uk)‖2L2(Ω)+γ osc2

k(Uk)).

Ingredients of the proof:

I Energy reduction: ‖∇(u−Uk+1)‖2 = ‖∇(u−Uk)‖2 −‖∇(Uk+1 −Uk)‖2

I Discrete lower bound: η2k(Mk) ≤ cL‖∇(Uk+1−Uk)‖2 +cL osc2

k(Uk,Mk)

I Oscillation estimate: the following is valid for all δ > 0

osc2k+1(Uk+1) ≤ (1 + δ)

(osc2

k(Uk)− 3

4osc2

k(Uk,Mk))

+ (1 + δ−1)C20‖∇(Uk+1 − Uk)‖2.

I Dorfler marking: η2k + osc2

k(Uk) ≥ θ2(ηk(Mk)2 + osc2

k(Uk,Mk))



Complexity of GapControlledRefine

• Basic structure of algorithm: if g ∈ N is the maximum level gap, then

I R = GapControlledRefine(R,M)for φ ∈MR = GapControlledSingleRefine(R, φ)

endforR = MakeAdmissible(R)

I R = GapControlledSingleRefine(R, φ)S := ψ ∈ H : `ψ = `φ − g, | supp(ψ) ∩ supp(φ)| > 0for ψ ∈ SR = GapControlledSingleRefine(R, ψ)

endforR = R∪ φ

• Theorem (complexity). There exists a constant C∗ only dependent on thepolynomial degree and maximum allowed gap such that

#Hk+1 ≤ #H0 + C∗

k∑j=0

#Mj

• Ingredients: use λ cost-function of Binev-DeVore, who existence proof isnon-trivial.



Approximation Classes

• Best N-term approximation of total error: Let H be the collection of allhierarchical B-splines spaces with fixed level gap and let σN (u, f) be thesmallest total error with N basis

σN (u, f) = infH∈H

dim(H)≤N

(infV ∈H

‖∇(u− V )‖+ oscH(V )).

• Approximation class As of order s:

(u, f) ∈ As if σN (u, f) . N−s, ∀N ∈ N.

• Maximum level gap: this does not restrict the approximation class, butchanges the constant in definition σN (u, f).

• Open problem: No characterization yet of membership in As in terms ofBesov spaces.



Optimality

Theorem (optimality): If (u, f) ∈ As, then

‖∇(u− Uk)‖+ osck(Uk)︸︷︷︸Ek

. dim(Hk)−s =(#Hk)−s.

Remarks:

I Same asymptotic decay s as dictated by approximation class AsI No need to know s

Ingredients: they are customary in AFEM theory

I Contraction property Ek+1 ≤ αEkI Control on the dimension of spaces

#Hk+1 ≤ #H0 + ck∑j=0

#Mj

I Minimality of Dorfler marking set Mj .


adaptive finite element methods lecture 5: extensions ii

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