Adaptive Finite Element Methods Lecture 4: Extensions

Adaptive Finite Element MethodsLecture 4: Extensions

Ricardo H. Nochetto

Department of Mathematics andInstitute for Physical Science and Technology

University of Maryland, USA

www2.math.umd.edu/˜rhn

7th Zurich Summer School, August 2012A Posteriori Error Control and Adaptivity

Outline Nonresidual Estimators Nonconforming Meshes DG Methods HDG Methods Discontinuous Coefficients

Outline

Nonresidual Estimators (w. J.M. Cascon)

Nonconforming Meshes (w. A. Bonito)

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

Hibridizable Discontinuous Galerkin Methods (HDG) (w. B. Cockburnand W. Zhang)

Discontinuous Coefficients (w. A. Bonito and R. DeVore)

Adaptive Finite Element Methods Lecture 4: Extensions Ricardo H. Nochetto


Outline








Continuous Model Problem

Consider the following linear elliptic symmetric PDE in weak form:

u ∈ V : B(u, v) :=∫

Ω

A∇u · ∇v dx = 〈f, v〉Ω ∀v ∈ V

where

I Ω ⊂ Rd (d ∈ N) is a polyhedral domain that is triangulated by T0;I A ∈ L∞(Ω; Rd×d) is pw. Lipschitz over T0 and uniformly spd;

I f ∈ L2(Ω) and 〈·, ·〉Ω is the L2 scalar product.

The bilinear form B defines the corresponding energy-norm by

|||v|||Ω := B(v, v)1/2,

which implies existence and uniqueness by the Lax-Milgram Theorem.



Continuous Model Problem

Consider the following linear elliptic symmetric PDE in weak form:

u ∈ V : B(u, v) :=∫

Ω

A∇u · ∇v dx = 〈f, v〉Ω ∀v ∈ V

where

I Ω ⊂ Rd (d ∈ N) is a polyhedral domain that is triangulated by T0;I A ∈ L∞(Ω; Rd×d) is pw. Lipschitz over T0 and uniformly spd;

I f ∈ L2(Ω) and 〈·, ·〉Ω is the L2 scalar product.

The bilinear form B defines the corresponding energy-norm by

|||v|||Ω := B(v, v)1/2,

which implies existence and uniqueness by the Lax-Milgram Theorem.



Discretization with Conforming Finite Elements

I Triangulations: We denote by T the set of all conforming refinementsof T0, that can be generated from T0 using refinement by bisection;

T0 T ∈ TI Finite Element Spaces: For T ∈ T we let V(T ) ⊂ V be the finite

element space of piecewise polynomials of degree n ≥ 1 over T .

The discrete problem reads:

UT ∈ V(T ) B(UT , V) = 〈f, V〉Ω ∀V ∈ V(T ).

Existence and uniqueness again follows by the Lax-Milgram Theorem.



Discretization with Conforming Finite Elements

I Triangulations: We denote by T the set of all conforming refinementsof T0, that can be generated from T0 using refinement by bisection;

T0 T ∈ TI Finite Element Spaces: For T ∈ T we let V(T ) ⊂ V be the finite

element space of piecewise polynomials of degree n ≥ 1 over T .

The discrete problem reads:

UT ∈ V(T ) B(UT , V) = 〈f, V〉Ω ∀V ∈ V(T ).

Existence and uniqueness again follows by the Lax-Milgram Theorem.



Adaptive cG: AFEM

We consider the most standard adaptive iteration

SOLVE −→ ESTIMATE −→ MARK −→ REFINE

I Module SOLVE: UT = SOLVE(T ) computes the Ritz projectionUT ∈ V(T ) to u:

I assuming exact integration;I assuming exact solution of the discrete system;

I Module ESTIMATE: ET (T )T∈T = ESTIMATE(UT , T ) computes(local) error indicators yielding the estimator:

I residual estimator (nonresidual estimators later);

I Module MARK:M = MARK(ET (T ), T T∈T ) selects a subsetM⊂ T of elements subject to refinement:

I Dorfler marking (single marking according to estimator);

I Module REFINE: T∗ = REFINE(T ,M) refines all elements inMand outputs a conforming refinement T∗ ≥ T :

I All elements of M are bisected b ≥ 1 (no interior node property).



Extension to Nonresidual Estimators

• Non-residual Estimators:I Hierarchical estimators (Bornemann, Kornhuber, Veeser)I Star estimators (Morin, Nochetto, Siebert)I Star estimators (Pares, Diez, Huerta)I H(div) estimators (Braess, Hoppe, Schoberl)I Gradient recovery (Zienkiewicz, Zhu)

• Star Estimator (MNS): Let U be piecewise linear Galerkin solution.Let P2

0 (ωz) be the space of piecewise quadratic polynomials on thestar ωz that vanish on ∂ωz. Let ξz ∈ P2

0 (ωz) solve∫ωz

A∇ξz∇ϕφz =∫

ωz

fϕφz −∫

ωz

A∇U∇ϕφz ∀ ϕ ∈ P20 (ωz)

The local indicator ηz is given by

η2z = η2

z(U, T ) =∫

ωz

A∇ξz∇ηzφz

and the global error estimator is given by ηT (U, T ) =( ∑

z∈N η2z

) 12

• Fractional Diffusion: similar estimator used by E. Otarola (Friday).







0 (ωz) solve∫ωz


ωz

fϕφz −∫

ωz



η2z = η2

z(U, T ) =∫

ωz

A∇ξz∇ηzφz


z∈N η2z

) 12








0 (ωz) solve∫ωz


ωz

fϕφz −∫

ωz



η2z = η2

z(U, T ) =∫

ωz

A∇ξz∇ηzφz


z∈N η2z

) 12




Properties of Estimators

• Global Upper Bound: there exists C1 depending on T0 so that

|||UT − u|||2Ω ≤ C1

(η2T (UT , T ) + osc2

T (UT , T ))

=: E2T (UT , T )

• Localized Upper Bound: if T ≤ T∗ and R = RT→T∗ is the set ofrefined elements of T to go to T∗, then

|||UT∗ − UT |||2Ω ≤ C1

(η2T (UT ,R) + osc2

T (UT ,R))

=: E2T (UT ,R)

• Discrete Local Lower Bound: if T ≤ T∗ and G`T→T∗ is the set of

simplices of RT→T∗ ⊂ T which are bisected at least ` times in T∗, then

C2η2T (UT ,G`

T→T∗) ≤ |||UT − UT∗ |||2Ω + osc2T (UT ,RT→T∗)

• Interior Node Property: this is guaranteed by the prescribed ` levelsof refinement (` = 3 for d = 2; ` = 6 for d = 3) Then UT and UT∗cannot be consecutive Galerkin solutions but rather obtained after `iterations of AFEM.



Properties of Estimators

• Global Upper Bound: there exists C1 depending on T0 so that

|||UT − u|||2Ω ≤ C1

(η2T (UT , T ) + osc2

T (UT , T ))

=: E2T (UT , T )

• Localized Upper Bound: if T ≤ T∗ and R = RT→T∗ is the set ofrefined elements of T to go to T∗, then

|||UT∗ − UT |||2Ω ≤ C1

(η2T (UT ,R) + osc2

T (UT ,R))

=: E2T (UT ,R)

• Discrete Local Lower Bound: if T ≤ T∗ and G`T→T∗ is the set of

simplices of RT→T∗ ⊂ T which are bisected at least ` times in T∗, then

C2η2T (UT ,G`

T→T∗) ≤ |||UT − UT∗ |||2Ω + osc2T (UT ,RT→T∗)

• Interior Node Property: this is guaranteed by the prescribed ` levelsof refinement (` = 3 for d = 2; ` = 6 for d = 3) Then UT and UT∗cannot be consecutive Galerkin solutions but rather obtained after `iterations of AFEM.



Nonresidual Estimators (Continued)

Module MARK: uses Dorfler marking with the total error indicatorsET (UT , T )T∈T to select M

M = MARK (ET (UT , T )T∈T , T )

Module REFINE: bisects elements ofM at least b ≥ 1 times andupdates the refinement flag ρT (T ) of all elements T ∈ T to guarantee `levels of refinement of marked elements

T∗, ρT∗(T )T∈T∗ = REFINE (T ,M, ρT (T )T∈T )

Theorem (Contraction Property of AFEM). There exists 0 < α < 1and γ > 0, depending T0, θ, b, C1, C2, and data A, such that

|||Uk+` − u|||2Ω + γ osc2k+`(Uk+`, Tk+`) ≤ α

(|||Uk − u|||2Ω + γ osc2

k(Uk, Tk)).

The proof is a combination of those by Mekchay-Nochetto ’05 andCascon-Kreuzer-Nochetto-Siebert ’08 (Lecture 2).



Nonresidual Estimators (Continued)

Module MARK: uses Dorfler marking with the total error indicatorsET (UT , T )T∈T to select M

M = MARK (ET (UT , T )T∈T , T )

Module REFINE: bisects elements ofM at least b ≥ 1 times andupdates the refinement flag ρT (T ) of all elements T ∈ T to guarantee `levels of refinement of marked elements

T∗, ρT∗(T )T∈T∗ = REFINE (T ,M, ρT (T )T∈T )

Theorem (Contraction Property of AFEM). There exists 0 < α < 1and γ > 0, depending T0, θ, b, C1, C2, and data A, such that

|||Uk+` − u|||2Ω + γ osc2k+`(Uk+`, Tk+`) ≤ α

(|||Uk − u|||2Ω + γ osc2

k(Uk, Tk)).

The proof is a combination of those by Mekchay-Nochetto ’05 andCascon-Kreuzer-Nochetto-Siebert ’08 (Lecture 2).



Comparison with Residual Estimators

I AFEM uses one Dorfler marking according to Ek(Uk, Tk) as before.

I AFEM enforces an interior node property after ` refinement steps(` = 2 for d = 2 and ` = 6 for d = 3).

I The estimator ηT (V, T ) does not dominate the error nor theoscillation. However

|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2

k(Uk, Tk)+osc2k(Uk, Tk) ≈ E2

k(Uk, Tk)

I The estimator ηT (V, T ) does not have a reduction property.

I Proof uses the discrete local lower bound.

I AFEM controls the quantity |||u− Uk|||2Ω + γ osc2k(Uk, Tk) instead of

|||u− Uk|||2Ω + γE2k(Uk, Tk).

I Kreuzer-Siebert’10 exploit property that all nonresidual estimatorsare equivalent to residual ones to prove a contraction property.







|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2


k(Uk, Tk)




|||u− Uk|||2Ω + γE2k(Uk, Tk).








|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2


k(Uk, Tk)




|||u− Uk|||2Ω + γE2k(Uk, Tk).








|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2


k(Uk, Tk)




|||u− Uk|||2Ω + γE2k(Uk, Tk).








|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2


k(Uk, Tk)




|||u− Uk|||2Ω + γE2k(Uk, Tk).








|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2


k(Uk, Tk)




|||u− Uk|||2Ω + γE2k(Uk, Tk).








|||u− Uk|||2Ω +osc2k(Uk, Tk) ≈ η2


k(Uk, Tk)




|||u− Uk|||2Ω + γE2k(Uk, Tk).




Quasi-Optimal Decay Rates

• REFINE creates a conforming refinement T∗ of T by bisection (or anon-conforming T∗ ≥ T with one hanging node per edge to bediscussed next), and guarantees that every marked element satisfies theinterior node property after ` refinement steps;

• MARK chooses a set M with minimal cardinality;

• The Dorfler parameter θ satisfies 0 < θ < θ∗ for a suitable θ∗ < 1.

• Triple (u, f,A) is in the approximation class As (0 < s ≤ 1/2):

supN≥#T0

Ns infT ∈TN

infV ∈V(T )

(|||u− V |||T + oscT (V, T )

)Theorem (optimal cardinality of AFEM) Let Tk, V(Tk), Uk∞k=0 bethe sequence of conforming meshes, conforming finite element spaces,and Galerkin solutions generated by AFEM. If (u, f,A) ∈ As, then

|||u− Uk|||Tk+ oscTk

(Uk, Tk) .(#Tk −#T0

)−s.



Quasi-Optimal Decay Rates

• REFINE creates a conforming refinement T∗ of T by bisection (or anon-conforming T∗ ≥ T with one hanging node per edge to bediscussed next), and guarantees that every marked element satisfies theinterior node property after ` refinement steps;

• MARK chooses a set M with minimal cardinality;



supN≥#T0

Ns infT ∈TN

infV ∈V(T )

(|||u− V |||T + oscT (V, T )

)Theorem (optimal cardinality of AFEM) Let Tk, V(Tk), Uk∞k=0 bethe sequence of conforming meshes, conforming finite element spaces,and Galerkin solutions generated by AFEM. If (u, f,A) ∈ As, then


(Uk, Tk) .(#Tk −#T0

)−s.



Outline








Extension to Nonconforming Meshes

I Hanging nodes: quad-refinement, red refinement, bisection showingdomain of influence of conforming node P .

P

PP

I Admissible meshes: domains of influence are comparable withelements contained in them (Ex: one hanging node per edge forquadrilaterals). This yields a fixed level of nonconformity.

I Quad refinements in 2d and 3d are used in deal.II (Bangerth,Hartmann, Kanschat). It does not require a suitable initial labeling.

I Contraction property on AFEM with hanging nodes: same result aswith conforming meshes and with either residual or non-residualestimators.



Extension to Nonconforming Meshes

I Hanging nodes: quad-refinement, red refinement, bisection showingdomain of influence of conforming node P .

P

PP

I Admissible meshes: domains of influence are comparable withelements contained in them (Ex: one hanging node per edge forquadrilaterals). This yields a fixed level of nonconformity.

I Quad refinements in 2d and 3d are used in deal.II (Bangerth,Hartmann, Kanschat). It does not require a suitable initial labeling.

I Contraction property on AFEM with hanging nodes: same result aswith conforming meshes and with either residual or non-residualestimators.



Complexity of Nonconforming Meshes

Bisection Method: g(T ′) ≤ g(T ) + 1 for all T ′ created by trying to refineT ∈ T (Binev-Dahmen-DeVore ’04 for d = 2, Stevenson ’06 for d ≥ 2)

T ’

0

5

=T

1

2

3

4

i

T

T

T

T

T

T

T

Refinement of Nonconforming Meshes: The basic recursive procedure[T∗,M∗] = MAKE ADMISSIBLE(T ,M, T ) refines T ∈ T once andperhaps other elements to keep the mesh admissible.

Lemma For all T ′ ∈ T∗\T created by MAKE ADMISSIBLE(T ,M, T )

g(T ′) ≤ g(T ) + 1.

Proposition (Binev-Dahmen-DeVore ’04) If inequality above holds, then

#Tk −#T0 4k−1∑j=0

#Mj .



Complexity of Nonconforming Meshes

Bisection Method: g(T ′) ≤ g(T ) + 1 for all T ′ created by trying to refineT ∈ T (Binev-Dahmen-DeVore ’04 for d = 2, Stevenson ’06 for d ≥ 2)

T ’

0

5

=T

1

2

3

4

i

T

T

T

T

T

T

T

Refinement of Nonconforming Meshes: The basic recursive procedure[T∗,M∗] = MAKE ADMISSIBLE(T ,M, T ) refines T ∈ T once andperhaps other elements to keep the mesh admissible.

Lemma For all T ′ ∈ T∗\T created by MAKE ADMISSIBLE(T ,M, T )

g(T ′) ≤ g(T ) + 1.

Proposition (Binev-Dahmen-DeVore ’04) If inequality above holds, then

#Tk −#T0 4k−1∑j=0

#Mj .



Optimal Cardinality with Nonconforming Meshes

• The matrix A is symmetric, uniformly positive definite and Lipschitz ineach element of T0;

• The forcing f = −div(A∇u) is in L2(Ω);• T∗ = REFINE(T ) creates a non-conforming mesh T∗ from T with a

fixed level of nonconformity (e.g. one hanging node per edge);

• M = MARK(ET (T )T∈T , T ) chooses a set M with minimalcardinality;



supN≥#T0

Ns infT ∈TN

infV ∈V(T )

(|||u− V |||T + oscT (V, T )

)Theorem (optimal cardinality) Let Tk, V(Tk), Uk∞k=0 be thesequence of nonconforming meshes, conforming finite element spaces,and Galerkin solutions generated by AFEM. If (u, f,A) ∈ As, then


(Uk, Tk) .(#Tk −#T0

)−s.



Optimal Cardinality with Nonconforming Meshes

• The matrix A is symmetric, uniformly positive definite and Lipschitz ineach element of T0;

• The forcing f = −div(A∇u) is in L2(Ω);• T∗ = REFINE(T ) creates a non-conforming mesh T∗ from T with a

fixed level of nonconformity (e.g. one hanging node per edge);

• M = MARK(ET (T )T∈T , T ) chooses a set M with minimalcardinality;



supN≥#T0

Ns infT ∈TN

infV ∈V(T )

(|||u− V |||T + oscT (V, T )

)Theorem (optimal cardinality) Let Tk, V(Tk), Uk∞k=0 be thesequence of nonconforming meshes, conforming finite element spaces,and Galerkin solutions generated by AFEM. If (u, f,A) ∈ As, then


(Uk, Tk) .(#Tk −#T0

)−s.



Outline








Extension to Discontinuous Galerkin (DG) Methods

Consider model problem −div(A∇u) = f in d dimensions. Given anonconforming mesh T as above, and a space V(T ) of discontinuous pwpolynomials of degree ≤ 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 ΣI [[·]] jump operator over set of interelement boundaries ΣI 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 ) = V0(T )⊕ V⊥(T ) w.r.t BT (·, ·)with V0(T ) := V(T ) ∩H1

0 (Ω).

• Continuous Galerkin Solution: U0 ∈ V0(T ) solves

U0 ∈ V0(T ) : BT (U0, V 0) = 〈f, V 0〉T ∀V 0 ∈ V0(T ).

• Nonconforming component: |||V ⊥|||T . δ12 ‖h− 1

2 [[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− 1

2 [[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 ).




• 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 byREFINE and 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),there exist 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-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 byREFINE and 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),there exist 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∗ > 1be explicit 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 nodeproperty).

I The non-monotone jump term ‖h−1 [[UT ]] ‖Σ is the trouble-makerbut it is controlled by the estimator. It does not enter in the upperbound.

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, div q = 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,div v)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

T h2T ‖qk − PeVk

qk‖2T + h2T ‖f − PeWk

f‖2T≈ hT ‖(qk − qk) · n‖2∂T + h2

T ‖f − fT ‖2T .



Main Results

Quasi-error: e2k := ‖q− qk‖2Ω + γζ2

div(f,qk, Tk)

Counterexample: ‖q− qk‖Ω is not monotone and a lower boundζdiv(f,qk, Tk) . ‖q− qk‖Ω is not valid.

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

e2k+1 + βζ2(f,qk+1, Tk+1) ≤ α

(e2k + βζ2(f,qk, Tk)

)

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

ek . (#Tk −#T0)−s.



Main Results


div(f,qk, Tk)



e2k+1 + βζ2(f,qk+1, Tk+1) ≤ α


)


ek . (#Tk −#T0)−s.



Main Results


div(f,qk, Tk)



e2k+1 + βζ2(f,qk+1, Tk+1) ≤ α


)


ek . (#Tk −#T0)−s.



Main Results


div(f,qk, Tk)



e2k+1 + βζ2(f,qk+1, Tk+1) ≤ α


)


ek . (#Tk −#T0)−s.



Outline








Setting

Consider elliptic PDE of the form −div(A∇u) = f with

• A = (aij(x))di,j=1 uniformly positive definite and bounded

λmin(A)|y|2 ≤ ytA(x)y ≤ λmax(A)|y|2 ∀ x ∈ Ω, y ∈ Rd;

• The discontinuities of A are not match by the sequence of meshes T ;

• The forcing f ∈W−1p (Ω) for some p > 2.

Goal: Design and study an AFEM able to handle such an A.

Difficulty: PDE perturbation results hinge on approximation of A in L∞

‖u− u‖H10 (Ω) ≤ λ−1

min(A)(‖f − f‖H−1(Ω) + ‖A− A‖L∞(Ω)‖f‖H−1(Ω)

)



Setting








‖u− u‖H10 (Ω) ≤ λ−1


)



Setting








‖u− u‖H10 (Ω) ≤ λ−1


)



Perturbation Argument

Theorem (perturbation). Let p ≥ 2, q = 2p/(p− 2) ∈ [2,∞] and∇u ∈ Lp(Ω). Then

‖u− u‖H10 (Ω) ≤ λ−1

min(A)(‖f − f‖H−1(Ω) + ‖A− A‖Lq(Ω)‖∇u‖Lp(Ω)

)

Question: can we guarantee that ∇u ∈ Lp(Ω) with p > 2?

Proposition (Meyers). Let K > 0 be so that the solution u of theLaplacian satisfies

‖∇u‖Lp(Ω) ≤ K‖f‖W−1p (Ω).

Then the solution u of −div(A∇u) = f satisfies

‖∇u‖Lp(Ω) ≤ K‖f‖W−1p (Ω)

if 2 ≤ p < p∗ and K = 1λmax(A)

eKη(p)

1− eKη(p)(1− λmin(A)

λmax(A)

) with η(p) =12−

1p

12−

1p∗

.





‖u− u‖H10 (Ω) ≤ λ−1


)



‖∇u‖Lp(Ω) ≤ K‖f‖W−1p (Ω).


‖∇u‖Lp(Ω) ≤ K‖f‖W−1p (Ω)


eKη(p)


λmax(A)

) with η(p) =12−

1p

12−

1p∗

.





‖u− u‖H10 (Ω) ≤ λ−1


)



‖∇u‖Lp(Ω) ≤ K‖f‖W−1p (Ω).


‖∇u‖Lp(Ω) ≤ K‖f‖W−1p (Ω)


eKη(p)


λmax(A)

) with η(p) =12−

1p

12−

1p∗

.



DISC: AFEM for Discontinuous Diffusion Matrices

Given ω > 0 explicit and β < 1, let

DISC(T0, ε1)k = 1LOOP

[T Fk , fk] = RHS(Tk−1, f, ωεk)

[T Ak , Ak] = COEFF(T F

k , A, ωεk)[Tk, Uk] = PDE(T A

k , Ak, fk, εk/2)εk+1 = βεk

k ← k + 1END LOOP

END DISC

• [T Fk , fk] = RHS(Tk−1, f, ωεk) gives a mesh T F

k ≥ Tk−1 and a pw po-lynonial approximation fk of f on T F

k such that ‖f−fk‖H−1(Ω) ≤ ωεk;

• [T Ak , Ak] = COEFF(T F

k , A, ωεk) gives a mesh T Ak ≥ T F

k and a pwpolynomial approximation Ak of A on T A

k such that‖A−Ak‖Lq(Ω) ≤ ωεk and its eigenvalues satisfy uniformly in k

C−1λmin(A) ≤ λ(Ak) ≤ Cλmax(A).Adaptive Finite Element Methods Lecture 4: Extensions Ricardo H. Nochetto


DISC: AFEM for Discontinuous Diffusion Matrices

Given ω > 0 explicit and β < 1, let

DISC(T0, ε1)k = 1LOOP

[T Fk , fk] = RHS(Tk−1, f, ωεk)

[T Ak , Ak] = COEFF(T F

k , A, ωεk)[Tk, Uk] = PDE(T A

k , Ak, fk, εk/2)εk+1 = βεk

k ← k + 1END LOOP

END DISC

• [T Fk , fk] = RHS(Tk−1, f, ωεk) gives a mesh T F

k ≥ Tk−1 and a pw po-lynonial approximation fk of f on T F

k such that ‖f−fk‖H−1(Ω) ≤ ωεk;

• [T Ak , Ak] = COEFF(T F

k , A, ωεk) gives a mesh T Ak ≥ T F

k and a pwpolynomial approximation Ak of A on T A

k such that‖A−Ak‖Lq(Ω) ≤ ωεk and its eigenvalues satisfy uniformly in k

C−1λmin(A) ≤ λ(Ak) ≤ Cλmax(A).Adaptive Finite Element Methods Lecture 4: Extensions Ricardo H. Nochetto


Optimality of DISC

Theorem (optimality). Assume that the right side f is in Bsf (H−1(Ω))with 0 < sf ≤ S, and that the diffusion matrix A is positive definite, inL∞(Ω) and inMsA(Lq(Ω)) for q := 2p

p−2 and 0 < sA ≤ S. Let T0 be the

initial subdivision and Uk ∈ V(Tk) be the Galerkin solution obtained atthe kth iteration of the algorithm. Then, whenever u ∈ As(H1

0 (Ω)) for0 < s ≤ S, we have for k ≥ 1

‖u− Uk‖H10 (Ω) ≤ εk,

and

#Tk −#T0 .(|A|1/s∗

Ms∗ (Lq(Ω)) + |f |1/s∗Bs∗ (H−1(Ω)) + |u|1/s∗

As∗ (H10 (Ω))

)ε−1/s∗k ,

with s∗ = min(s, sA, sf ).

Counterexample: s cannot be achieved if sA, sf < s.



Optimality of DISC

Theorem (optimality). Assume that the right side f is in Bsf (H−1(Ω))with 0 < sf ≤ S, and that the diffusion matrix A is positive definite, inL∞(Ω) and inMsA(Lq(Ω)) for q := 2p

p−2 and 0 < sA ≤ S. Let T0 be the

initial subdivision and Uk ∈ V(Tk) be the Galerkin solution obtained atthe kth iteration of the algorithm. Then, whenever u ∈ As(H1

0 (Ω)) for0 < s ≤ S, we have for k ≥ 1

‖u− Uk‖H10 (Ω) ≤ εk,

and

#Tk −#T0 .(|A|1/s∗

Ms∗ (Lq(Ω)) + |f |1/s∗Bs∗ (H−1(Ω)) + |u|1/s∗

As∗ (H10 (Ω))

)ε−1/s∗k ,

with s∗ = min(s, sA, sf ).

Counterexample: s cannot be achieved if sA, sf < s.


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Adaptive Finite Element Methods Lecture 4: Extensions