Unified primal formulation-based a priori and a posteriori error analysis … · 2017. 6. 12. ·...

Transcript of Unified primal formulation-based a priori and a posteriori error analysis … · 2017. 6. 12. ·...

Unified primal formulation-baseda priori and a posteriori error analysis

of mixed finite element methods

Martin Vohralík

Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)

Austin, April 30, 2008
Framework A priori est. A posteriori est. Remarks Exp. C.

IntroductionModel problem (S inhomogeneous and anisotropic)

u = −S∇p,∇ · u = f in Ω −∇ · (S∇p) = f in Ωp = 0 on ∂Ω p = 0 on ∂Ω

Mixed finite elements(S−1uh,vh)− (ph,∇ · vh) = 0 ∀vh ∈ Vh,

(∇ · uh, φh) = (f , φh) ∀φh ∈ ΦhTraditional analysis

weak mixed formulation(S−1u,v)− (p,∇ · v) = 0

∀v ∈ H(div,Ω),(∇ · u, φ) = (f , φ) ∀φ ∈ L2(Ω)

inf–sup condition∇ · Vh = Φh

Presented analysisclassical weak formulation

(S∇p,∇ϕ) = (f , ϕ)∀ϕ ∈ H10 (Ω)

postprocessing and discreteFriedrichs inequality∇ · Vh = Φh

unified and optimal a priori and a posteriori error analysisM. Vohralík Unified a priori and a posteriori analysis of MFEs






∀v ∈ H(div,Ω),(∇ · u, φ) = (f , φ) ∀φ ∈ L2(Ω)



(S∇p,∇ϕ) = (f , ϕ)∀ϕ ∈ H10 (Ω)








∀v ∈ H(div,Ω),(∇ · u, φ) = (f , φ) ∀φ ∈ L2(Ω)



(S∇p,∇ϕ) = (f , ϕ)∀ϕ ∈ H10 (Ω)








∀v ∈ H(div,Ω),(∇ · u, φ) = (f , φ) ∀φ ∈ L2(Ω)



(S∇p,∇ϕ) = (f , ϕ)∀ϕ ∈ H10 (Ω)



Outline1 General framework

An abstract result for the flux variablePostprocessing for the scalar variable

2 A priori error estimatesLowest-order Raviart–Thomas caseGeneral case

3 A posteriori error estimatesEstimates for the fluxEstimates for the potentialLocal efficiency

4 RemarksComments on the estimatesL2(Ω) estimatesRT0 and pure diffusion problems

5 Numerical experiments6 Conclusions and future work

M. Vohralík Unified a priori and a posteriori analysis of MFEs
Framework A priori est. A posteriori est. Remarks Exp. C. Flux variable Scalar variable








Bilinear forms and weak solution

Definition (Bilinear form B)

B(p, ϕ) :=∑

K∈Th

(S∇p,∇ϕ)K , p, ϕ ∈ H1(Th).

Definition (Bilinear form A)

A(u,v) :=∑

K∈Th

(u,S−1v)K , u,v ∈ L2(Ω).

Definition (Weak solution)

p ∈ H10 (Ω) such that B(p, ϕ) = (f , ϕ) ∀ϕ ∈ H10 (Ω) ;or

p ∈ H10 (Ω) such that A(S∇p,S∇ϕ) = (f , ϕ) ∀ϕ ∈ H10 (Ω).




B(p, ϕ) :=∑

K∈Th

(S∇p,∇ϕ)K , p, ϕ ∈ H1(Th).


A(u,v) :=∑

K∈Th

(u,S−1v)K , u,v ∈ L2(Ω).







B(p, ϕ) :=∑

K∈Th

(S∇p,∇ϕ)K , p, ϕ ∈ H1(Th).


A(u,v) :=∑

K∈Th

(u,S−1v)K , u,v ∈ L2(Ω).





Energy norms

Definition (Energy semi-norm)

|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈ H1(Th).

Definition (Energy norm)

|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈W k0 (Th).


|||v|||2∗ := A(v,v), v ∈ L2(Ω).

Definition (Energy–div norm)

|||v|||2∗,div := |||v|||2∗ + ‖∇ · v‖2, v ∈ H(div,Ω).


Energy norms


|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈ H1(Th).


|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈W k0 (Th).


|||v|||2∗ := A(v,v), v ∈ L2(Ω).


|||v|||2∗,div := |||v|||2∗ + ‖∇ · v‖2, v ∈ H(div,Ω).


Energy norms


|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈ H1(Th).


|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈W k0 (Th).


|||v|||2∗ := A(v,v), v ∈ L2(Ω).


|||v|||2∗,div := |||v|||2∗ + ‖∇ · v‖2, v ∈ H(div,Ω).


Energy norms


|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈ H1(Th).


|||ϕ|||2 := B(ϕ,ϕ), ϕ ∈W k0 (Th).


|||v|||2∗ := A(v,v), v ∈ L2(Ω).


|||v|||2∗,div := |||v|||2∗ + ‖∇ · v‖2, v ∈ H(div,Ω).


An abstract result for the flux variable

Theorem (Abstract framework (scheme-independent))

Let v,w, t ∈ L2(Ω) be arbitrary. Then|||v−w|||∗ ≤ |||w− t|||∗ +

∣∣∣∣A(v−w, v− t|||v− t|||∗)∣∣∣∣ .

A priori error estimate|||u− uh|||∗ ≤ |||u− Πhu|||∗A posteriori error estimate

put v = u, w = uh, t = −S∇s with s ∈ H10 (Ω) arbitrary:

|||u− uh|||∗ ≤ |||uh + S∇s|||∗ +∣∣∣∣A(u− uh, u + S∇s|||u + S∇s|||∗

)∣∣∣∣notice that A(u,−S∇ϕ) = (f , ϕ) (here ϕ = p − s/|||p − s|||)notice that A(uh,−S∇ϕ) = (πl(f ), ϕ)

get |||u−uh|||∗≤ infs∈H10 (Ω)

|||uh +S∇s|||∗+

{∑K∈Th

CPh2KcS,K

‖f−πl(f )‖2K

}12





∣∣∣∣A(v−w, v− t|||v− t|||∗)∣∣∣∣ .

A priori error estimateput v = uh, w = u, t = Πhu:

|||uh − u|||∗ ≤ |||u− Πhu|||∗ +∣∣∣∣A(uh − u, uh − Πhu|||uh − Πhu|||∗

)∣∣∣∣notice that A(uh − u,uh − Πhu) = 0 in MFEsget |||u− uh|||∗ ≤ |||u− Πhu|||∗

A priori error estimate|||u− uh|||∗ ≤ |||u− Πhu|||∗

A posteriori error estimateput v = u, w = uh, t = −S∇s with s ∈ H10 (Ω) arbitrary:




|||uh +S∇s|||∗+

{∑K∈Th

CPh2KcS,K

‖f−πl(f )‖2K

}12





∣∣∣∣A(v−w, v− t|||v− t|||∗)∣∣∣∣ .






|||uh +S∇s|||∗+

{∑K∈Th

CPh2KcS,K

‖f−πl(f )‖2K

}12





∣∣∣∣A(v−w, v− t|||v− t|||∗)∣∣∣∣ .






|||uh +S∇s|||∗+

{∑K∈Th

CPh2KcS,K

‖f−πl(f )‖2K

}12


Postprocessing for the scalar variablePostprocessing in mixed finite elements

Arnold and Brezzi ’85: Mixed and nonconforming finiteelement methods: implementation, postprocessing anderror estimatesBramble and Xu ’89: A local post-processing technique forimproving the accuracy in mixed finite-elementapproximationsStenberg ’91: Postprocessing schemes for some mixedfinite elementsArbogast and Chen ’95: On the implementation of mixedmethods as nonconforming methods for second-orderelliptic problemsChen ’96: Equivalence between and multigrid algorithmsfor nonconforming and mixed methods for second-orderelliptic problems


Postprocessing for the scalar variable

Postprocessing in mixed finite elements

Usually used in order to implement MFEMs and getsuperconvergence for the postprocessed variable.Usually not used in order to get a priori or a posteriori errorestimates.


Postprocessing for the scalar variable

Postprocessing in mixed finite elements

Usually used in order to implement MFEMs and getsuperconvergence for the postprocessed variable.Usually not used in order to get a priori or a posteriori errorestimates.


Lowest-order Raviart–Thomas case

Definition (Postprocessed scalar variable p̃h)

We define p̃h such that, separately on each K ∈ Th,−SK∇p̃h|K = uh|K (flux of p̃h is uh),(p̃h,1)K/|K | = pK (mean of p̃h on K is pK ).

Properties of p̃hp̃h exists and is unique (it is a pw second-order polynomial)p̃h is nonconforming, 6∈ H10 (Ω), only ∈ H1(Th) in generalmeans of traces of p̃h on the sides continuous, p̃h∈W 00 (Th)the means are equal to the Lagrange multipliers from thehybridization

Remarksexact (not weak) connection of p̃h and uhonly valid in the lowest-order case on simplices or, when Sis diagonal, on rectangular parallelepipeds






Proof: 0 = −(∇p̃h,vσK ,L)K∪L − (p̃h,∇ · vσK ,L)K∪L= −〈vσK ,L · n, p̃h〉∂K − 〈vσK ,L · n, p̃h〉∂L= 〈vσK ,L · nK , p̃h|L − p̃h|K 〉σK ,L



General postprocessing

Definition (Postprocessed scalar variable p̃h (Arbogast & Chen))

We define p̃h such that, separately on each K ∈ Th,(p̃h, φh)K = (ph, φh)K ∀φh ∈ Φh(K ).(p̃h, µh)σ = (λh, µh)σ ∀µh ∈ Λh(σ) ∀σ ∈ E inth .

Properties of p̃hp̃h exists and is uniquep̃h is nonconforming, 6∈ H10 (Ω), but p̃h ∈W k0 (Th)p̃h is in general a nonconforming polynomial plus a bubblep̃h satisfies −(S−1uh,vh)K = (∇p̃h,vh)K ∀vh ∈ Vh(K )

Remarksuh is a PVh,S−1 projection of −S∇p̃h onto Vh, weakconnection of p̃h and uhbasis of our a priori and a posteriori error estimates

Framework A priori est. A posteriori est. Remarks Exp. C. Lowest-order Raviart–Thomas case General case










|||p − p̃h||| = |||u− uh|||∗ ≤ |||u− Πhu|||∗ ≤ Chp̃h ∈W 00 (Th): discrete Friedrichs inequality

‖p − p̃h‖ ≤ C12DF

{∑K∈Th

‖∇(p − p̃h)‖2K

} 12

optimal value of CDF (only depends on the shape regularityparameter and infb∈Rd{thickb(Ω)}): Vohralík, NFAO 2005

consequently:{∑

K∈Th ‖p − p̃h‖21,K} 1

2 ≤ Ch

superconvergence: ‖p − p̃h‖ ≤ Ch2





‖p − p̃h‖ ≤ C12DF

{∑K∈Th

‖∇(p − p̃h)‖2K

} 12


consequently:{∑

K∈Th ‖p − p̃h‖21,K} 1

2 ≤ Ch






‖p − p̃h‖ ≤ C12DF

{∑K∈Th

‖∇(p − p̃h)‖2K

} 12


consequently:{∑

K∈Th ‖p − p̃h‖21,K} 1

2 ≤ Ch






‖p − p̃h‖ ≤ C12DF

{∑K∈Th

‖∇(p − p̃h)‖2K

} 12


consequently:{∑

K∈Th ‖p − p̃h‖21,K} 1

2 ≤ Ch






‖p − p̃h‖ ≤ C12DF

{∑K∈Th

‖∇(p − p̃h)‖2K

} 12


consequently:{∑

K∈Th ‖p − p̃h‖21,K} 1

2 ≤ Ch



General case

General case

a little bit more complicated since we only haveuh = −PVh,S−1(S∇p̃h) instead of uh = −S∇p̃hone still easily recovers all the known a priori errorestimates for mixed finite elements

Framework A priori est. A posteriori est. Remarks Exp. C. Estimates for the flux Estimates for the potential Efficiency








What is/should be an a posteriori error estimate

Usual form‖p − ph‖2 .

∑K∈Th ηK (ph)

2.Can be used to determine mesh elements with large error.We can then refine these elements: mesh adaptivity.

Reliability‖p − ph‖2 ≤ C

∑K∈Th ηK (ph)

2

Guaranteed upper bound‖p − ph‖2 ≤

∑K∈Th ηK (ph)

2

Local efficiencyηK (ph)2 ≤ C2eff,K

∑L close to K ‖p − ph‖2L

Asymptotic exactness∑K∈Th ηK (ph)

2/‖p − ph‖2 → 1Robustness

independence of the data variation or mesh propertiesNegligible evaluation cost

estimators which can be evaluated locallyM. Vohralík Unified a priori and a posteriori analysis of MFEs



∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2


∑K∈Th ηK (ph)

2









∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2

Problems:What is C?What does it depend on?How does it depend on data?


∑K∈Th ηK (ph)

2









∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2


∑K∈Th ηK (ph)

2









∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2


∑K∈Th ηK (ph)

2









∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2


∑K∈Th ηK (ph)

2









∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2


∑K∈Th ηK (ph)

2









∑K∈Th ηK (ph)



∑K∈Th ηK (ph)

2


∑K∈Th ηK (ph)

2







Previous works on a posteriori analysis for MFEMs

Previous works . . .Alonso ’96Braess and Verfürth ’96Carstensen ’97Hoppe and Wohlmuth ’97, ’99Kirby ’03El Alaoui and Ern ’04Wheeler and Yotov ’05Lovadina and Stenberg ’06

. . . do not cover

evaluation of the constants (guaranteed upper bound )robustnessasymptotic exactnessan analysis of the convection–diffusion case


Previous works on a posteriori analysis for MFEMs

Previous works . . .Alonso ’96Braess and Verfürth ’96Carstensen ’97Hoppe and Wohlmuth ’97, ’99Kirby ’03El Alaoui and Ern ’04Wheeler and Yotov ’05Lovadina and Stenberg ’06

. . . do not cover

evaluation of the constants (guaranteed upper bound )robustnessasymptotic exactnessan analysis of the convection–diffusion case


A first abstract estimate for the flux

Theorem (A first abstract estimate for the flux and its efficiency)

Let u be the weak flux and let uh ∈ H(div,Ω) be arbitrary. Then

|||u− uh|||2∗ ≤ infs∈H10 (Ω)

|||uh + S∇s|||2∗ +CF,Ωh2Ω

cS,Ω‖f −∇ · uh‖2

≤ |||u− uh|||2∗ +CF,Ωh2Ω

cS,Ω‖f −∇ · uh‖2.

PropertiesGuaranteed upper bound (no undetermined constant).|||uh + S∇s|||∗ penalizes uh 6= −S∇s for some s ∈ H10 (Ω).Advantage: scheme-independent (promoted by Repin).Disadvantage: scheme-independent (no information fromthe computation used).Disadvantage: C1/2F,Ω hΩ/c

1/2S,Ω‖f −∇ · uh‖ too big.


A first abstract estimate for the flux

Theorem (A first abstract estimate for the flux and its efficiency)

Let u be the weak flux and let uh ∈ H(div,Ω) be arbitrary. Then

|||u− uh|||2∗ ≤ infs∈H10 (Ω)

|||uh + S∇s|||2∗ +CF,Ωh2Ω

cS,Ω‖f −∇ · uh‖2

≤ |||u− uh|||2∗ +CF,Ωh2Ω

cS,Ω‖f −∇ · uh‖2.

PropertiesGuaranteed upper bound (no undetermined constant).|||uh + S∇s|||∗ penalizes uh 6= −S∇s for some s ∈ H10 (Ω).Advantage: scheme-independent (promoted by Repin).Disadvantage: scheme-independent (no information fromthe computation used).Disadvantage: C1/2F,Ω hΩ/c

1/2S,Ω‖f −∇ · uh‖ too big.


An improved abstract estimate for the flux

Theorem (An improved abstract estimate for the flux and itsefficiency)

Let u be the weak flux and let uh ∈ H(div,Ω) such that∇ · uh = πl(f ) be arbitrary. Then

|||u− uh|||2∗ ≤ infs∈H10 (Ω)

|||uh + S∇s|||2∗ + η2R ≤ |||u− uh|||2∗ + η2R,

whereηR :=

{∑K∈Th

CPh2KcS,K

‖f − πl(f )‖2K

} 12

.

PropertiesNo global Galerkin orthogonality needed, just localconservativity.ηR is in general a higher-order term for RT methods.ηR is not in general a higher-order term for BDM methods.


An improved abstract estimate for the flux

Theorem (An improved abstract estimate for the flux and itsefficiency)


|||u− uh|||2∗ ≤ infs∈H10 (Ω)

|||uh + S∇s|||2∗ + η2R ≤ |||u− uh|||2∗ + η2R,

whereηR :=

{∑K∈Th

CPh2KcS,K

‖f − πl(f )‖2K

} 12

.

PropertiesNo global Galerkin orthogonality needed, just localconservativity.ηR is in general a higher-order term for RT methods.ηR is not in general a higher-order term for BDM methods.


An energy–div norm abstract estimate for the flux

Theorem (An energy–div norm abstract estimate for the fluxand its efficiency)

Let u be the weak flux and let uh ∈ H(div,Ω) such that∇ · uh = πl(f ) be arbitrary. Then|||u− uh|||2∗,div ≤ inf

s∈H10 (Ω)|||uh + S∇s|||2∗ + ‖f − πl(f )‖2 + η2R

≤ |||u− uh|||2∗,div + η2R.

PropertiesηR gets always a higher-order term.


An energy–div norm abstract estimate for the flux

Theorem (An energy–div norm abstract estimate for the fluxand its efficiency)

Let u be the weak flux and let uh ∈ H(div,Ω) such that∇ · uh = πl(f ) be arbitrary. Then|||u− uh|||2∗,div ≤ inf

s∈H10 (Ω)|||uh + S∇s|||2∗ + ‖f − πl(f )‖2 + η2R

≤ |||u− uh|||2∗,div + η2R.

PropertiesηR gets always a higher-order term.


A fully computable estimate for the flux

Theorem (A fully computable estimate for the flux)


|||u− uh|||2∗ ≤∑

K∈Th

(η2P,K + η

2R,K

),

|||u− uh|||2∗,div ≤∑

K∈Th

(η2P,K + η

2R,K + η

2D,K

).

potential estimatorηP,K := |||uh + S∇(IOs(p̃h))|||∗,KIOs(p̃h): Oswald interpolate Pn(Th)→ Pn(Th) ∩ H10 (Ω)

residual estimatorηR,K :=

C1/2P hKc1/2S,K‖f − πl (f )‖K

divergence estimatorηD,K := ‖f − πl (f )‖K





|||u− uh|||2∗ ≤∑

K∈Th

(η2P,K + η

2R,K

),

|||u− uh|||2∗,div ≤∑

K∈Th

(η2P,K + η

2R,K + η

2D,K

).









|||u− uh|||2∗ ≤∑

K∈Th

(η2P,K + η

2R,K

),

|||u− uh|||2∗,div ≤∑

K∈Th

(η2P,K + η

2R,K + η

2D,K

).









|||u− uh|||2∗ ≤∑

K∈Th

(η2P,K + η

2R,K

),

|||u− uh|||2∗,div ≤∑

K∈Th

(η2P,K + η

2R,K + η

2D,K

).






An abstract estimate for the potential

Theorem (Abstract a posteriori estimate for the potential and itsefficiency)

Let p be the weak potential and let p̃h ∈ H1(Th) be arbitrary.Then

|||p − p̃h|||2 ≤ infs∈H10 (Ω)

|||p̃h − s|||2

+ inft∈H(div,Ω)

supϕ∈H10 (Ω), |||ϕ|||=1

((f−∇·t, ϕ)−(S∇p̃h +t,∇ϕ))2

≤2|||p − p̃h|||2.

PropertiesGuaranteed upper bound, quasi-exact, and robust.Holds uniformly for any mesh (anisotropic) and polynomialdegree of ph.


An abstract estimate for the potential

Theorem (Abstract a posteriori estimate for the potential and itsefficiency)

Let p be the weak potential and let p̃h ∈ H1(Th) be arbitrary.Then

|||p − p̃h|||2 ≤ infs∈H10 (Ω)

|||p̃h − s|||2

+ inft∈H(div,Ω)

supϕ∈H10 (Ω), |||ϕ|||=1

((f−∇·t, ϕ)−(S∇p̃h +t,∇ϕ))2

≤2|||p − p̃h|||2.

PropertiesGuaranteed upper bound, quasi-exact, and robust.Holds uniformly for any mesh (anisotropic) and polynomialdegree of ph.


A first computable estimate for the potential

Theorem (A first computable estimate for the potential)

Let p be the weak potential and let p̃h ∈ H1(Th) be arbitrary.Take any th ∈ H(div,Ω) and any sh ∈ H10 (Ω). Then

|||p − p̃h|||2 ≤|||p̃h − sh|||2+

(C1/2F,Ω hΩ

c1/2S,Ω‖f −∇·th‖+|||S∇p̃h + th|||∗

)2.

Properties|||S∇p̃h + th|||∗ penalizes −S∇p̃h 6∈ H(div,Ω).|||p̃h − sh||| penalizes p̃h 6∈ H10 (Ω).Advantage: scheme-independent.

Disadvantage: C1/2F,Ω hΩ/c1/2S,Ω‖f −∇ · th‖ too big.


A first computable estimate for the potential

Theorem (A first computable estimate for the potential)

Let p be the weak potential and let p̃h ∈ H1(Th) be arbitrary.Take any th ∈ H(div,Ω) and any sh ∈ H10 (Ω). Then

|||p − p̃h|||2 ≤|||p̃h − sh|||2+

(C1/2F,Ω hΩ

c1/2S,Ω‖f −∇·th‖+|||S∇p̃h + th|||∗

)2.

Properties|||S∇p̃h + th|||∗ penalizes −S∇p̃h 6∈ H(div,Ω).|||p̃h − sh||| penalizes p̃h 6∈ H10 (Ω).Advantage: scheme-independent.

Disadvantage: C1/2F,Ω hΩ/c1/2S,Ω‖f −∇ · th‖ too big.


A fully computable estimate for the potential

Theorem (A fully computable estimate for the potential)

Let p be the weak potential and let p̃h ∈ H1(Th) anduh ∈ H(div,Ω) such that ∇ · uh = πl(f ) be arbitrary. Then

|||p − p̃h|||2 ≤∑

K∈Th

{η2NC,K + (ηR,K + ηDF,K )

2}.

nonconformity estimatorηNC,K := |||p̃h − IOs(p̃h)|||K

diffusive flux estimatorηDF,K := |||uh + S∇p̃h|||∗,K

residual estimator

ηR,K :=C1/2P hK

c1/2S,K‖f − πl (f )‖K





|||p − p̃h|||2 ≤∑

K∈Th


2}.



residual estimator

ηR,K :=C1/2P hK

c1/2S,K‖f − πl (f )‖K





|||p − p̃h|||2 ≤∑

K∈Th


2}.



residual estimator

ηR,K :=C1/2P hK

c1/2S,K‖f − πl (f )‖K





|||p − p̃h|||2 ≤∑

K∈Th


2}.



residual estimator

ηR,K :=C1/2P hK

c1/2S,K‖f − πl (f )‖K


Local efficiency of the estimates

Theorem (Local efficiency of the estimates)Let p,u be the weak potential and flux, respectively, and let uhbe the MFE flux and p̃h the postprocessed potential. Then

ηDF,K ≤ |||u− uh|||∗,K + |||p − p̃h|||K ,ηP,K ≤ ηDF,K + ηNC,K ,

ηNC,K ≤ C

√CS,KcS,TK

|||p − p̃h|||TK ,

ηR,K ≤ C

√CS,KcS,K

|||u− uh|||∗,K ,

where C depends only on the space dimension d, the maximalpolynomial degree n of p̃h, the shape regularity parameter κT ,and the polynomial degree m of f .



Proof for ηNC,K .

Oswald interpolate (Karakashian and Pascal ’03, Burmanand Ern ’07):

‖∇(ϕh − IOs(ϕh))‖2K ≤ C∑σ∈ẼK

h−1σ ‖[[ϕh]]‖2σ

Achdou, Bernardi, Coquel ’03:

h− 12σ ‖[[p̃h]]‖σ ≤ C

∑L;σ∈EL

‖∇(p̃h − ϕ)‖L

η2NC,K = |||p̃h − IOs(p̃h)|||2K ≤ CCS,K∑σ∈ẼK

h−1σ ‖[[p̃h]]‖2σ

≤ CCS,K∑L∈TK

‖∇(p − p̃h)‖2L ≤ CCS,KcS,TK

∑L∈TK

|||p − p̃h|||2L



Proof for ηR,K .

‖f − πl(f )‖K = ‖f −∇ · uh‖K ≤ CC1/2S,K h

−1K |||u− uh|||∗,K

element bubble functionsequivalence of norms on finite-dimensional spacesweak solution definitionGreen theoremCauchy–Schwarz inequalityenergy norm definitioninverse inequality

residual estimator is always efficient (also for BDM)

Framework A priori est. A posteriori est. Remarks Exp. C. Comments L2(Ω) estimates RT0 and pure diffusion problems








Comments on the estimates

General comments

p ∈ H1(Ω), no additional regularityno convexity of Ω neededno saturation assumptionno Helmholtz decompositionno shape-regularity needed for the upper bounds (only forthe efficiency proofs)polynomial degree-independent upper boundno “monotonicity” hypothesis on inhomogeneitiesdistributionthe only important tool: optimal Poincaré–Friedrichs andtrace inequalitiesholds from diffusion to convection–diffusion–reaction cases


L2(Ω) estimates

Theorem (Estimate for p̃h in the L2(Ω)-norm)

Let p be the weak potential and let p̃h ∈W 00 (Th) anduh ∈ H(div,Ω) such that ∇ · uh = πl(f ) be arbitrary. Then

‖p − p̃h‖2 ≤CDFcS,Ω

∑K∈Th


2}.

Theorem (Estimate for ph in the L2(Ω)-norm)

Let p be the weak potential and let ph ∈ Φh, p̃h ∈W 00 (Th), anduh ∈ H(div,Ω) such that ∇ · uh = πl(f ) be arbitrary. Then

‖p−ph‖ ≤

CDFcS,Ω ∑K∈Th{η2NC,K + (ηR,K + ηDF,K )

2}

12

+‖p̃h−ph‖.


L2(Ω) estimates

Theorem (Estimate for p̃h in the L2(Ω)-norm)

Let p be the weak potential and let p̃h ∈W 00 (Th) anduh ∈ H(div,Ω) such that ∇ · uh = πl(f ) be arbitrary. Then

‖p − p̃h‖2 ≤CDFcS,Ω

∑K∈Th


2}.

Theorem (Estimate for ph in the L2(Ω)-norm)

Let p be the weak potential and let ph ∈ Φh, p̃h ∈W 00 (Th), anduh ∈ H(div,Ω) such that ∇ · uh = πl(f ) be arbitrary. Then

‖p−ph‖ ≤

CDFcS,Ω ∑K∈Th{η2NC,K + (ηR,K + ηDF,K )

2}

12

+‖p̃h−ph‖.


Some additional comments

Some additional comments

We believe that L2(Ω) norm is not optimal for a posteriorierror estimates in mixed finite elements.We believe that trying to directly and only derive estimatesfor ph in the L2(Ω)-norm was the bottleneck of a lot ofprevious works.|||uh + S∇p̃h|||∗,K or |||uh + S∇(IOs(p̃h))|||∗,K (ourestimates): clear physical meaninghK |||uh + S∇ph|||∗,K = hK |||uh|||∗,K in RT0 (some previousworks): no good sense


Pure diffusion problem −∇ · (S∇p) = f , p = 0 on ∂ΩTheorem (Mixed FEM for the diffusion problem)There holds|||p − p̃h||| ≤ inf

s∈H10 (Ω)|||p̃h − s|||+

{∑K∈Th

CPh2K

cS,K‖f − fK‖2K

} 12

.

Theorem (Galerkin FEM for the diffusion problem)There holds |||p − ph||| ≤ inf

sh∈Vh|||p − sh|||.

Mixed FEM 1D:no nonconformity, p̃h ∈ H10 (Ω)|||p − p̃h||| ≤ Ch2 when f ∈ H1(Th)p̃h = p, the exact solution, for pw constant S (arbitraryinhomogeneities) and pw constant f

Galerkin FEM 1D:|||p − p̃h||| ≤ Ch



s∈H10 (Ω)|||p̃h − s|||+

{∑K∈Th

CPh2K


} 12

.


sh∈Vh|||p − sh|||.





s∈H10 (Ω)|||p̃h − s|||+

{∑K∈Th

CPh2K


} 12

.


sh∈Vh|||p − sh|||.





s∈H10 (Ω)|||p̃h − s|||+

{∑K∈Th

CPh2K


} 12

.


sh∈Vh|||p − sh|||.



Framework A priori est. A posteriori est. Remarks Exp. C. Inhomogeneous diffusion Dominating convection








Problem with discontinuous and inhomogeneousdiffusion tensor

consider the pure diffusion equation

−∇ · (S∇p) = 0 in Ω = (−1,1)× (−1,1)discontinuous and inhomogeneous S, two cases:

−1 0 1−1

0

1

s1=5s

2=1

s3=5 s

4=1

−1 0 1−1

0

1

s1=100s

2=1

s3=100 s

4=1

analytical solution: singularity at the origin

p(r , θ)|Ωi = rα(ai sin(αθ) + bi cos(αθ))

(r , θ) polar coordinates in Ωai , bi constants depending on Ωiα regularity of the solution


Analytical solutions


Estimated and actual error distribution on anadaptively refined mesh, case 1

0.005

0.01

0.015

0.02

0.025

0.03

0.035

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0.005

0.01

0.015

0.02

0.025

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Approximate solution and the correspondingadaptively refined mesh, case 2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

xy

p


Estimated and actual error against the number ofelements in uniformly/adaptively refined meshes

102

103

104

105

10−2

10−1

100

101

Number of triangles

Ene

rgy

erro

r

error uniformestimate uniformerror adapt.estimate adapt.

102

103

104

105

100

101

102

Number of triangles

Ene

rgy

erro

r



Global efficiency of the estimates

102

103

104

105

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

Number of triangles

Effi

cien

cy

efficiency uniformefficiency adapt.

102

103

104

105

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

Number of triangles

Effi

cien

cy



Convection-dominated problem

consider the convection–diffusion–reaction equation

−ε4p +∇ · (p(0,1)) + p = f in Ω = (0,1)× (0,1)

analytical solution: layer of width a

p(x , y) = 0.5(

1− tanh(0.5− x

a

))consider

ε = 1, a = 0.5ε = 10−2, a = 0.05ε = 10−4, a = 0.02

unstructured grid of 46 elements given,uniformly/adaptively refined


Analytical solutions, ε = 1, a = 0.5 and ε = 10−4,a = 0.02

0.2

0.3

0.4

0.5

0.6

0.7

0.8

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0.2

0.3

0.4

0.5

0.6

0.7

0.8

xy

p

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.2

0.4

0.6

0.8

1

xy

p


Estimated and actual error distribution, ε = 1, a = 0.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

−3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

1

1.5

2

2.5

3

x 10−3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Modified Oswald interpolate: estimated and actualerror against the number of elements and globalefficiency of the estimates, ε = 1, a = 0.5

101

102

103

104

105

10−5

10−4

10−3

10−2

10−1

100

Number of triangles

Ene

rgy

erro

r

error uniformest. uniformest. res. uniformest. nonc. uniformest. upw. uniformest. conv. uniform

101

102

103

104

105

1.45

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

Number of triangles

Effi

cien

cy

efficiency uniform


Oswald interpolate: estimated and actual error againstthe number of elements and global efficiency of theestimates, ε = 1, a = 0.5

101

102

103

104

105

10−5

10−4

10−3

10−2

10−1

100

Number of triangles

Ene

rgy

erro

r

error uniformest. uniformest. res. uniformest. nonc. uniformest. upw. uniformest. conv. uniform

101

102

103

104

105

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Number of triangles

Effi

cien

cy

efficiency uniform


Estimated and actual error distribution, ε = 10−2,a = 0.05

2

4

6

8

10

12

x 10−3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

−3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Approximate solution and the correspondingadaptively refined mesh, ε = 10−4, a = 0.02

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.2

0.40.6

0.81

00.2

0.40.6

0.810

0.2

0.4

0.6

0.8

1

xy

p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


Estimated and actual error against the number ofelements in uniformly/adaptively refined meshes,ε = 10−2, a = 0.05 and ε = 10−4, a = 0.02

101

102

103

104

105

10−3

10−2

10−1

100

101

Number of triangles

Ene

rgy

erro

r


101

102

103

104

105

10−2

10−1

100

101

Number of triangles

Ene

rgy

erro

r



Global efficiency of the estimates, ε = 10−2, a = 0.05and ε = 10−4, a = 0.02

101

102

103

104

105

2

3

4

5

6

7

8

9

10

11

12

Number of triangles

Effi

cien

cy


101

102

103

104

105

10

20

30

40

50

60

70

80

90

100

Number of triangles

Effi

cien

cy



Conclusions and future work

Conclusionsunified framework for a priori and a posteriori error controlin mixed finite elementsoptimality of the framework for a posteriori error estimation:guaranteed upper bound, local efficiency, asymptoticexactness, robustness, negligible evaluation costdirectly implementable—all constants evaluatedparallel work for finite volumes, discontinuous Galerkinfinite elements, and continuous finite elements

Future workfull asymptotic exactness and robustnessnonlinear (degenerate) casesextensions to other types of problems (Stokes,Navier–Lamé, Maxwell)systems of equations



Conclusionsunified framework for a priori and a posteriori error controlin mixed finite elementsoptimality of the framework for a posteriori error estimation:guaranteed upper bound, local efficiency, asymptoticexactness, robustness, negligible evaluation costdirectly implementable—all constants evaluatedparallel work for finite volumes, discontinuous Galerkinfinite elements, and continuous finite elements

Future workfull asymptotic exactness and robustnessnonlinear (degenerate) casesextensions to other types of problems (Stokes,Navier–Lamé, Maxwell)systems of equations


BibliographyPapers

VOHRALÍK M., Unified primal formulation-based a priori and aposteriori error analysis of mixed finite element methods, to besubmitted.VOHRALÍK M., A posteriori error estimates for lowest-order mixedfinite element discretizations of convection–diffusion–reactionequations, SIAM J. Numer. Anal. 45 (2007), 1570–1599.VOHRALÍK M., Residual flux-based a posteriori error estimatesfor finite volume discretizations of inhomogeneous, anisotropic,and convection-dominated problems, submitted to Numer. Math.ERN A., STEPHANSEN, A. F., VOHRALÍK M., Improved energynorm a posteriori error estimation based on flux reconstructionfor discontinuous Galerkin methods, submitted to SIAM J.Numer. Anal.VOHRALÍK M., Guaranteed and fully robust a posteriori errorestimates for conforming discretizations of diffusion problemswith discontinuous coefficients, submitted to Math. Comp.

Thank you for your attention!M. Vohralík Unified a priori and a posteriori analysis of MFEs

General frameworkAn abstract result for the flux variablePostprocessing for the scalar variable

A priori error estimatesLowest-order Raviart--Thomas caseGeneral case

A posteriori error estimatesEstimates for the fluxEstimates for the potentialLocal efficiency

RemarksComments on the estimatesL2() estimatesRT0 and pure diffusion problems

Numerical experiments

Unified primal formulation-based a priori and a posteriori error analysis … · 2017. 6. 12. ·...

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