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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
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
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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).
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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 ).
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
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
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
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
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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).
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
ηk(Uk)2 ≤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
ηk(Uk)2 ≤j+N∑k=j+1
ρ2 ηk−1(Uk−1)2 + C2ηj(Uj)2.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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!
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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 .
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
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
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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)
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d = 1
I B`: B-spline basis of level `
(Adaptive) hierarchical basis (idea)
I use B-splines from different levels
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d = 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Hierarchical Basis: Case d > 1
wϕ = supp(ϕ)
Children of a B–spline
ϕ ∈ B` ch(ϕ) = ψ ∈ B`+1 : wψ ⊂ wϕ
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
I H` = HN for all ` ≥ N (RN = ∅) H := HN
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
I H` = HN for all ` ≥ N (RN = ∅) H := HN
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
I H` = HN for all ` ≥ N (RN = ∅) H := HN
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Admissible Replacement Sequence: Definition
R` is admissible ifϕ ∈ chR`−1 : supp(ϕ) ⊂
⋃ψ∈R`
supp(ψ)⊂R`
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Admissible Replacement Sequence: Definition
R` is admissible ifϕ ∈ chR`−1 : supp(ϕ) ⊂
⋃ψ∈R`
supp(ψ)⊂R`
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Admissible Replacement Sequence: Definition
R` is admissible ifϕ ∈ chR`−1 : supp(ϕ) ⊂
⋃ψ∈R`
supp(ψ)⊂R`
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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)
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖
.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖
.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖
.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖
.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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 .
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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 .
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖,
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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‖,
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Upper Bound
‖∇e‖2 .(∑ϕ∈H
η2(ϕ)
) 12
‖∇e‖+
∑ϕ∈H
h2ϕ‖R−Mϕ(R)‖2ϕ
12
‖∇e‖
Upper error estimate:
‖∇e‖L2(D) .
(∑ϕ∈H
η2(ϕ)
) 12
︸ ︷︷ ︸estimator
+
∑ϕ∈H
h2ϕ‖R−Mϕ(R)‖2ϕ
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
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
Upper Bound
Upper error estimate:
‖∇e‖L2(D) .
(∑ϕ∈H
η2(ϕ)
) 12
︸ ︷︷ ︸estimator
+
∑ϕ∈H
h2ϕ‖R−Mϕ(R)‖2ϕ
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
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
• Estimating term I:
I ≤ ‖∇e‖ϕ‖∇ζ‖ϕ = ‖∇e‖ϕ η(ϕ).
• Estimating term II:
II ≤(∫|∇e|2ϕ
) 12(∫
ζ2 |∇ϕ|2
ϕ
) 12
.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
• Estimating term I:
I ≤ ‖∇e‖ϕ‖∇ζ‖ϕ = ‖∇e‖ϕ η(ϕ).
• Estimating term II:
II ≤(∫|∇e|2ϕ
) 12(∫
ζ2 |∇ϕ|2
ϕ
) 12
.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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).
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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).
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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))
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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.
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto
Outline DG Methods HDG Methods Quasi-Orthogonality Property Adaptive Hierarchical B-Splines
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 .
Lecture 5: Quasi-Orthogonality Property Ricardo H. Nochetto