A POSTERIORI ERROR ESTIMATES, STOPPING CRITERIA, AND ...

A POSTERIORI ERROR ESTIMATES,STOPPING CRITERIA,

AND INEXPENSIVE IMPLEMENTATIONS

for error control and efficiencyin numerical simulations

Martin Vohralík

Habilitation à diriger des recherches

Paris, December 13, 2010

I Guar. & rob. est. Stop. crit. Impl., rel., & loc. postpr. C

Outline1 Introduction2 Guaranteed and robust estimates for model problems

Inhomogeneous diffusionDominant reactionDominant convectionHeat equationStokes equationMultiscale, multinumerics, and mortarsSystem of variational inequalities

3 Stopping criteria for iterative solvers and linearizationsLinearization errorAlgebraic errorTwo-phase flows

4 Implementations, relations, and local postprocessingPrimal formulation-based a priori analysis of MFEInexpensive implementations of MFE, their link to FV

5 Conclusions and future directionsM. Vohralík A posteriori estimates, stopping criteria, and implementations

Basic CV

joined Université Pierre et Marie Curie, LaboratoireJacques-Louis Lions as a “maître de conférences” inSeptember 2006January 2005–August 2006: post-doc at CNRS, UniversitéParis-SudDecember 2004: Ph.D. at Czech Technical University inPrague & Université Paris-Sud

M. Vohralík A posteriori estimates, stopping criteria, and implementations

Topics of the habilitationTopics of the habilitationPart 1

a posteriori error estimatesguaranteed and robust error controlunified frameworks

Part 2stopping criteriaequilibration of error componentsadaptive algorithms

Part 3a priori error estimatesinexpensive implementationsdevelopment of scientific calculation codes

Background

GNR MoMaS project Mathematical Modeling andNumerical Simulation for Nuclear Waste ManagementProblemsERT project with the French Petroleum Institute Enhancedoil recovery and geological sequestration of CO2: meshadaptivity, a posteriori error control, and other advancedtechniquesHydroExpert society, simulations of flow and contaminanttransport in underground porous media (code TALISMAN)

Co-supervision of Ph.D. candidates

Nancy Chalhoubframework of the GNR MoMaS projectco-supervision with Alexandre Ern (ENPC) and Toni Sayah(Université Saint-Joseph, Beirut, Lebanon)general framework for a posteriori error estimation ininstationary convection–diffusion–reaction problems

Soleiman Yousef, Carole Widmerframework of the ERT project with the French PetroleumInstituteco-supervision with Daniele Di Pietro (French PetroleumInstitute) and Vivette Girault (LJLL)a posteriori estimates, stopping criteria, and adaptivealgorithms for the Stefan problem (S. Yousef) and aposteriori estimates and adaptivity for cell-centered finitevolume discretizations of two-phase flows (C. Widmer)

Co-supervision of Ph.D. candidates

Nancy Chalhoubframework of the GNR MoMaS projectco-supervision with Alexandre Ern (ENPC) and Toni Sayah(Université Saint-Joseph, Beirut, Lebanon)general framework for a posteriori error estimation ininstationary convection–diffusion–reaction problems

Soleiman Yousef, Carole Widmerframework of the ERT project with the French PetroleumInstituteco-supervision with Daniele Di Pietro (French PetroleumInstitute) and Vivette Girault (LJLL)a posteriori estimates, stopping criteria, and adaptivealgorithms for the Stefan problem (S. Yousef) and aposteriori estimates and adaptivity for cell-centered finitevolume discretizations of two-phase flows (C. Widmer)

Numerical approximation of a nonlinear, instationaryPDE

Exact and approximate solution

let p be the weak solution of Ap = F , A nonlinear,instationarylet ph be its approximate numerical solution, Ahph = Fh

Solution algorithm

introduce a temporal mesh of (0,T ) given by tn, 0 ≤ n ≤ Nintroduce a spatial mesh T n

h of Ω on each tn

on each tn, solve a nonlinear algebraic problem Anhpn

h = F nh

Numerical approximation of a nonlinear, instationaryPDE

Exact and approximate solution

let p be the weak solution of Ap = F , A nonlinear,instationarylet ph be its approximate numerical solution, Ahph = Fh

Solution algorithm

introduce a temporal mesh of (0,T ) given by tn, 0 ≤ n ≤ Nintroduce a spatial mesh T n

h of Ω on each tn

on each tn, solve a nonlinear algebraic problem Anhpn

h = F nh

Iterative linearization and iterative algebraic solvers

Iterative linearization of Anhpn

h = F nh

An,(i−1)L,h pn,(i)

h = F n,(i−1)L,h : discrete Newton or fixed-point

linearizationloop in iwhen do we stop?

Iterative algebraic system solution on each tn and foreach i

h = F n,(i−1)L,h is a linear algebraic system

we only solve it inexactly by, e.g., some iterative methodwhen do we stop?

Approximate solutionthe approximate solution pa

h that we have as an outcomedoes not solve Ahpa

h = Fh

how big is the overall error ‖p − pah‖Ω×(0,T )?

h = F nh

h = Fh

h = F nh

h = Fh

h = F nh

h = Fh

h = F nh

h = Fh

h = F nh

h = Fh

h = F nh

h = Fh

h = F nh

h = Fh

A posteriori error estimates: 5 optimal properties

Guaranteed upper bound (global error upper bound)‖p − pa

h‖2Ω×(0,T ) ≤∑N

n=1∑

K∈T nhηn

K (pah)2

no undetermined constant: error controlLocal efficiency (local error lower bound)

ηnK (pa

h)2 ≤ C2eff,K ,n

∑L close to K ‖p − pa

h‖2L×(tn−1,tn)

enables to predict the overall error distributionAsymptotic exactness∑N

n=1∑

K∈T nhηn

K (pah)2/‖p − pa

h‖2Ω×(0,T ) → 1overestimation factor goes to one with meshes size

RobustnessCeff,K ,n does not depend on coefficients, their relative sizeand variation, solution regularity, domain Ω, final time Testimators equally good in all situations

Negligible evaluation costestimators can be evaluated locally

h‖2Ω×(0,T ) ≤∑N

n=1∑

K∈T nhηn

K (pah)2

ηnK (pa

h)2 ≤ C2eff,K ,n

h‖2L×(tn−1,tn)

n=1∑

K∈T nhηn

h‖2Ω×(0,T ) ≤∑N

n=1∑

K∈T nhηn

K (pah)2

ηnK (pa

h)2 ≤ C2eff,K ,n

h‖2L×(tn−1,tn)

n=1∑

K∈T nhηn

h‖2Ω×(0,T ) ≤∑N

n=1∑

K∈T nhηn

K (pah)2

ηnK (pa

h)2 ≤ C2eff,K ,n

h‖2L×(tn−1,tn)

n=1∑

K∈T nhηn

h‖2Ω×(0,T ) ≤∑N

n=1∑

K∈T nhηn

K (pah)2

ηnK (pa

h)2 ≤ C2eff,K ,n

h‖2L×(tn−1,tn)

n=1∑

K∈T nhηn

Aims and benefits of this workAims of this work

give a guaranteed and robust upper bound on the overallerror ‖p − pa

h‖Ω×(0,T ), if possible asymptotically exactensure local efficiency (optimal mesh refinement)develop unified frameworksdistinguish the algebraic/linearization errors, due to inexactsolution of linear/nonlinear problems, and the space andtime discretization errors, due to mesh size, time step, andnumerical schemestop the iterative solvers whenever algebraic/linearizationerrors do not affect the overall error significantlyequilibrate the space and time error components

Benefitsoptimal computable overall error boundimprovement of approximation precisionimportant computational savings

Previous results

Continuous finite elements

Babuška and Rheinboldt (1978), introductionLadevèze and Leguillon (1983), equilibrated fluxesestimates (equality of Prager and Synge (1947))Zienkiewicz and Zhu (1987), averaging-based estimatesVerfürth (1996, book), residual-based estimatesRepin (1997), functional a posteriori error estimatesDestuynder and Métivet (1999), equilibrated fluxesestimatesAinsworth and Oden (2000, book), equilibrated residualestimatesLuce and Wohlmuth (2004), equilibrated fluxes estimatesBraess and Schöberl (2008), equilibrated fluxes estimates

Previous results

Finite volumesOhlberger (2001), non-energy norm estimatesAchdou, Bernardi, Coquel (2003), links FV–FENicaise (2005, 2006), postprocessing

Discontinuous Galerkin finite elementsBecker, Hansbo, Larson (2003), residual-based estimatesKarakashian and Pascal (2003), residual-based estimatesAinsworth (2007), reconstruction of side fluxesKim (2007), Cochez-Dhondt and Nicaise (2008),reconstruction of equilibrated H(div,Ω)-conforming fluxes

Mixed finite elementsBraess and Verfürth (1996)Carstensen (1997)Hoppe and Wohlmuth (1997)Lovadina and Stenberg (2006)Kim (2007)

Previous results

Previous results: inhomogeneous diffusion, reaction

Diffusion with discontinuous coefficients

Dörfler and Wilderotter, conforming finite elementsBernardi and Verfürth (2000), conforming finite elementsPetzoldt (2002), conforming finite elementsAinsworth (2005), nonconforming finite elements

Reaction-dominated problems

Verfürth (1998), residual estimatesAinsworth and Babuška (1999), equilibrated residualestimatesGrosman (2006), equilibrated residual estimates,anisotropic meshes

Previous results: inhomogeneous diffusion, reaction

Diffusion with discontinuous coefficients

Dörfler and Wilderotter, conforming finite elementsBernardi and Verfürth (2000), conforming finite elementsPetzoldt (2002), conforming finite elementsAinsworth (2005), nonconforming finite elements

Reaction-dominated problems

Verfürth (1998), residual estimatesAinsworth and Babuška (1999), equilibrated residualestimatesGrosman (2006), equilibrated residual estimates,anisotropic meshes

Previous results: parabolic problemsContinuous finite elements

Bieterman and Babuška (1982), introductionEriksson and Johnson (1991), rigorous analysisPicasso (1998), evolving meshesStrouboulis, Babuška, and Datta (2003), guaranteedestimatesVerfürth (2003), efficiency, robustness with respect to thefinal timeMakridakis and Nochetto (2003), elliptic reconstruction

Finite volumesOhlberger (2001), non energy-norm estimatesAmara, Nadau, and Trujillo (2004), energy-norm estimates

Discontinuous Galerkin finite elementsGeorgoulis and Lakkis (2009)

Nonconforming finite elementsNicaise and Soualem (2005)

Previous results: algebraic error

A posteriori estimates accounting for algebraic error

Repin (1997)

Stopping criteria for iterative solvers

Becker, Johnson, and Rannacher (1995)Maday and Patera (2000)Arioli (2004)Meidner, Rannacher, Vihnarev (2009)

Algebraic energy error estimation in the conjugategradient method

Meurant (1997)Strakoš and Tichý (2002)

Repin (1997)

Previous results: nonlinear problemsContinuous finite elements

Han (1994), general frameworkVerfürth (1994), residual estimatesBarrett and Liu (1994), quasi-norm estimatesLiu and Yan (2001), quasi-norm estimatesVeeser (2002), convergence p-LaplacianCarstensen and Klose (2003), guaranteed estimatesChaillou and Suri (2006, 2007), distinguishingdiscretization and linearization errors (only fixed-point, onelinearized problem (not an iterative loop))Diening and Kreuzer (2008), linear cvg p-Laplacian

Other methodsLiu and Yan (2001), quasi-norm estimates for thenonconforming finite element methodKim (2007), guaranteed estimates for locally conservativemethods

Previous results: nonlinear problemsContinuous finite elements

Han (1994), general frameworkVerfürth (1994), residual estimatesBarrett and Liu (1994), quasi-norm estimatesLiu and Yan (2001), quasi-norm estimatesVeeser (2002), convergence p-LaplacianCarstensen and Klose (2003), guaranteed estimatesChaillou and Suri (2006, 2007), distinguishingdiscretization and linearization errors (only fixed-point, onelinearized problem (not an iterative loop))Diening and Kreuzer (2008), linear cvg p-Laplacian

Other methodsLiu and Yan (2001), quasi-norm estimates for thenonconforming finite element methodKim (2007), guaranteed estimates for locally conservativemethods

Previous results: various

Variational inequalities

Hlavácek, Haslinger, Necas, and Lovíšek (1982)Ainsworth, Oden, and Lee (1993)Chen and Nochetto (2000)Wohlmuth (2007)

Stokes problem

Verfürth (1989)Dörfler and Ainsworth (2005)

Unified frameworks

Ainsworth (2005)Carstensen (2005–2009)

Stokes problem

Unified frameworks

Stokes problem

Unified frameworks

Previous results: error components equilibration

Error components equilibration

engineering literature, since 1950’sLadevèze (since 1980’s)Verfürth (2003), space and time error equilibrationBraack and Ern (2003), estimation of model errorsBabuška, Oden (2004), verification and validation. . .

Previous results: implementations, relations

Implementations, mixed methods

Arnold and Brezzi (1985)Arbogast and Chen (1995)Cockburn and Gopalakrishnan (2004, 2005)

Relations between different methods

Russell and Wheeler (1983)Younès, Ackerer, Chavent (1999–2004)

Previous results: implementations, relations

Implementations, mixed methods

Arnold and Brezzi (1985)Arbogast and Chen (1995)Cockburn and Gopalakrishnan (2004, 2005)

Relations between different methods

Russell and Wheeler (1983)Younès, Ackerer, Chavent (1999–2004)

Papers of this habilitation (a posteriori estimates)

Co-authors No. Year Journal Problem Num. meth. Main results

— [A13] 2010 JSC D FE, VCFV, CCFV, FD guar. & rob. estimates w.r.t.the jumps in dif. coef.

Cheddadi, Fucík, Prieto [A4] 2009 M2AN RD FE, VCFV guar. & rob. estimates w.r.t.the reaction

Ern, Stephansen [A6] 2010 JCAM CRD DG guar. & rob. estimates w.r.t.the convection and reaction

— [A11] 2007 SINUM CRD MFE guar. estimates— [A12] 2008 NM CRD FV guar. estimatesErn [A7] 2010 SINUM heat DG, MFE, VCFV, CCFV unified framework

FCFV, FE, NCFEHilhorst [A9] 2010 CMAME CRD instat. VCFV guar. (& rob.) estimatesHannukainen, Stenberg [B2] 2010 submt. Stokes DG, MFE, FV unified framework

FE, FES, NCFEPencheva, Wheeler, Wildey [B3] 2010 submt. D DG, MFE, FV extension to multiscale,

multinumerics, and mortarsBen Belgacem, Bernardi, Blouza [B1] 2010 submt. syst. var. ineq. FE optimal a posteriori estimates

Papers of this habilitation (stopping criteria)

Jiránek, Strakoš [A10] 2010 SISC D CCFV, MFE algebraic error, stopping criteriaEl Alaoui, Ern [A5] 2010 CMAME monot. nonlin. FE guar. & rob. estimates w.r.t.

the nonlinearity;linealization error, stoping criteria

— [B4] 2010 prep. two-phase DG, MFE, VCFV a posteriori estimates, stoping criteriaCCFV, FCFV

Papers of this habilitation (implementations, relationsbetween methods, postprocessing, a priori estimates)

Eymard, Hilhorst [A8] 2010 NMPDE CRD par. deg. VCFV convergence proofBen Belgacem, Bernardi, Blouza [A1] 2009 M2AN syst. var. ineq. FE well-posedness, a priori estimates,

residual a posteriori estimatesBen Belgacem, Bernardi, Blouza [A2] 2009 MMNP syst. var. ineq. FE inexpensive implementation— [A14] 2010 MC D MFE unified a priori and a posteriori analysisWohlmuth [B5] 2010 submt. D MFE inexpensive implementation,

relation to FV methods

I Guar. & rob. est. Stop. crit. Impl., rel., & loc. postpr. C Dif. React. Conv. Heat Stokes MsMnM Syst. var. ineq.

A model problem with discontinuous coefficients

Model problem with discontinuous coefficients

−∇·(S∇p) = f in Ω,

p = 0 on ∂Ω

AssumptionsΩ ⊂ Rd , d = 2,3, is a polygonal domainS is a piecewise constant scalar, inhomogeneous

Energy norm

|||ϕ|||2 := ‖S 12∇ϕ‖2, ϕ ∈ H1

0 (Ω)

VOHRALÍK

Guaranteed and fully robust a posteriori error estimates forconforming discretizations of diffusion problems withdiscontinuous coefficientsJ. Sci. Comput. 2010 [A13]

−∇·(S∇p) = f in Ω,

p = 0 on ∂Ω

Energy norm

|||ϕ|||2 := ‖S 12∇ϕ‖2, ϕ ∈ H1

0 (Ω)

VOHRALÍK

−∇·(S∇p) = f in Ω,

p = 0 on ∂Ω

Energy norm

|||ϕ|||2 := ‖S 12∇ϕ‖2, ϕ ∈ H1

0 (Ω)

VOHRALÍK

−∇·(S∇p) = f in Ω,

p = 0 on ∂Ω

Energy norm

|||ϕ|||2 := ‖S 12∇ϕ‖2, ϕ ∈ H1

0 (Ω)

VOHRALÍK

Properties of the weak solution

0.90.80.70.60.50.40.30.20.10.0

-0.21.0

exact solution

Solution p is in H10 (Ω)

0.90.80.70.60.50.40.30.20.10.0

-1.51.0

-exact flux

Flux −S∇p is in H(div,Ω)

Approximate solution and approximate flux

0.90.80.70.60.50.40.30.20.10.0

-0.21.0

exact solutionapproximate solution

Approximate solution ph is inH1

0 (Ω)

0.90.80.70.60.50.40.30.20.10.0

-1.51.0

-exact flux-approximate flux

Approximate flux −S∇ph is notin H(div,Ω)

Approximate solution and postprocessed flux

0.90.80.70.60.50.40.30.20.10.0

-0.21.0

exact solutionapproximate solution

Approximate solution ph is inH1

0 (Ω)

0.90.80.70.60.50.40.30.20.10.0

-1.51.0

-postprocessed flux-approximate flux-exact flux

Construct a postprocessed fluxth in H(div,Ω) (Prager–Synge

(1947))

A posteriori error estimate for −∇·(S∇p) = f

Theorem (A posteriori error estimate)Let

p be the weak solution,ph ∈ H1

0 (Ω) be arbitrary,Dh = Dint

h ∪ Dexth be a partition of Ω,

th ∈ H(div,Ω) be arbitrary but such that

(∇·th,1)D = (f ,1)D for all D ∈ Dinth .

|||p − ph||| ≤∑

D∈Dh

(ηR,D + ηDF,D)2

Estimators

diffusive flux estimator

ηDF,D := ‖S 12∇ph + S−

12 th‖D

penalizes the fact that −S∇ph 6∈ H(div,Ω)

residual estimator

ηR,D := mD,S‖f −∇·th‖Dresidue evaluated for thm2

D,S := CP,DhD2/cS,D for D ∈ Dint

h , CP,D = 1/π2 if D convexm2

D,S := CF,DhD2/cS,D for D ∈ Dext

h , CF,D = 1 in generalcS,D is the smallest value of S on DCP,D and CF,D can be replaced by 1/π2 for appropriateconstruction of th (always done in practice)

Estimators

ηDF,D := ‖S 12∇ph + S−

12 th‖D

residual estimator

Estimators

ηDF,D := ‖S 12∇ph + S−

12 th‖D

residual estimator

Estimators

ηDF,D := ‖S 12∇ph + S−

12 th‖D

residual estimator

Main steps of the proof

Main steps of the proof, cf. Prager–Synge equality (1947).energy norm characterization:

|||p − ph||| = supϕ∈H1

0 (Ω), |||ϕ|||=1(S∇(p − ph),∇ϕ)

adding and subtracting th ∈ H(div,Ω), Green theorem:

|||p−ph||| = infth∈H(div,Ω)

supϕ∈H1

0 (Ω), |||ϕ|||=1|(f−∇·th,ϕ)|+|(S∇ph+th,∇ϕ)|

Cauchy–Schwarz inequality:

|(S∇ph + th,∇ϕ)| ≤∥∥S

12∇ph + S−

12 th∥∥ =

D∈Dh

η2DF,D

local conservation property (∇·th,1)D = (f ,1)D ∀D ∈ Dinth ,

Poincaré and Friedrichs inequalities:

|(f −∇·th, ϕ)| ≤∑

D∈Dh

η2R,D

|||p − ph||| = supϕ∈H1

0 (Ω), |||ϕ|||=1(S∇(p − ph),∇ϕ)

supϕ∈H1

0 (Ω), |||ϕ|||=1|(f−∇·th,ϕ)|+|(S∇ph+th,∇ϕ)|

|(S∇ph + th,∇ϕ)| ≤∥∥S

12∇ph + S−

12 th∥∥ =

D∈Dh

η2DF,D

|(f −∇·th, ϕ)| ≤∑

D∈Dh

η2R,D

|||p − ph||| = supϕ∈H1

0 (Ω), |||ϕ|||=1(S∇(p − ph),∇ϕ)

supϕ∈H1

0 (Ω), |||ϕ|||=1|(f−∇·th,ϕ)|+|(S∇ph+th,∇ϕ)|

|(S∇ph + th,∇ϕ)| ≤∥∥S

12∇ph + S−

12 th∥∥ =

D∈Dh

η2DF,D

|(f −∇·th, ϕ)| ≤∑

D∈Dh

η2R,D

|||p − ph||| = supϕ∈H1

0 (Ω), |||ϕ|||=1(S∇(p − ph),∇ϕ)

supϕ∈H1

0 (Ω), |||ϕ|||=1|(f−∇·th,ϕ)|+|(S∇ph+th,∇ϕ)|

|(S∇ph + th,∇ϕ)| ≤∥∥S

12∇ph + S−

12 th∥∥ =

D∈Dh

η2DF,D

|(f −∇·th, ϕ)| ≤∑

D∈Dh

η2R,D

Construction of th

Practical construction of th

relies on the local conservation property of the givennumerical methoduses Raviart–Thomas–Nédélec spaces, two types

direct prescription of the degrees of freedom by averagingthe normal fluxes ∇ph·nsolution of local Neumann problems by mixed finiteelements (Bank and Weiser (1985), Ern and Vohralík(2009))

Local efficiency of the estimates for −∇·(S∇p) = f

Theorem (Local efficiency)Suppose that

th is constructed from ph by direct prescription or solutionof local Neumann MFE problemsthe discontinuities are aligned with the dual mesh Dh;harmonic averaging was used in the scheme;harmonic averaging was used in the construction of th.

ThenηR,D + ηDF,D ≤ C|||p − ph|||TVD

where C depends only on the space dimension d, on the shaperegularity parameter κT , and on the polynomial degree m of f .

Main steps of the proof.

ηDF,D ≤ C

σ∈ED

‖[[S∇ph·n]]‖2σ

ηR,D ≤ C(|||p − ph|||D + ηDF,D)

Philosophy

our estimates are, up to a generic constant, a lower boundfor the residual oneswe then can use the results for the residual estimates (R.Verfürth)

ηDF,D ≤ C

σ∈ED

‖[[S∇ph·n]]‖2σ

ηR,D ≤ C(|||p − ph|||D + ηDF,D)

Philosophy

ηDF,D ≤ C

σ∈ED

‖[[S∇ph·n]]‖2σ

ηR,D ≤ C(|||p − ph|||D + ηDF,D)

Philosophy

Estimates for −∇·(S∇p) = f , FE, VCFV, CCFV, FD

Properties

guaranteed upper boundlocal efficiencyfull robustness, singular cases included (nomonotonicity-like assumption as in Dörfler and Wilderotter(2000), Bernardi and Verfürth (2000), Petzoldt (2002),Ainsworth (2005), Chen and Dai (2002), or Cai and Zhang(2009))almost asymptotically exactnegligible evaluation cost

Vertex-centered finite volumes in 1D

Model problem

−p′′ = π2sin(πx) in ]0,1[,

p = 0 in 0,1

Exact solutionp(x) = sin(πx)

Vertex-centered finite volumes in 1D

Model problem

−p′′ = π2sin(πx) in ]0,1[,

p = 0 in 0,1

Exact solutionp(x) = sin(πx)

Estimated and actual errors

10−6

10−4

10−2

Number of vertices

error uniformestimate uniformres. est. uniformdif. flux est. uniform

Actual error and estimator andits components

Number of verticesE

ity in

effectivity ind. uniform

Effectivity index

Discontinuous diffusion tensor and vertex-centeredfinite 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

s3=5 s

−1 0 1−1

s1=100s

s3=100 s

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

Case 1 Case 2

Error distribution on a uniformly ref. mesh, case 1

Estimated error distribution Exact error distribution

Error distribution on an adaptively ref. mesh, case 2

Estimated error distribution Exact error distribution

Approximate solutions on adaptively refined meshes

−0.50

0.5−0.5

−0.5

Case 1

−0.50

−0.5

Case 2

Estimated and actual errors in uniformly/adaptivelyrefined meshes

10−1

Number of dual volumes

error uniformestimate uniformerror adapt.estimate adapt.

Case 1

error uniformestimate uniformerror adapt.estimate adapt.

Case 2

Effectivity indices in uniformly/adaptively refinedmeshes

effectivity ind. uniformeffectivity ind. adapt.

Case 1

effectivity ind. uniformeffectivity ind. adapt.

Case 2

A reaction–diffusion problem

Problem

−4p + rp = f in Ω,

p = 0 on ∂Ω

AssumptionsΩ ⊂ Rd , d = 2,3, is a polygonal domainr ∈ L∞(Ω) such that for each D ∈ Dh, 0 ≤ cr ,D ≤ r ≤ Cr ,D,a.e. in D

Energy norm

|||ϕ|||2Ω := ‖∇ϕ‖2 + ‖r1/2ϕ‖2, ϕ ∈ H10 (Ω)

CHEDDADI, FUCÍK, PRIETO, VOHRALÍK

Guaranteed and robust a posteriori error estimates forsingularly perturbed reaction–diffusion problemsM2AN Math. Model. Numer. Anal. 2009 [A4]

Problem

−4p + rp = f in Ω,

p = 0 on ∂Ω

Energy norm

|||ϕ|||2Ω := ‖∇ϕ‖2 + ‖r1/2ϕ‖2, ϕ ∈ H10 (Ω)

Problem

−4p + rp = f in Ω,

p = 0 on ∂Ω

Energy norm

|||ϕ|||2Ω := ‖∇ϕ‖2 + ‖r1/2ϕ‖2, ϕ ∈ H10 (Ω)

Problem

−4p + rp = f in Ω,

p = 0 on ∂Ω

Energy norm

|||ϕ|||2Ω := ‖∇ϕ‖2 + ‖r1/2ϕ‖2, ϕ ∈ H10 (Ω)

A posteriori error estimate for −4p + rp = f

p be the weak solution,ph ∈ H1

0 (Ω) be arbitrary,Dh = Dint

th ∈ H(div,Ω) be arbitrary but such that

(∇·th + rph,1)D = (f ,1)D for all D ∈ Dinth .

|||p − ph||| ≤∑

D∈Dh

(ηR,D + ηDF,D)2

Residual and diffusive flux estimators

Estimatorsresidual estimator

ηR,D := mD‖f −∇·th − rph‖Ddiffusive flux estimator

ηDF,D := minη(1)

DF,D, η(2)DF,D

η(1)DF,D :=‖∇ph + th‖D

η(2)DF,D :=

K∈SD

(mK‖4ph +∇·th − (4ph +∇·th)K‖K

+ m12K

σ∈EK∩Ginth

C12t,K ,σ‖(∇ph + th)·n‖σ

mD, mK , mK : cutoff factors as in Verfürth (1998)M. Vohralík A posteriori estimates, stopping criteria, and implementations

Residual and diffusive flux estimators

ηR,D := mD‖f −∇·th − rph‖Ddiffusive flux estimator

ηDF,D := minη(1)

DF,D, η(2)DF,D

η(1)DF,D :=‖∇ph + th‖D

η(2)DF,D :=

K∈SD

(mK‖4ph +∇·th − (4ph +∇·th)K‖K

+ m12K

σ∈EK∩Ginth

C12t,K ,σ‖(∇ph + th)·n‖σ

mD, mK , mK : cutoff factors as in Verfürth (1998)M. Vohralík A posteriori estimates, stopping criteria, and implementations

Local efficiency of the estimates for −4p + rp = f

Theorem (Local efficiency)Suppose that th was constructed from ph by direct prescription.Then there holds

ηR,D + ηDF,D ≤ C|||p − ph|||D,

where C depends only on d, κT , m, and Cr ,D/cr ,D.

Properties

guaranteed upper boundlocal efficiencyrobustnessalmost asymptotically exactnegligible evaluation cost

Local efficiency of the estimates for −4p + rp = f

Theorem (Local efficiency)Suppose that th was constructed from ph by direct prescription.Then there holds

ηR,D + ηDF,D ≤ C|||p − ph|||D,

where C depends only on d, κT , m, and Cr ,D/cr ,D.

Properties

guaranteed upper boundlocal efficiencyrobustnessalmost asymptotically exactnegligible evaluation cost

Problem and exact solution

Problem

−4p + rp = 0 in Ω,

p = p0 on ∂Ω

Solution

p0(x , y) = e−√

rx + e−√

Effectivity indices in dependence on r

10−6

10−4

10−2

reaction term r

ity in

uniform grid, 32 triangles

jump. est.min. est.

Mesh with 32 triangles

10−6

10−4

10−2

reaction term ref

uniform grid, 131072 triangles

jump. est.min. est.

Mesh with 131072 triangles

Estimated and actual errors in uniformly/adaptivelyrefined meshes and effectivity indices

number of triangles

min. est., uniformmin. est., adaptiveexact error, uniformexact error, adaptive

Est. and act. errors, r = 106

number of triangles

ity in

min. est., uniformmin. est., adaptive

Effectivity indices, r = 106

Error distribution on an adaptively refined mesh,r = 106

IsoValue1.145493.436475.727458.0184310.309412.600414.891417.182319.473321.764324.055326.346328.637230.928233.219235.510237.801240.092142.383144.6741

Estimated Error Distribution

Estimated error distribution

IsoValue0.05938840.1781650.2969420.4157180.5344950.6532720.7720480.8908251.00961.128381.247151.365931.484711.603481.722261.841041.959812.078592.197372.31614

Exact Error Distribution

Exact error distribution

A convection–diffusion–reaction problem

A model convection–diffusion–reaction problem

−∇·(S∇p) + w·∇p + rp = f in Ω,

p = 0 on ∂Ω

Energy normSet B = BS + BA, where

BS(p, ϕ) := (S∇p,∇ϕ) +((

r − 12∇·w

)p, ϕ

BA(p, ϕ) :=(w·∇p + 1

2(∇·w)p, ϕ)

BS is symmetric on H1(Th); put |||ϕ|||2 := BS(ϕ,ϕ)BA is skew-symmetric on H1

0 (Ω)

ERN, STEPHANSEN, VOHRALÍK

Guaranteed and robust discontinuous Galerkin a posteriorierror estimates for convection–diffusion–reaction problemsJ. Comput. Appl. Math. 2010 [A6]

−∇·(S∇p) + w·∇p + rp = f in Ω,

p = 0 on ∂Ω

BS(p, ϕ) := (S∇p,∇ϕ) +((

r − 12∇·w

)p, ϕ

BA(p, ϕ) :=(w·∇p + 1

2(∇·w)p, ϕ)

0 (Ω)

−∇·(S∇p) + w·∇p + rp = f in Ω,

p = 0 on ∂Ω

BS(p, ϕ) := (S∇p,∇ϕ) +((

r − 12∇·w

)p, ϕ

BA(p, ϕ) :=(w·∇p + 1

2(∇·w)p, ϕ)

0 (Ω)

A dual norm augmented by the convective derivative

for p, ϕ ∈ H1(Th) define

BD(p, ϕ) := −∑

σ∈Eh

〈w·n[[p]], Π0ϕ〉σ

introduce the augmented norm

|||p|||⊕ := |||p|||+ supϕ∈H1

0 (Ω), |||ϕ|||=1BA(p, ϕ) + BD(p, ϕ)

when p ∈ H10 (Ω) and ∇·w = 0, recover the augmented

norm introduced by Verfürth ’05BD contribution is new and specific to the nonconformingcase

BD(p, ϕ) := −∑

σ∈Eh

〈w·n[[p]], Π0ϕ〉σ

|||p|||⊕ := |||p|||+ supϕ∈H1

0 (Ω), |||ϕ|||=1BA(p, ϕ) + BD(p, ϕ)

BD(p, ϕ) := −∑

σ∈Eh

〈w·n[[p]], Π0ϕ〉σ

|||p|||⊕ := |||p|||+ supϕ∈H1

0 (Ω), |||ϕ|||=1BA(p, ϕ) + BD(p, ϕ)

BD(p, ϕ) := −∑

σ∈Eh

〈w·n[[p]], Π0ϕ〉σ

|||p|||⊕ := |||p|||+ supϕ∈H1

0 (Ω), |||ϕ|||=1BA(p, ϕ) + BD(p, ϕ)

Augmented norm estimate and its efficiency

Theorem (Fully robust a posteriori estimate)Let p be the weak solution and ph its DG approximation. Thenthere holds