Upload
hakien
View
217
Download
0
Embed Size (px)
Citation preview
Estimations d’erreur a posteriori pour lesvolumes finis et les éléments finis mixtes
Martin Vohralík
Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)
Séminaire du LJLL, 15 décembre 2006
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Motivation
Problems−∇ · S∇p = f
flow in porous mediaS inhomogeneous andanisotropic
−∇ · S∇p +∇ · (pw) + rp = ftransport in porous mediaconvection-dominated
Mixed finite elements / finitevolumes
accurate for inhomogeneousand anisotropic pbsupwinding forconvection-dominated pbs
need for good a posteriori error control
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Motivation
Problems−∇ · S∇p = f
flow in porous mediaS inhomogeneous andanisotropic
−∇ · S∇p +∇ · (pw) + rp = ftransport in porous mediaconvection-dominated
Mixed finite elements / finitevolumes
accurate for inhomogeneousand anisotropic pbsupwinding forconvection-dominated pbs
need for good a posteriori error control
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Motivation
Problems−∇ · S∇p = f
flow in porous mediaS inhomogeneous andanisotropic
−∇ · S∇p +∇ · (pw) + rp = ftransport in porous mediaconvection-dominated
Mixed finite elements / finitevolumes
accurate for inhomogeneousand anisotropic pbsupwinding forconvection-dominated pbs
need for good a posteriori error control
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What is an a posteriori error estimate
A posteriori error estimate
Let p be a weak solution of a PDE.Let ph be its approximate numerical solution.A priori error estimate: ‖p − ph‖Ω ≤ f (p)hq. Dependent onp, not computable. Useful in theory.A posteriori error estimate: ‖p − ph‖Ω . f (ph). Only usesph, computable. Great in practice.
Usual formf (ph)
2 =∑
K∈ThηK (ph)
2, where ηK (ph) is an elementindicator.Can be used to determine mesh elements with large error.We can then refine these elements: mesh adaptivity.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What is an a posteriori error estimate
A posteriori error estimate
Let p be a weak solution of a PDE.Let ph be its approximate numerical solution.A priori error estimate: ‖p − ph‖Ω ≤ f (p)hq. Dependent onp, not computable. Useful in theory.A posteriori error estimate: ‖p − ph‖Ω . f (ph). Only usesph, computable. Great in practice.
Usual formf (ph)
2 =∑
K∈ThηK (ph)
2, where ηK (ph) is an elementindicator.Can be used to determine mesh elements with large error.We can then refine these elements: mesh adaptivity.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
(Mathematical) reliability (global upper bound)‖p − ph‖2
Ω ≤ C∑
K∈ThηK (ph)
2
Problems:What is C?What does it depend on?How does it depend on data?
Mathematicians often do not care about constants...(Real-life) reliability (global upper bound)
‖p − ph‖2Ω ≤
∑K∈Th
ηK (ph)2
Real life depends on constants very much!Global efficiency (global lower bound)∑
K∈ThηK (ph)
2 ≤ C2eff,Ω‖p − ph‖2
Ω
Local efficiency (local lower bound)ηK (ph)
2 ≤ C2eff,K
∑L close to K ‖p − ph‖2
L
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
What an a posteriori error estimate should fulfill
Semi-robustnessCeff,K only depends on inhomogeneities and anisotropiesaround K (Bernardi and Verfürth ’00, Galerkin FEMs)Ceff,K affine dependence on the local Péclet numberaround K (energy norm), PeK := hK
|wK ||SK | , (Verfürth ’98,
Galerkin FEMs)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Previous works on a posteriori error estimates forFVMs and MFEMs
Previous works on FVMs . . .Ohlberger ’01Lazarov and Tomov ’02Achdou, Bernardi, andCoquel ’03Nicaise ’05
Previous works on MFEMs . . .Alonso ’96Braess and Verfürth ’96Carstensen ’97Kirby ’03El Alaoui and Ern ’04Wheeler and Yotov ’05Lovadina and Stenberg ’06
. . . do not cover
evaluation of the constants (with the exception of Nicaise)a proper analysis of inhomogeneous (with the exception ofEl Alaoui and Ern ’04) and anisotropic problemsa proper analysis of the convection–diffusion case
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Previous works on a posteriori error estimates forFVMs and MFEMs
Previous works on FVMs . . .Ohlberger ’01Lazarov and Tomov ’02Achdou, Bernardi, andCoquel ’03Nicaise ’05
Previous works on MFEMs . . .Alonso ’96Braess and Verfürth ’96Carstensen ’97Kirby ’03El Alaoui and Ern ’04Wheeler and Yotov ’05Lovadina and Stenberg ’06
. . . do not cover
evaluation of the constants (with the exception of Nicaise)a proper analysis of inhomogeneous (with the exception ofEl Alaoui and Ern ’04) and anisotropic problemsa proper analysis of the convection–diffusion case
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Previous works on a posteriori error estimates forFVMs and MFEMs
Previous works on FVMs . . .Ohlberger ’01Lazarov and Tomov ’02Achdou, Bernardi, andCoquel ’03Nicaise ’05
Previous works on MFEMs . . .Alonso ’96Braess and Verfürth ’96Carstensen ’97Kirby ’03El Alaoui and Ern ’04Wheeler and Yotov ’05Lovadina and Stenberg ’06
. . . do not cover
evaluation of the constants (with the exception of Nicaise)a proper analysis of inhomogeneous (with the exception ofEl Alaoui and Ern ’04) and anisotropic problemsa proper analysis of the convection–diffusion case
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
A convection–diffusion–reaction problem
Problem
−∇ · S∇p +∇ · (pw) + rp = f in Ω ,
p = g on ΓD ,
−S∇p · n = u on ΓN
AssumptionsΩ ⊂ Rd , d = 2, 3, is a polygonal domaindata related to a basic triangulation Th of Ω
S|K is a constant SPD matrix, cS,K its smallest, and CS,K
its largest eigenvalue on each K ∈ Th(12∇ ·w + r
)|K ≥ cw,r ,K ≥ 0 (from pure diffusion to
convection–diffusion–reaction cases)Difficulties
S inhomogeneous and anisotropicw dominates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Bilinear form, weak solution, energy (semi-)norm
Definition (Bilinear form B)
We define a bilinear form B for p, ϕ ∈ H1(Th) by
B(p, ϕ) :=∑
K∈Th
(S∇p,∇ϕ)K + (∇ · (pw), ϕ)K + (rp, ϕ)K
.
Definition (Weak solution)
Weak solution: p ∈ H1(Ω) with p|ΓD = g such that
B(p, ϕ) = (f , ϕ)Ω − 〈u, ϕ〉ΓN ∀ϕ ∈ H1D(Ω).
Definition (Energy (semi-)norm)
We define the energy (semi-)norm for ϕ ∈ H1(Th) by
|||ϕ|||2Ω :=∑
K∈Th
|||ϕ|||2K , |||ϕ|||2K :=∥∥∥S
12∇ϕ
∥∥∥2
K+∥∥∥(1
2∇·w+r
) 12ϕ∥∥∥2
K.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Bilinear form, weak solution, energy (semi-)norm
Definition (Bilinear form B)
We define a bilinear form B for p, ϕ ∈ H1(Th) by
B(p, ϕ) :=∑
K∈Th
(S∇p,∇ϕ)K + (∇ · (pw), ϕ)K + (rp, ϕ)K
.
Definition (Weak solution)
Weak solution: p ∈ H1(Ω) with p|ΓD = g such that
B(p, ϕ) = (f , ϕ)Ω − 〈u, ϕ〉ΓN ∀ϕ ∈ H1D(Ω).
Definition (Energy (semi-)norm)
We define the energy (semi-)norm for ϕ ∈ H1(Th) by
|||ϕ|||2Ω :=∑
K∈Th
|||ϕ|||2K , |||ϕ|||2K :=∥∥∥S
12∇ϕ
∥∥∥2
K+∥∥∥(1
2∇·w+r
) 12ϕ∥∥∥2
K.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Bilinear form, weak solution, energy (semi-)norm
Definition (Bilinear form B)
We define a bilinear form B for p, ϕ ∈ H1(Th) by
B(p, ϕ) :=∑
K∈Th
(S∇p,∇ϕ)K + (∇ · (pw), ϕ)K + (rp, ϕ)K
.
Definition (Weak solution)
Weak solution: p ∈ H1(Ω) with p|ΓD = g such that
B(p, ϕ) = (f , ϕ)Ω − 〈u, ϕ〉ΓN ∀ϕ ∈ H1D(Ω).
Definition (Energy (semi-)norm)
We define the energy (semi-)norm for ϕ ∈ H1(Th) by
|||ϕ|||2Ω :=∑
K∈Th
|||ϕ|||2K , |||ϕ|||2K :=∥∥∥S
12∇ϕ
∥∥∥2
K+∥∥∥(1
2∇·w+r
) 12ϕ∥∥∥2
K.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
General finite volume scheme
Definition (FV scheme for −∇ · S∇p +∇ · (pw) + rp = f )
Find pK , K ∈ Th, such that∑σ∈EK
SK ,σ +∑σ∈EK
WK ,σ + rK pK |K | = fK |K | ∀K ∈ Th .
SK ,σ : diffusive fluxWK ,σ : convective flux
no specific form,
just conservativity neededrK := (r , 1)/|K |fK := (f , 1)/|K |
Example
SK ,σ = −sK ,L|σK ,L|dK ,L
(pL − pK )
WK ,σ = pσ〈w · n, 1〉σ: weighted-upwind
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
General finite volume scheme
Definition (FV scheme for −∇ · S∇p +∇ · (pw) + rp = f )
Find pK , K ∈ Th, such that∑σ∈EK
SK ,σ +∑σ∈EK
WK ,σ + rK pK |K | = fK |K | ∀K ∈ Th .
SK ,σ : diffusive fluxWK ,σ : convective flux
no specific form,
just conservativity neededrK := (r , 1)/|K |fK := (f , 1)/|K |
Example
SK ,σ = −sK ,L|σK ,L|dK ,L
(pL − pK )
WK ,σ = pσ〈w · n, 1〉σ: weighted-upwind
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
General finite volume scheme
Definition (FV scheme for −∇ · S∇p +∇ · (pw) + rp = f )
Find pK , K ∈ Th, such that∑σ∈EK
SK ,σ +∑σ∈EK
WK ,σ + rK pK |K | = fK |K | ∀K ∈ Th .
SK ,σ : diffusive fluxWK ,σ : convective flux
no specific form,
just conservativity neededrK := (r , 1)/|K |fK := (f , 1)/|K |
Example
SK ,σ = −sK ,L|σK ,L|dK ,L
(pL − pK )
WK ,σ = pσ〈w · n, 1〉σ: weighted-upwind
σK,L K
L
• xK
•xL
·dK,L
nK
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
First-order system reformulation
Second-order PDE
−∇ · S∇p +∇ · (pw) + rp = f
⇓First-order system
u = −S∇p ,
∇ · u +∇ · (pw) + rp = f
ΓD = ∂Ω and g = 0 for the sake of simplicity
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
A centered scheme
Definition (A centered mixed finite element scheme)
Find uh ∈ RT0(Th) and ph ∈ Φ(Th) such that
(S−1uh, vh)Ω − (ph,∇ · vh)Ω = 0 ∀vh ∈ RT0(Th) ,
(∇ · uh, φh)Ω − (S−1uhw, φh)Ω + ((r +∇ ·w)ph, φh)Ω= (f , φh)Ω ∀φh ∈ Φ(Th) .
Basis of RT0(Th)
σvσ
Basis of Φ(Th)
φK = 1|K
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Problem FV schemes MFE schemes
An upwind-weighted scheme
Definition (An upwind-weighted mixed finite element scheme)
Find uh ∈ RT0(Th) and ph ∈ Φ(Th) such that
(S−1uh, vh)Ω − (ph,∇ · vh)Ω = 0 ∀vh ∈ RT0(Th) ,
(∇ · uh, φh)Ω +∑
K∈Th
∑σ∈EK
pσwK ,σφK + (rph, φh)Ω
= (f , φh)Ω ∀φh ∈ Φh .
wK ,σ := 〈w · n, 1〉σ: flux of w through a side σpσ: weighted upwind value,
pσ :=
(1− νσ)pK + νσpL if wK ,σ ≥ 0(1− νσ)pL + νσpK if wK ,σ < 0
νσ: coefficient of the amount of upstream weighting, e.g.
νσ := mincS,σ|σ|/(hσ|wK ,σ|), 1/2
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,−SK∇ph|K = uh|K (flux of ph is uh),(ph, 1)K /|K | = pK (mean of ph on K is pK ).
Properties of ph
ph exists and is unique (it is a pw second-order polynomial)ph is nonconforming, 6∈ H1
0 (Ω), only ∈ H1(Th) in generalmeans of traces of ph on the sides continuous, ph ∈ W0(Th)
the means are equal to the Lagrange multipliers from thehybridized forms of the schemes
Remarksph is a connection of ph and uh
basis of our a posteriori error estimatesM. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,−SK∇ph|K = uh|K (flux of ph is uh),(ph, 1)K /|K | = pK (mean of ph on K is pK ).
Properties of ph
ph exists and is unique (it is a pw second-order polynomial)ph is nonconforming, 6∈ H1
0 (Ω), only ∈ H1(Th) in generalmeans of traces of ph on the sides continuous, ph ∈ W0(Th)
the means are equal to the Lagrange multipliers from thehybridized forms of the schemes
Remarksph is a connection of ph and uh
basis of our a posteriori error estimatesM. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,−SK∇ph|K = uh|K (flux of ph is uh),(ph, 1)K /|K | = pK (mean of ph on K is pK ).
Properties of ph
ph exists and is unique (it is a pw second-order polynomial)ph is nonconforming, 6∈ H1
0 (Ω), only ∈ H1(Th) in generalmeans of traces of ph on the sides continuous, ph ∈ W0(Th)
the means are equal to the Lagrange multipliers from thehybridized forms of the schemes
Remarksph is a connection of ph and uh
basis of our a posteriori error estimatesM. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,−SK∇ph|K = uh|K (flux of ph is uh),(ph, 1)K /|K | = pK (mean of ph on K is pK ).
Properties of phph exists and is unique (it is a pw second-order polynomial)ph is nonconforming, 6∈ H1
0 (Ω), only ∈ H1(Th) in generalmeans of traces of ph on the sides continuous, ph ∈ W0(Th)the means are equal to the Lagrange multipliers from thehybridized forms of the schemes
Proof: 0 = −(∇ph, vσK ,L)K∪L − (ph,∇ · vσK ,L)K∪L
= −〈vσK ,L · n, ph〉∂K − 〈vσK ,L · n, ph〉∂L
= 〈vσK ,L · nK , ph|L − ph|K 〉σK ,L
Remarksph is a connection of ph and uhbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,−SK∇ph|K = uh|K (flux of ph is uh),(ph, 1)K /|K | = pK (mean of ph on K is pK ).
Properties of ph
ph exists and is unique (it is a pw second-order polynomial)ph is nonconforming, 6∈ H1
0 (Ω), only ∈ H1(Th) in generalmeans of traces of ph on the sides continuous, ph ∈ W0(Th)
the means are equal to the Lagrange multipliers from thehybridized forms of the schemes
Remarksph is a connection of ph and uh
basis of our a posteriori error estimatesM. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,−SK∇ph|K = uh|K (flux of ph is uh),(ph, 1)K /|K | = pK (mean of ph on K is pK ).
Properties of ph
ph exists and is unique (it is a pw second-order polynomial)ph is nonconforming, 6∈ H1
0 (Ω), only ∈ H1(Th) in generalmeans of traces of ph on the sides continuous, ph ∈ W0(Th)
the means are equal to the Lagrange multipliers from thehybridized forms of the schemes
Remarksph is a connection of ph and uh
basis of our a posteriori error estimatesM. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Theorem (A posteriori error estimate for the centered scheme)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K + ηC,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IMO(ph)|||KIMO(ph): modified Oswald int. (preserves means of traces)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 1
cw,r,K
convection estimator
ηC,K := min‖∇·(vw)− 1
2 v∇·w‖K√cw,r,K
,(
CP,d h2K‖∇·(vw)‖2
KcS,K
+‖v∇·w‖2
K4cw,r,K
)12
v = ph − IMO(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Theorem (A posteriori error estimate for the centered scheme)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K + ηC,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IMO(ph)|||KIMO(ph): modified Oswald int. (preserves means of traces)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 1
cw,r,K
convection estimator
ηC,K := min‖∇·(vw)− 1
2 v∇·w‖K√cw,r,K
,(
CP,d h2K‖∇·(vw)‖2
KcS,K
+‖v∇·w‖2
K4cw,r,K
)12
v = ph − IMO(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Theorem (A posteriori error estimate for the centered scheme)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K + ηC,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IMO(ph)|||KIMO(ph): modified Oswald int. (preserves means of traces)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 1
cw,r,K
convection estimator
ηC,K := min‖∇·(vw)− 1
2 v∇·w‖K√cw,r,K
,(
CP,d h2K‖∇·(vw)‖2
KcS,K
+‖v∇·w‖2
K4cw,r,K
)12
v = ph − IMO(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Theorem (A posteriori error estimate for the centered scheme)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K + ηC,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IMO(ph)|||KIMO(ph): modified Oswald int. (preserves means of traces)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 1
cw,r,K
convection estimator
ηC,K := min‖∇·(vw)− 1
2 v∇·w‖K√cw,r,K
,(
CP,d h2K‖∇·(vw)‖2
KcS,K
+‖v∇·w‖2
K4cw,r,K
)12
v = ph − IMO(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof.
s ∈ H10 (Ω) arbitrary: |||p − ph|||Ω ≤ |||p − s|||Ω + |||s − ph|||Ω
(p − s) ∈ H10 (Ω): coercivity of B implies
|||p − s|||Ω ≤ B(p − s, p − s)
|||p − s|||Ω≤ sup
ϕ∈H10 (Ω), |||ϕ|||Ω=1
B(p − s, ϕ)
= supϕ∈H1
0 (Ω), |||ϕ|||Ω=1
B(p − ph, ϕ) + B(ph − s, ϕ)
Green theorem and weak solution definition: B(p − ph, ϕ)
= (f , ϕ)Ω −∑
K∈Th
(S∇ph,∇ϕ)K +
(∇ · (phw), ϕ
)K + (r ph, ϕ)K
=∑
K∈Th
(f +∇ · SK∇ph −∇ · (phw)− r ph, ϕ
)K +
∑K∈Th
〈uh · n, ϕ〉∂K
=∑
K∈Th
(f +∇ · SK∇ph −∇ · (phw)− r ph, ϕ
)K
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof.
s ∈ H10 (Ω) arbitrary: |||p − ph|||Ω ≤ |||p − s|||Ω + |||s − ph|||Ω
(p − s) ∈ H10 (Ω): coercivity of B implies
|||p − s|||Ω ≤ B(p − s, p − s)
|||p − s|||Ω≤ sup
ϕ∈H10 (Ω), |||ϕ|||Ω=1
B(p − s, ϕ)
= supϕ∈H1
0 (Ω), |||ϕ|||Ω=1
B(p − ph, ϕ) + B(ph − s, ϕ)
Green theorem and weak solution definition: B(p − ph, ϕ)
= (f , ϕ)Ω −∑
K∈Th
(S∇ph,∇ϕ)K +
(∇ · (phw), ϕ
)K + (r ph, ϕ)K
=∑
K∈Th
(f +∇ · SK∇ph −∇ · (phw)− r ph, ϕ
)K +
∑K∈Th
〈uh · n, ϕ〉∂K
=∑
K∈Th
(f +∇ · SK∇ph −∇ · (phw)− r ph, ϕ
)K
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof.
s ∈ H10 (Ω) arbitrary: |||p − ph|||Ω ≤ |||p − s|||Ω + |||s − ph|||Ω
(p − s) ∈ H10 (Ω): coercivity of B implies
|||p − s|||Ω ≤ B(p − s, p − s)
|||p − s|||Ω≤ sup
ϕ∈H10 (Ω), |||ϕ|||Ω=1
B(p − s, ϕ)
= supϕ∈H1
0 (Ω), |||ϕ|||Ω=1
B(p − ph, ϕ) + B(ph − s, ϕ)
Green theorem and weak solution definition: B(p − ph, ϕ)
= (f , ϕ)Ω −∑
K∈Th
(S∇ph,∇ϕ)K +
(∇ · (phw), ϕ
)K + (r ph, ϕ)K
=∑
K∈Th
(f +∇ · SK∇ph −∇ · (phw)− r ph, ϕ
)K +
∑K∈Th
〈uh · n, ϕ〉∂K
=∑
K∈Th
(f +∇ · SK∇ph −∇ · (phw)− r ph, ϕ
)K
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof (cont.)definition of the centered scheme:(
f +∇ · SK∇ph −∇ · (phw)− r ph, ϕK)
K = 0 ∀K ∈ Th
hence:B(p−ph, ϕ) =
∑K∈Th
(f +∇·SK∇ph−∇·(phw)−r ph, ϕ−ϕK
)K
remain the nonconformity and convection terms
Remarks
completely based on the primal abstract frameworkgeneralization of the techniques due to Verfürth ’98 forGalerkin FEMsnonconformity (techniques due to El Alaoui and Ern ’04,Karakashian and Pascal ’03)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof (cont.)definition of the centered scheme:(
f +∇ · SK∇ph −∇ · (phw)− r ph, ϕK)
K = 0 ∀K ∈ Th
hence:B(p−ph, ϕ) =
∑K∈Th
(f +∇·SK∇ph−∇·(phw)−r ph, ϕ−ϕK
)K
remain the nonconformity and convection terms
Remarks
completely based on the primal abstract frameworkgeneralization of the techniques due to Verfürth ’98 forGalerkin FEMsnonconformity (techniques due to El Alaoui and Ern ’04,Karakashian and Pascal ’03)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof (cont.)definition of the centered scheme:(
f +∇ · SK∇ph −∇ · (phw)− r ph, ϕK)
K = 0 ∀K ∈ Th
hence:B(p−ph, ϕ) =
∑K∈Th
(f +∇·SK∇ph−∇·(phw)−r ph, ϕ−ϕK
)K
remain the nonconformity and convection terms
Remarks
completely based on the primal abstract frameworkgeneralization of the techniques due to Verfürth ’98 forGalerkin FEMsnonconformity (techniques due to El Alaoui and Ern ’04,Karakashian and Pascal ’03)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the centered scheme
Sketch of proof (cont.)definition of the centered scheme:(
f +∇ · SK∇ph −∇ · (phw)− r ph, ϕK)
K = 0 ∀K ∈ Th
hence:B(p−ph, ϕ) =
∑K∈Th
(f +∇·SK∇ph−∇·(phw)−r ph, ϕ−ϕK
)K
remain the nonconformity and convection terms
Remarks
completely based on the primal abstract frameworkgeneralization of the techniques due to Verfürth ’98 forGalerkin FEMsnonconformity (techniques due to El Alaoui and Ern ’04,Karakashian and Pascal ’03)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the upwind-weightedscheme
Theorem (A posteriori error estimate for the upwind-weightedscheme)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K + ηC,K + ηU,K )2
12
.
upwinding estimatorηU,K :=
∑σ∈EK
mσ‖(pσ − pσ)w · n‖σ
pσ: the weighted upwind valuepσ: the mean of ph over the side σmσ: function of cS,K , cw,r ,K =
( 12∇ ·w + r
)|K , d , hK , |σ|, |K |
all dependencies evaluated explicitly
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
A posteriori error estimate for the upwind-weightedscheme
Theorem (A posteriori error estimate for the upwind-weightedscheme)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K + ηC,K + ηU,K )2
12
.
upwinding estimatorηU,K :=
∑σ∈EK
mσ‖(pσ − pσ)w · n‖σ
pσ: the weighted upwind valuepσ: the mean of ph over the side σmσ: function of cS,K , cw,r ,K =
( 12∇ ·w + r
)|K , d , hK , |σ|, |K |
all dependencies evaluated explicitly
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Local efficiency of the estimates
Theorem (Local efficiency of the residual estimator)There holds
ηR,K ≤ |||p − ph|||K C
max
1,Cw,r ,K
cw,r ,K
+ min
PeK , %K
.
residual estimator is locally efficient (lower bound forerror on K ), robust with respect to S, and semi-robustwith respect to wCeff,K :
C independent of hK , S, w, and rno dependency on inhomogeneities and anisotropiesCw,r,Kcw,r,K
≤ 2 for r nonnegativeCeff,K depends affinely on PeK
%K := |w|K |√cw,r,K
√cS,Kprevents Ceff,K from exploding in
convection-dominated cases on rough grids
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Local efficiency of the estimates
Theorem (Local efficiency of the residual estimator)There holds
ηR,K ≤ |||p − ph|||K C
max
1,Cw,r ,K
cw,r ,K
+ min
PeK , %K
.
residual estimator is locally efficient (lower bound forerror on K ), robust with respect to S, and semi-robustwith respect to wCeff,K :
C independent of hK , S, w, and rno dependency on inhomogeneities and anisotropiesCw,r,Kcw,r,K
≤ 2 for r nonnegativeCeff,K depends affinely on PeK
%K := |w|K |√cw,r,K
√cS,Kprevents Ceff,K from exploding in
convection-dominated cases on rough grids
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Local efficiency of the estimates
Theorem (Local efficiency of the nonconformity and convectionestimators)There holds
η2NC,K +η2
C,K ≤ α∑
L;L∩K 6=∅
|||p−ph|||2L+β infsh∈P2(Th)∩H1
0 (Ω)
∑L;L∩K 6=∅
‖p−sh‖2L .
nonconformity and convection estimators are locallyefficient (up to higher-order terms if cw,r ,K 6= 0) andsemi-robustCeff,K :
depends on maximal ratio of inhomogeneities around Kdepends on anisotropy in each L around K by CS,L
cS,L
again min
PeL, %L
in each L around K prevents Ceff,K fromexploding in convection-dominated cases on rough grids
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates Efficiency
Local efficiency of the estimates
Theorem (Local efficiency of the nonconformity and convectionestimators)There holds
η2NC,K +η2
C,K ≤ α∑
L;L∩K 6=∅
|||p−ph|||2L+β infsh∈P2(Th)∩H1
0 (Ω)
∑L;L∩K 6=∅
‖p−sh‖2L .
nonconformity and convection estimators are locallyefficient (up to higher-order terms if cw,r ,K 6= 0) andsemi-robustCeff,K :
depends on maximal ratio of inhomogeneities around Kdepends on anisotropy in each L around K by CS,L
cS,L
again min
PeL, %L
in each L around K prevents Ceff,K fromexploding in convection-dominated cases on rough grids
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,
−∇ · S∇ph =1|K |
∑σ∈EK
SK ,σ ,
(1− µK )(ph, 1)K /|K |+ µK ph(xK ) = pK ,
−S∇ph|K · n = SK ,σ/|σ| ∀σ ∈ EK .
Properties of ph
ph exists and is uniqueflux of ph is given by SK ,σ, point or mean value by pK
ph is nonconforming, 6∈ H1(Ω), 6∈ W0(Th), only ∈ H1(Th)
for simplices or rectangular parallelepipeds when S isdiagonal: ph is a piecewise second-order polynomialbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,
−∇ · S∇ph =1|K |
∑σ∈EK
SK ,σ ,
(1− µK )(ph, 1)K /|K |+ µK ph(xK ) = pK ,
−S∇ph|K · n = SK ,σ/|σ| ∀σ ∈ EK .
Properties of ph
ph exists and is uniqueflux of ph is given by SK ,σ, point or mean value by pK
ph is nonconforming, 6∈ H1(Ω), 6∈ W0(Th), only ∈ H1(Th)
for simplices or rectangular parallelepipeds when S isdiagonal: ph is a piecewise second-order polynomialbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,
−∇ · S∇ph =1|K |
∑σ∈EK
SK ,σ ,
(1− µK )(ph, 1)K /|K |+ µK ph(xK ) = pK ,
−S∇ph|K · n = SK ,σ/|σ| ∀σ ∈ EK .
Properties of ph
ph exists and is uniqueflux of ph is given by SK ,σ, point or mean value by pK
ph is nonconforming, 6∈ H1(Ω), 6∈ W0(Th), only ∈ H1(Th)
for simplices or rectangular parallelepipeds when S isdiagonal: ph is a piecewise second-order polynomialbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,
−∇ · S∇ph =1|K |
∑σ∈EK
SK ,σ ,
(1− µK )(ph, 1)K /|K |+ µK ph(xK ) = pK ,
−S∇ph|K · n = SK ,σ/|σ| ∀σ ∈ EK .
Properties of ph
ph exists and is uniqueflux of ph is given by SK ,σ, point or mean value by pK
ph is nonconforming, 6∈ H1(Ω), 6∈ W0(Th), only ∈ H1(Th)
for simplices or rectangular parallelepipeds when S isdiagonal: ph is a piecewise second-order polynomialbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,
−∇ · S∇ph =1|K |
∑σ∈EK
SK ,σ ,
(1− µK )(ph, 1)K /|K |+ µK ph(xK ) = pK ,
−S∇ph|K · n = SK ,σ/|σ| ∀σ ∈ EK .
Properties of ph
ph exists and is uniqueflux of ph is given by SK ,σ, point or mean value by pK
ph is nonconforming, 6∈ H1(Ω), 6∈ W0(Th), only ∈ H1(Th)
for simplices or rectangular parallelepipeds when S isdiagonal: ph is a piecewise second-order polynomialbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A locally postprocessed scalar variable ph
Definition (Postprocessed scalar variable ph)
We define ph such that, separately on each K ∈ Th,
−∇ · S∇ph =1|K |
∑σ∈EK
SK ,σ ,
(1− µK )(ph, 1)K /|K |+ µK ph(xK ) = pK ,
−S∇ph|K · n = SK ,σ/|σ| ∀σ ∈ EK .
Properties of ph
ph exists and is uniqueflux of ph is given by SK ,σ, point or mean value by pK
ph is nonconforming, 6∈ H1(Ω), 6∈ W0(Th), only ∈ H1(Th)
for simplices or rectangular parallelepipeds when S isdiagonal: ph is a piecewise second-order polynomialbasis of our a posteriori error estimates
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A posteriori error estimate
Theorem (A posteriori error estimate)There holds
|||p−ph|||Ω≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K +ηC,K +ηU,K +ηRQ,K +ηΓN,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IOs(ph)|||KIOs(ph): Oswald int. operator (Burman and Ern ’07)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 4
cw,r,K
convection estimator
ηC,K := min
2‖∇·(vw)‖K + 12‖v∇·w‖K√
cw,r,K,(CP,d h2
K‖∇·(vw)‖2K
cS,K+
‖v∇·w‖2K
4cw,r,K
)12
v = ph − IOs(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A posteriori error estimate
Theorem (A posteriori error estimate)There holds
|||p−ph|||Ω≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K +ηC,K +ηU,K +ηRQ,K +ηΓN,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IOs(ph)|||KIOs(ph): Oswald int. operator (Burman and Ern ’07)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 4
cw,r,K
convection estimator
ηC,K := min
2‖∇·(vw)‖K + 12‖v∇·w‖K√
cw,r,K,(CP,d h2
K‖∇·(vw)‖2K
cS,K+
‖v∇·w‖2K
4cw,r,K
)12
v = ph − IOs(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A posteriori error estimate
Theorem (A posteriori error estimate)There holds
|||p−ph|||Ω≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K +ηC,K +ηU,K +ηRQ,K +ηΓN,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IOs(ph)|||KIOs(ph): Oswald int. operator (Burman and Ern ’07)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 4
cw,r,K
convection estimator
ηC,K := min
2‖∇·(vw)‖K + 12‖v∇·w‖K√
cw,r,K,(CP,d h2
K‖∇·(vw)‖2K
cS,K+
‖v∇·w‖2K
4cw,r,K
)12
v = ph − IOs(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A posteriori error estimate
Theorem (A posteriori error estimate)There holds
|||p−ph|||Ω≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
(ηR,K +ηC,K +ηU,K +ηRQ,K +ηΓN,K )2
12
.
nonconformity estimatorηNC,K := |||ph − IOs(ph)|||KIOs(ph): Oswald int. operator (Burman and Ern ’07)
residual estimatorηR,K := mK‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K
m2K := min
CP,d
h2K
cS,K, 4
cw,r,K
convection estimator
ηC,K := min
2‖∇·(vw)‖K + 12‖v∇·w‖K√
cw,r,K,(CP,d h2
K‖∇·(vw)‖2K
cS,K+
‖v∇·w‖2K
4cw,r,K
)12
v = ph − IOs(ph)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C A postprocessed scalar variable Estimates
A posteriori error estimate
upwinding estimatorηU,K :=
∑σ∈EK\EN
hmσ‖(pσ − IOs(ph)σ)w · n‖σ
pσ: the weighted upwind valueIOs(ph)σ: the mean of IOs(ph) over the side σmσ: function of cS,K , cw,r ,K =
( 12∇ ·w + r
)|K , d , hK , |σ|, |K |
all dependencies evaluated explicitly
reaction quadrature estimatorηRQ,K := 1√
cw,r,K‖rK pK − (r ph, 1)K |K |−1‖K
disappears when r pw constant and ph fixed by mean
Neumann boundary estimatorηΓN,K := 0 + 1√cS,K
∑σ∈EK∩EN
h
√Ct,K ,σ
√hK‖uσ − u‖σ
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Comments on the estimates and their efficiency
no saturation assumptionp ∈ H1(Ω), no additional regularityno convexity of Ω neededno “monotonicity” hypothesis on inhomogeneitiesdistribution as El Alaoui and Ern ’04 or Bernardi andVerfürth ’00the only important tool: optimal discretePoincaré–Friedrichs inequalities (Vohralík, NFAO 2005)holds from diffusion to convection–diffusion–reaction casesno efficiency for the upwinding estimator (since theupwind-weighted scheme does not change to the centeredone; improvement: smooth transition upwind-weighted →centered scheme)no efficiency for the reaction quadrature and Neumannboundary estimators
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Comparison with Galerkin FEMs
residual estimator (‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K )
very good sense in MFEM/FVM (ph is piecewise quadratic)∇ · SK∇ph|K = 0 in piecewise linear Galerkin FEM
edge mass balance estimator (‖[S∇ph · n]‖σ)not present in MFEM/FVM (mass conservation)typical in Galerkin FEM
nonconformity estimator (|||ph − I(ph)|||K )
essential in MFEM/FVMnot present in conforming Galerkin FEM
upwinding estimator (‖(pσ − pσ)w · n‖σ)
when local Péclet number gets small, disappears in MFEMand gets a higher-order term in FVMSUPG (Verfürth ’98): different form, efficient
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Comparison with Galerkin FEMs
residual estimator (‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K )
very good sense in MFEM/FVM (ph is piecewise quadratic)∇ · SK∇ph|K = 0 in piecewise linear Galerkin FEM
edge mass balance estimator (‖[S∇ph · n]‖σ)not present in MFEM/FVM (mass conservation)typical in Galerkin FEM
nonconformity estimator (|||ph − I(ph)|||K )
essential in MFEM/FVMnot present in conforming Galerkin FEM
upwinding estimator (‖(pσ − pσ)w · n‖σ)
when local Péclet number gets small, disappears in MFEMand gets a higher-order term in FVMSUPG (Verfürth ’98): different form, efficient
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Comparison with Galerkin FEMs
residual estimator (‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K )
very good sense in MFEM/FVM (ph is piecewise quadratic)∇ · SK∇ph|K = 0 in piecewise linear Galerkin FEM
edge mass balance estimator (‖[S∇ph · n]‖σ)not present in MFEM/FVM (mass conservation)typical in Galerkin FEM
nonconformity estimator (|||ph − I(ph)|||K )
essential in MFEM/FVMnot present in conforming Galerkin FEM
upwinding estimator (‖(pσ − pσ)w · n‖σ)
when local Péclet number gets small, disappears in MFEMand gets a higher-order term in FVMSUPG (Verfürth ’98): different form, efficient
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Comparison with Galerkin FEMs
residual estimator (‖f +∇ · SK∇ph −∇ · (phw)− r ph‖K )
very good sense in MFEM/FVM (ph is piecewise quadratic)∇ · SK∇ph|K = 0 in piecewise linear Galerkin FEM
edge mass balance estimator (‖[S∇ph · n]‖σ)not present in MFEM/FVM (mass conservation)typical in Galerkin FEM
nonconformity estimator (|||ph − I(ph)|||K )
essential in MFEM/FVMnot present in conforming Galerkin FEM
upwinding estimator (‖(pσ − pσ)w · n‖σ)
when local Péclet number gets small, disappears in MFEMand gets a higher-order term in FVMSUPG (Verfürth ’98): different form, efficient
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Theorem (A posteriori error estimate for the diffusion problem)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
η2R,K
12
.
nonconformity estimator
ηNC,K := |||ph − I(ph)|||K = ‖S 12∇(ph − I(ph))‖K
I(ph): Oswald or modified Oswald interpolate
residual estimatorη2
R,K := CP,dh2
KcS,K
‖f − fK‖2K (fK is the mean of f over K )
only dependent on sources f
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Theorem (A posteriori error estimate for the diffusion problem)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
η2R,K
12
.
nonconformity estimator
ηNC,K := |||ph − I(ph)|||K = ‖S 12∇(ph − I(ph))‖K
I(ph): Oswald or modified Oswald interpolate
residual estimatorη2
R,K := CP,dh2
KcS,K
‖f − fK‖2K (fK is the mean of f over K )
only dependent on sources f
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Theorem (A posteriori error estimate for the diffusion problem)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2NC,K
12
+
∑K∈Th
η2R,K
12
.
nonconformity estimator
ηNC,K := |||ph − I(ph)|||K = ‖S 12∇(ph − I(ph))‖K
I(ph): Oswald or modified Oswald interpolate
residual estimatorη2
R,K := CP,dh2
KcS,K
‖f − fK‖2K (fK is the mean of f over K )
only dependent on sources f
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Theorem (A posteriori error estimate for the diffusion problem)There holds
|||p− ph|||Ω ≤ infs∈H1
0 (Ω)|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
asymptotically efficient with constant 1 for however largeinhomogeneities and anisotropies
infs∈H1
0 (Ω)|||ph − s|||Ω ≤ 1|||p − ph|||Ω
MFEM: is in fact also an a priori estimate:
infs∈H1
0 (Ω)|||ph − s|||Ω ≤ |||ph − IMO(ph)|||Ω ≤ Ch
using that ph ∈ W0(Th) (cont. means of traces)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Theorem (A posteriori error estimate for the diffusion problem)There holds
|||p− ph|||Ω ≤ infs∈H1
0 (Ω)|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
asymptotically efficient with constant 1 for however largeinhomogeneities and anisotropies
infs∈H1
0 (Ω)|||ph − s|||Ω ≤ 1|||p − ph|||Ω
MFEM: is in fact also an a priori estimate:
infs∈H1
0 (Ω)|||ph − s|||Ω ≤ |||ph − IMO(ph)|||Ω ≤ Ch
using that ph ∈ W0(Th) (cont. means of traces)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Theorem (A posteriori error estimate for the diffusion problem)There holds
|||p− ph|||Ω ≤ infs∈H1
0 (Ω)|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
asymptotically efficient with constant 1 for however largeinhomogeneities and anisotropies
infs∈H1
0 (Ω)|||ph − s|||Ω ≤ 1|||p − ph|||Ω
MFEM: is in fact also an a priori estimate:
infs∈H1
0 (Ω)|||ph − s|||Ω ≤ |||ph − IMO(ph)|||Ω ≤ Ch
using that ph ∈ W0(Th) (cont. means of traces)
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0Theorem (Mixed FEM for the diffusion problem)There holds|||p− ph|||Ω ≤ inf
s∈H10 (Ω)
|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
Theorem (Galerkin FEM for the diffusion problem)There holds
|||p − ph|||Ω ≤ infsh∈Vh
|||p − sh|||Ω .
Mixed FEM 1D:no nonconformity, ph ∈ H1
0 (Ω)|||p − ph|||Ω ≤ Ch2 when f ∈ H1(Th)ph = p, the exact solution, for pw constant S (arbitraryinhomogeneities) and pw constant f
Galerkin FEM 1D:|||p − ph|||Ω ≤ Ch
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0Theorem (Mixed FEM for the diffusion problem)There holds|||p− ph|||Ω ≤ inf
s∈H10 (Ω)
|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
Theorem (Galerkin FEM for the diffusion problem)There holds
|||p − ph|||Ω ≤ infsh∈Vh
|||p − sh|||Ω .
Mixed FEM 1D:no nonconformity, ph ∈ H1
0 (Ω)|||p − ph|||Ω ≤ Ch2 when f ∈ H1(Th)ph = p, the exact solution, for pw constant S (arbitraryinhomogeneities) and pw constant f
Galerkin FEM 1D:|||p − ph|||Ω ≤ Ch
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0Theorem (Mixed FEM for the diffusion problem)There holds|||p− ph|||Ω ≤ inf
s∈H10 (Ω)
|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
Theorem (Galerkin FEM for the diffusion problem)There holds
|||p − ph|||Ω ≤ infsh∈Vh
|||p − sh|||Ω .
Mixed FEM 1D:no nonconformity, ph ∈ H1
0 (Ω)|||p − ph|||Ω ≤ Ch2 when f ∈ H1(Th)ph = p, the exact solution, for pw constant S (arbitraryinhomogeneities) and pw constant f
Galerkin FEM 1D:|||p − ph|||Ω ≤ Ch
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0Theorem (Mixed FEM for the diffusion problem)There holds|||p− ph|||Ω ≤ inf
s∈H10 (Ω)
|||ph − s|||Ω +
∑K∈Th
CP,dh2
KcS,K
‖f − fK‖2K
12
.
Theorem (Galerkin FEM for the diffusion problem)There holds
|||p − ph|||Ω ≤ infsh∈Vh
|||p − sh|||Ω .
Mixed FEM 1D:no nonconformity, ph ∈ H1
0 (Ω)|||p − ph|||Ω ≤ Ch2 when f ∈ H1(Th)ph = p, the exact solution, for pw constant S (arbitraryinhomogeneities) and pw constant f
Galerkin FEM 1D:|||p − ph|||Ω ≤ Ch
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Definition (Generalized weak solution)
Generalized weak solution: p ∈ W0(Th) such that
B(p, ϕ) = (f , ϕ)Ω ∀ϕ ∈ W0(Th).
Theorem (A posteriori error estimate for mixed FEM and thegeneralized weak solution of the pure diffusion problem)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2R,K
12
.
residual estimatorη2
R,K := CP,dh2
KcS,K
‖f − fK‖2K
only dependent on sources ff ∈ H1(Th) ⇒ |||p − ph|||Ω ≤ Ch2
if f piecewise constant ⇒ ph coincides with p for arbitraryinhomogeneities and anisotropies
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Definition (Generalized weak solution)
Generalized weak solution: p ∈ W0(Th) such that
B(p, ϕ) = (f , ϕ)Ω ∀ϕ ∈ W0(Th).
Theorem (A posteriori error estimate for mixed FEM and thegeneralized weak solution of the pure diffusion problem)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2R,K
12
.
residual estimatorη2
R,K := CP,dh2
KcS,K
‖f − fK‖2K
only dependent on sources ff ∈ H1(Th) ⇒ |||p − ph|||Ω ≤ Ch2
if f piecewise constant ⇒ ph coincides with p for arbitraryinhomogeneities and anisotropies
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Pure diffusion problem −∇ · S∇p = f , ΓD = ∂Ω, g = 0
Definition (Generalized weak solution)
Generalized weak solution: p ∈ W0(Th) such that
B(p, ϕ) = (f , ϕ)Ω ∀ϕ ∈ W0(Th).
Theorem (A posteriori error estimate for mixed FEM and thegeneralized weak solution of the pure diffusion problem)There holds
|||p − ph|||Ω ≤
∑K∈Th
η2R,K
12
.
residual estimatorη2
R,K := CP,dh2
KcS,K
‖f − fK‖2K
only dependent on sources ff ∈ H1(Th) ⇒ |||p − ph|||Ω ≤ Ch2
if f piecewise constant ⇒ ph coincides with p for arbitraryinhomogeneities and anisotropies
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C Comments Diffusion problem Relations MFEs–FVs
Relations of mixed finite elements to finite volumes
MFE matrix problem:
(A CB D
)(UP
)=
(FG
)local flux expression formula Vohralík (M2AN, 2006): locallinear problem MV U int
V = FV − AV PV for each vertex V⇒ the matrix problem can be exactly, without anynumerical integration, rewritten as SP = Hlowest-order RT MFEM is thus equivalent to a particularmulti-point FV schemeS is sparse, in general positive definite but nonsymmetricworks from pure diffusion to convection–diffusion–reactioncases, for upwind schemes, in 2D and 3D, for nonlinearcases
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
Problem with discontinuous and inhomogeneousdiffusion tensor: finite volumes
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) = rα(ai sin(αθ) + bi cos(αθ))
(r , θ) polar coordinates in Ωai , bi constants depending on Ωiα regularity of the solution
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
Analytical solutions
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
error uniformestimate uniformerror adapt.estimate adapt.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
efficiency uniformefficiency adapt.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
Convection-dominated problem: mixed finite elements
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
error uniformestimate uniformerror adapt.estimate adapt.
101
102
103
104
105
10−2
10−1
100
101
Number of triangles
Ene
rgy
erro
r
error uniformestimate uniformerror adapt.estimate adapt.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C FVs & inhomogeneous diffusion MFEs & dominating convection
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
efficiency uniformefficiency adapt.
101
102
103
104
105
10
20
30
40
50
60
70
80
90
100
Number of triangles
Effi
cien
cy
efficiency uniformefficiency adapt.
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Outline1 Introduction2 Problem and schemes
A convection–diffusion–reaction problemFinite volume schemesMixed finite element schemes
3 A posteriori error estimates for mixed finite elementsA postprocessed scalar variableEstimatesLocal efficiency
4 A posteriori error estimates for finite volumes5 Remarks
Comments on the estimates and their efficiencyPure diffusion problemRelation of mixed finite elements to finite volumes
6 Numerical experiments7 Conclusions and future work
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Conclusions and future work
Conclusionsasymptotically exact, fully reliable, and locally efficientestimates for inhomogeneous and anisotropic andconvection-dominated problemsdirectly computable—all constants evaluatedworks for mixed finite elements and finite volumesbased on conformity of the flux variableconnections mixed finite elements – finite volumes
Future worka unified a priori and a posteriori error analysis of mixedfinite element / finite volume methodsnonlinear case
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Conclusions and future work
Conclusionsasymptotically exact, fully reliable, and locally efficientestimates for inhomogeneous and anisotropic andconvection-dominated problemsdirectly computable—all constants evaluatedworks for mixed finite elements and finite volumesbased on conformity of the flux variableconnections mixed finite elements – finite volumes
Future worka unified a priori and a posteriori error analysis of mixedfinite element / finite volume methodsnonlinear case
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM
I Pb Estimates MFEs Estimates FVs Remarks Exp C
Bibliography
PapersVOHRALÍK M., Asymptotically exact a posteriori errorestimates for mixed finite element discretizations ofconvection–diffusion–reaction equations, to appearin SIAM J. Numer. Anal.VOHRALÍK M., Residual flux-based a posteriori errorestimates for finite volume discretizations ofinhomogeneous, anisotropic, and convection-dominatedproblems, submitted to Numer. Math.
Merci de votre attention !
M. Vohralík Estimations d’erreur a posteriori pour les VF et les EFM