Introduction

Les méthodes hybrides discontinues ont fait l'objet d'un développement soutenu des an-nées 2000 à nos jours. Plus récemment (voir [44,56]), une fonction de stabilisation se fondant sur la projection de la variable uh dans l'espace des traces a été introduite dans le but d'in-tégrer la technique de post-processing au processus de résolution numérique du problème de diusion.

Ainsi, an d'étendre la technique de sur-stabilisation de la méthode H-RT à cette nouvelle fonction de stabilisation, ce chapitre introduit une nouvelle méthode de résolution numérique,

appelée projective Hybridizable RaviartThomas (H-RTp), à travers un article scientique

accepté dans la revue Applied Mathematical Modeling (AMM).

Cette nouvelle méthode de résolution numérique permet ainsi de porter la vitesse de convergence de la variable uh à l'ordre p + 2 pour p ≥ 0. La méthode H-RTp est alors évaluée dans des situations de diusion en milieu homogène et isotrope ainsi qu'en milieu fortement hétérogène et anisotrope. Cette méthode est alors comparée aux méthodes H-RT et HdG (H-LdG) classiques.

ContentslistsavailableatScienceDirect

Applied Mathematical Modelling

journalhomepage:www.elsevier.com/locate/apm

A projective hybridizable discontinuous Galerkin mixed

method for second-order diffusion problems

Loic Dijouxa, Vincent Fontainea,∗, Thierry Alex Marab

a Laboratoire Piment, Université de La Réunion, France

b Joint Research Centre, European Commission, Ispra, VA, Italy

a r t i c l e i n f o

Article history:

Received 30 October 2017 Revised 19 May 2019 Accepted 29 May 2019 Available online 3 June 2019

Keywords:

Hybridizable discontinuous Galerkin Projective-based stabilization function Higher-order Raviart–Thomas space Simplified postprocessing

h and p refinements

a b s t r a c t

In this paper, we present a hybridizable discontinuous Galerkin (HDG) mixed method for second-orderdiffusion problemsusinga projectivestabilizationfunction andbroken Raviart–Thomas functions to approximate the dual variable. The proposed HDG mixed method isinspired by the primal HDG scheme withreduced stabilizationsuggested by Lehrenfeld and Schöberl in 2010, and the standard hybridized version of the Raviart– Thomas(H-RT) method.Indeed,weusethebrokenRaviart–Thomasspace ofdegreek≥ 0 fortheflux,apiecewisepolynomialofdegreek+1forthepotential,andapiecewise poly-nomialofdegreekforitsnumericaltrace.Thisunconventionalpolynomialcombinationis madepossiblebytheprojectiveLehrenfeld–Schöberl(LS)stabilizationfunction.Its introduc-tionandtheuseofRaviart-Thomasspaceswillhavebeneficial effects:no postprocessing isrequiredtoimprovetheaccuracyofthepotentialuh,andastraightforwardflux recon-structionissufficienttoobtaina H (div)–conformingfluxvariable.Theconvergenceand ac-curacyofourmethodareinvestigatedthroughnumericalexperimentsintwo-dimensional spacebyusinghandprefinementstrategies.Anoptimalconvergenceorder(k+1)forthe H (div)-conformingfluxandsuperconvergence(k+2) forthepotential isobserved. Com-parativetestswiththeclassicalH-RTandthewell-knownhybridizablelocaldiscontinuous Galerkin(H-LDG)mixedmethodsarealsoperformedandexposedintermsofCPUtime.

© 2019ElsevierInc.Allrightsreserved.

1. Introduction

Considerthesecond-orderdiffusionmodelproblem:

∇·(−κ−1∇u)= f in , (1a)

u=g on ∂. (1b)

where⊂ Rd isaboundedpolyhedral domain(d≥ 2)with boundary∂⊂ Rd−1.Theboldfacefontsare usedthroughout thispaper tocharacterizeanyvector-valued or matrix-valuedfunctions.Toensurethat theproblem (1)iswellposed, the followingassumptionsareassumedtobesatisfied:κ∈[L^∞()]d,disamatrix-valuedfunctionthatissymmetricandpositive definitein,f∈L2(),andg∈H1/2(∂)isaprescribedDirichletboundarycondition.System (1)can beusedtomodel,for

∗Corresponding author.

E-mail addresses: loic.dijoux@univ-reunion.fr (L. Dijoux), vincent.fontaine@univ-reunion.fr (V. Fontaine),thierry.mara@jrc.ec.europa.eu (T.A. Mara). https://doi.org/10.1016/j.apm.2019.05.054

664 L. Dijoux, V. Fontaine and T.A. Mara / Applied Mathematical Modelling 75 (2019) 663–677

instance,thegroundwaterflowinporousmediainwhichthepotentialurepresentsthepressurehead,fasink/sourceterm andκ−1the permeabilitytensor[1].Inthisphysicalframework, itisconvenienttointroducetheDarcyvelocityσ (i.e., the flux)asasupplementaryunknownsuchthattheproblem(1)canberewrittenas afirst-ordersystem:

κσ=−∇u in , (2a)

∇·σ=f in , (2b)

u=g on ∂. (2c)

TheH(div)-conformingmixed finiteelement (MFE) methods are verywidely usedfor solvingthe system (2) (see,e.g., [[1–3], and the referencestherein]). Several successful combinations of compatible polynomialinterpolation functions for the flux σ∈H(div; ) and the potentialu∈L²()satisfying the discrete inf-sup condition have been proposed inthe lit-erature(e.g., the Raviart–Thomas (RT)[3] andthe Brezzi–Douglas–Marini (BDM)[2]families). These formulationsprovide numericalsolutions(σh,u_h)withoptimalconvergenceinL2andcontinuousnormalcomponentsforσh attheinterelement boundaries[1,2,4].However,MFEmethodsleadtoasaddle-pointproblemwithlargecoupleddegreesof freedom,whichis quitechallengingandtime-consumingtosolve(see,e.g.,[2,5],andthereferencestherein).

To circumvent these issues, mixed methods using the nonconforming Galerkin finite elements have been extensively studied over the last two decades for stationary diffusion problems (see, e.g., [6], and the references therein). The term

nonconformingmeans that no regularityassumptions are made onthe discrete variables.These methods were developed

within the general framework of the discontinuous Galerkin (DG) formalism and consider a combination of completely discontinuous approximationspaces forboth the flux and potential [7]. Among the DG methods, the local discontinuous Galerkin(LDG)methodisofparticularinterestduetoitslocalconservationpropertiesanditsflexibilityinhandlingadaptive

hprefinement.Incontrasttothe standardMFEmethods,thediscretefluxvariableσh canbe easilyeliminatedlocally,thus reducingthelinearsystemtoonlythe primalvariableu_h unknowns.However,relativetostandard H(div)-conformingMFE methods,theLDGmethodprovidesalessaccurateapproximationofdiscretevariables(u_h,σh).AccordingtoCorkburnetal. [8],thislackof robustnessoccursbecausethenumericaltraceuˆ_h isexpressedsolelyintermsof u_h,independentofσh.

Inresponse tothesedrawbacks,thehybridizationtechniquehasbeenintroducedin[9,10]usingLagrangemultipliersand staticcondensation,inadditiontoreducingthe globalcoupledlinearsystem toonlyunknownslocatedonthe skeletonof themesh[11].Thisconditionisachievedbyintroducinganewdiscretesingle-valuedvariableuˆ_hthatismerelytheso-called

numericaltraceof thepotentialdefinedat theelement boundaries[10].This techniqueallowsonetoestablishlocal solvers

ineachelementwithuˆ_h playingtheroleoftheDirichletboundaryconditions.Therefore,theinteriorunknowns(σh,u_h)can beeasilyeliminatedbystaticcondensationfromthesetofalgebraicequationsthatarenowexpressedintermsofLagrange multipliersuˆ_h only.The problemis thenclosed,andtheglobal algebraiclinearsystemisassembledbyimposingtransmission

conditions throughoutthe Lagrangemultipliers.Thehybridizable discontinuousGalerkin (HDG)formalism, introduced first

by Cockburnet al. in [10], representsaunified framework thatincludes the well-establishedhybridized version of mixed finite element (H-MFE) methods by using a null-stabilization function [2,3,9], the hybridizable LDG (H-LDG) methods by usingamultiplicativestabilizationfunction[8,10–12],andtherecentlydevelopedprimalhybridhigher-order(HHO)methods by using aprojective stabilization function [13–16].All of thesehybrid methods are established by suitablychoosing the stabilizationfunctionandsubsequentlythelocalapproximationspacesinadditiontothepostprocessingstep.Notethatthe HDG formalism has several advantageous features and iswell suited for parallel computing.Since its introduction, HDG methods havebenefitedfrom intensive researchanddevelopment (see,e.g.,[[11,12,16,17],andthe referencestherein])and havebeenappliedtoalargeclassofphysicalproblems[18–22].

The starting point for the present method isthe primal HDG method with reduced stabilization initially designed by LehrenfeldandSchöberlin[18] (see,e.g.,[12,17,23]).In allstandard HDGmethods, piecewisepolynomials of equaldegree are used for approximations of the state variable and its trace to provide optimal convergences. The motivation of the reducedstabilizationfunctionisthatitusesasuitableL2-projectionoperator,allowingfortheconsiderationof polynomials of higherdegree fortheapproximationof the potentialto obtainabetterestimated discrete variable.Theprojective HDG mixedmethod thatwe develop inthispaper can be considered as an inspired variation of the primal HDGscheme with reducedstabilizationandthestandardhybridizedversionoftheRaviart–Thomasmixedfiniteelement(H-RT)method.Thus, oneachsimplexAofthemesh,thehigher-orderRaviart–ThomasspaceRT_k(A)withk≥ 0forthefluxσhisused,apiecewise polynomialof degree k+1 forthe potential u_h and apiecewise polynomialof degreek forits numericaltrace uˆ_h on ∂A. Fromanotherperspective,theproposedmethodcanalsobeconsideredasapenalizedvariantoftheH-RTmethodbyadding the reduced(LS)stabilization function inthe definitionof the normal trace of the fluxonthe mesh skeleton.As aresult, theproposedmethodfundamentallydiffersfromthestandardH-RTformulationandindeedbelongstothegeneralclassof HDGmixedmethods:itisclosetotheprimalHDGschemedevelopedin[17,18]butrecasthereinamixedformatandusing Raviart–Thomas basis functions forthe approximation of the dual variable.Thus, the introduction of the LS stabilization functionandtheuse ofRaviart-Thomasfunctionswill havebeneficialeffects: nopostprocessingisrequiredtoimprovethe accuracy of the potential u_h since it converges naturally at the order k+2, and a straightforward flux reconstruction is sufficient toobtaina H(div)–conforming variableσ

h convergingat the orderk+1. Notealso thatthe approach presented herecanbeextendedtoanyothercompatiblemixedfiniteelementsuchasthoseproposedbytheauthorsin[2](e.g.,BDM orBDFMspaces)andtoanyother stabilizationfunction(e.g.,multiplicative-type).

Therestofthispaperisorganizedasfollows.InSection2,wedescribethegeneralformalismofHDGmixedmethods, de-finethelocalsolversandtheglobalproblemandpresentthestaticcondensationfromanalgebraicviewpoint.InSection3, we reviewthe selected existingstabilization functions andprovidedetails concerning thepostprocessing of σh andu_h.In Section 4, we describe ourprojective HDG method; inparticular,we review theLS stabilization function, define thelocal approximationspaces anddescribethe simplifiedpostprocessingtoestimatethe flux.InSection 5, numericalexperiments areinvestigated usingh-andp-refinement strategies,andcomparisonswith boththeH-RTandH-LDGmixedmethods are exposedintermsofCPUtime.Finally,weendwithconcludingremarksandperspectives.

2. HybridizablediscontinuousGalerkinmethods

2.1. Notations

Letusnowintroducethenotationthatwillbeusedthroughoutthispaper.Wedenotebyh:={A}(themesh)a triangu-lationof thedomain intoaffine-mapped simplexes (trianglesifd=2,and tetrahedronifd=3)andby∂h:={∂A}. Let Ea

h =Ei

h∪Eb

h (themeshskeleton)be thesetof allfacets(edgesifd=2, andfacesifd=3)of h,where Ei

h andEb

h denote the set of interior andboundary facets,respectively. We saythat F isan interior facet F∈Ei

h ifthere existA₁ and A₂ in

h suchthatF =A₁∩A₂.Furthermore, we assumethatthe (d− 1)-Lebesguemeasure of F isnotnull.For allA∈h and F∈Ea

h, |A|and |F| represent the measure of A and F,respectively. We set h_A:=diam(A)and h:=max_A∈h(h_A), i.e.,the maximalelementdiameter. Hereafter,foranydomainD⊂ Rd, we denoteby (· , · )D andby· , ·_∂Dthe standardL2-inner productsinL2(D)andL2(∂D),respectively.Letus introducethefollowing notations associatedwiththe weak formulation:

(·,·) h:= ^ A∈h (·,·)A, ·,·_∂h:= ^ A∈h ·,·_∂A and ·,·Es h := ^ F∈Es h ·,·F, (3)

wheretheindexs={a,i,b}.LetP_k^d(D)denotethespaceofpolynomialfunctionsofdegreenotexceedingk≥ 0onD,andlet Pd

k(D):=[Pd

k(D)]d.Similarly,wesetL2(h):=[L2(h)]d.Letusnowintroducethejump[[·]]andweightedaverage{· }_ω trace operatorsdefinedoninteriorfacetsasfollows:

[[v]]:=v1· n1+v2· n2, (4)

{ϕ}ω :=ω1ϕ1+ω2ϕ2, (5)

forall(v,ϕ)∈L2(h)× L2(h),wherethevectorof weightsω:=(ω1,ω2)issuchthatω1+ω2=1. Similarly,we introduce thecomplementofωdenotedω definedasfollows:

ω:=1−ω=(ω2,ω1). (6)

If ω=(1/2,1/2), we then recoverthe classical averageoperator; we omit the subscript ω inits definition. For the HDG discretization,twotypesoffiniteelementspacesarerequired.Thefirsttypeforσh andu_h isdefinedinsidetheelements:

Vh:={v∈L²(h):v|A∈V(A), ∀A∈h}, (7) Qh:={q∈L2(h):q|A∈Q(A), ∀A∈h}, (8)

andthesecondtypeforthetracefunctionuˆ_h isdefinedonthemeshskeleton:

g

h:={μ∈L2(Ea

h):μ|F∈(F) ∀F∈Ei

h and μ|F=hg ∀F∈Eb

h}, (9)

whereh denotestheL2-projection onto(F).Following(9),thenumericaltraceofthepotentialisdefinedasfollows:

ˆ u_h=  λh on Ei h hg on Eb h , (10) whereλh∈0

h denotes theLagrangemultipliers.Forconvenience, letus introducethelocalspaceassociatedwith the de-greesoffreedomof uˆ_h ontheboundaryof theelementA,

(∂A):={μ∈L2(∂A):μ|F∈(F) ∀F⊂∂A}. (11)

Furthermore,weassumeforthemomentthatV(A),Q(A)and(∂A)arethesuitablychosenlocalapproximationspacesof finitedimensionsthatwedefinepreciselybelow.

666 L. Dijoux, V. Fontaine and T.A. Mara / Applied Mathematical Modelling 75 (2019) 663–677

2.2. Localsolvers&Globalproblem

TheHDGmethodsprovideanapproximationof(σh,u_h)intermsofsolutionsofalocalDirichletboundaryvalueproblem oneach elementof the mesh, whichare thenpatched togetherby transmissionconditions acrossinterelement boundaries. In other words, forall A∈h, we assume that we know the numerical trace of the potential uˆ_h on ∂A, and we initially determine(σh,u_h)intermsof(uˆ_h,f)bysolvingthelocalproblem:

(κσh,vh)A−(u_h,∇·vh)A+uˆ_h,vh· n_∂A=0,

−(σh,∇q_h)A+σˆh· n,q_h∂A=(f,q_h)A, (12)

forall(q_h,v_h)∈Q(A)× V(A).Here,thenumericaltraceσˆh· nrepresentsanapproximationofσ· nover∂A,andweassume thatthistrace isconsistentanddependslinearly onσh,u_h anduˆ_h.Following[12], wethen assumethatthe tracehas the followingsimpleform:

ˆ

σh· n:=σh· n+τ(u_h− ˆu_h) on ∂A, (13)

whereτ isalinear localstabilization function thatwe describe below. Inpractice,its choiceisquitedelicate sinceitcan stronglyaffectthe stabilityandaccuracyof the HDGmethod.Inserting(13) into(12),the weakform of thelocalproblem ineachelementA∈h isfoundtobeasfollows:foranygivenuˆ_h∈(∂A),find(u_h,σh)∈Q(A)× V(A)suchthat

(κσh,v_h)A−(u_h,∇·v_h)A+uˆ_h,v_h· n∂A=0,

(∇·σh,q_h)A+τ(u_h− ˆu_h),q_h∂A=(f,q_h)A, (14)

forall (q_h,v_h)∈Q(A)× V(A).Note thatthe localproblem (14)can besolved inan element-by-element fashionto derive thevariables(u_h, σh)solelyintermsof (uˆ_h,f):this keystepcorrespondsto thestaticcondensationtechnique.Theglobal problemisthenclosed byweaklyimposingthetransmissionconditions,i.e.,thesingle-valuednessof thenormalcomponent ofthediscretevariableσˆh oneachinterior facetofthemeshasfollows:finduˆ_h∈g

h suchthat

σh· n+τ(u_h− ˆu_h),μh∂h =0, (15)

forall μh∈0

h, where σh and u_h are solutions of the localproblem (14). Substitutingthe solutions of (14) into (15), we obtainthe globallinear systemthatinvolvesonlythenumericaltrace uˆ_h.This stepcompletes thepresentationof theHDG methods.Allhybridmethodsreferencedintheliterature,suchashybridizedBrezzi–Douglas–Marini(H-BDM)orH-RT meth-ods[2,9],H-LDGmethods[8,10,24]or,morerecently,HHOmethods[14–16],areestablishedbyproperlychoosingthe stabi-lizationfunction τ andsubsequentlythelocalspaces V(A), Q(A) and(F) inaddition tothepostprocessing step.Before describingsome of thesechoices,let usbrieflyreview thewell-knowntechnique ofstatic condensationusedtoreduce the stiffnessmatrixof theHDGmethods.

2.3. Staticcondensation

Notethatthe HDGmethodjustdescribedaboveconsistsof seekingan approximation(u_h,σh,uˆ_h)∈Qh× Vh×g h satis-fyingtheequations

_a(σ h,vh)+b(u_h,vh)− c(uˆ_h,vh)=0, b(q_h,σh)+dτ(u_h,q_h)− rτ(uˆ_h,q_h)=(f,q_h) h, c(μh,σh)+rτ(μh,u_h)+sτ(uˆ_h,λh)=0, (16) forall(q_h,v_h,μh)∈Qh× Vh×0

h.Notably,thebilinearfunctionalsusedin(16)canbedecomposedintotwoclasses:those thatare independentof thestabilizationfunctionτ suchas

a(σh,vh):=−(κσh,vh) h,

b(u_h,vh):=(u_h,∇·vh) h,

c(uˆ_h,vh):=uˆ_h,vh· n∂h, andthosedepending onit:

dτ(u_h,q_h):=τ(u_h),q_h∂h,

rτ(uˆ_h,q_h):=τ(uˆ_h),q_h∂h,

sτ(μh,uˆ_h):=−τ(uˆ_h),μh_∂h.

Hence,thecorrespondingalgebraicsystemcanbewrittenasfollows:  A Bt −Ct B Dτ −Rt τ C Rτ Sτ  ·  h U_h ˆ U_h  =  G F H  , (17)

whereh,U_h andU^ˆ_h arevectors of degrees of freedomassociatedwiththe discrete variablesσh,u_h andλh,respectively. Toeliminate(σh,u_h)locally,weintroducethecorrespondingvectorofinteriordegrees offreedom˜t

h=[t h,Ut

h].Then,the linearsystem(17)canberewritteninacompactform:

 ˜ A − ˜Bt ˜ B Sτ  ·  ˜ h ˆ U_h  =  ˜ F H  , (18)

whereA˜,B˜andF˜ aredefinedasfollows:

˜ A=  A Bt B Dτ  , B˜^t=  Ct Rt τ  , and F˜ =  G F  . (19)

TheassociatedmatrixA˜ hasablock-diagonalstructureduetothe discontinuousnatureof Vh andQh,andits inverse can be easily computed by using aCholesky factorization. Thus, the elimination of ˜h follows immediately, andwe obtaina singlematrix equationonlyforthemultipliers_Uˆ_h,

A_Uˆ_h=H˜, (20)

whereAisasymmetricpositivedefinitematrixandH˜,thecorrespondingright-handside,isgivenby

A=Sτ+B˜A˜⁻¹B˜t and H˜=H− ˜BA˜⁻¹F˜. (21)

Once the solution uˆ_h is obtained, the discrete interior variables (σh, u_h) can be computed by solving the local problem (14)oneachelementofthemesh.ThisstepcompletesthereductiontechniquebystaticcondensationoftheHDGmethods. Itisclear thatthe choiceof localspaces V(A), Q(A) and (F) hasa directimpacton thedimension of thelocalsolvers andof theglobalproblem(20).Forinstance, thesizeofA isdirectlyproportional tothedimensionof (F).However,no assumptionshave beenmadeaboutlocalspaces noronthe stabilizationfunction τ usedtocharacterizetheHDGmethod. Thispointisdiscussedindetailinthenextsection.

3. Stabilizationfunctionsandpostprocessingtechniques

3.1. Localstabilizationfunctions

Thechoiceofthestabilization functionremainsanopen question.Severalformulationshavebeenproposedinthe liter-ature,somemoreintuitive[9,10],andothersmoresophisticated[15,16,18],butalloftheminitiallyassumethatτ isalinear, nonnegativefunctionandthatthelocalspacesQ(A)andV(A) cannotbechosenarbitrarilyandmustsatisfyinclusion con-straints. A remarkable feature of these propertiesis that HDGmethods can be well-defined completely independently of thechoiceof thelocalapproximationspace(∂A),but inpractice,itwill beassumedto besufficientlyrich.As suggested by Cockburnin[12], “therole ofτ isto transformthe discrepancy betweenu_h and uˆ_h on ∂Ainto anenergy” toenhance the stabilityof HDGmethods and toensurethatthe localproblem(14) and, hence,the globalproblem (16)are well-defined. Wethenfocusontwowell-knownvariantsthatwedescribe indetailbelow.

3.1.1. Nullstabilization

Inthesimplestformof thenull–stabilizationfunction,we supposethat

τn(u_h− ˆu_h):=0 on ∂h. (22)

Assuming(22),the bilinear functionals dτ, rτ and sτ become nulls, and the HDGformulation (17) consistsof seeking an approximation(u_h,σh,uˆ_h)∈Qh× Vh×g

h satisfyingtheequations _(κσ

h,vh) h−(u_h,∇·vh) h+uˆ_h,vh· n_∂h=0,

(∇·σh,q_h) h=(f,q_h),

σh· n,μh∂h=0, ⁽²³⁾

for all (q_h,v_h,μh)∈Qh× Vh×0

h. This simplified formulation istypically considered in the so-called hybridized version of mixedfinite element methods [1,2].Inthis case, the choiceof appropriatefinite dimensionalspaces Qh,Vh andh is notevident,butstability conditionsprovide manysuccessfulcombinations ofthem suchas theRTand BDMfamilies [2,3]. Hence,thelocalproblems(14)aswellastheglobalproblem(23)arewell-definedwithoutanystabilizationprocedure.The expectedconvergenceratesof u_h andσh forthe H-RT_k andH-BDM_k methodsaresummarized inTable2.

3.1.2. Multiplicativestabilization

Themultiplicative–stabilizationfunctionwassuggestedfirstbyCockburnetal.[10]toestablishtheoriginalH-LDGmethod,

namely,

668 L. Dijoux, V. Fontaine and T.A. Mara / Applied Mathematical Modelling 75 (2019) 663–677

Table 1

Admissible polynomial spaces V (A) , Q (A) and

(F) for the H-LDG method with k ≥ 1. H-LDG V (A) Q (A) (F) Variant 1. _P d k−1(A) P d k(A) P d−1 k (F) Variant 2. P d k(A) P d k(A) P d−1 k (F) Variant 3. P d k(A) P d k−1(A) P d−1 k (F) Table 2

Comparison of various hybrid mixed methods using the same local approximation space (F) := P d−1

k (F) . These methods are defined by their local approximation spaces Q (A) and V (A) and the stabilization function τn , τm or τLS . The columns u h , σh , u ^h and σ

h indicate the expected convergence rate of u − u h L2(h) , σ−σh L2(h) , u − u  h L2(h) and σ−σ h L2(h) , respectively. Method Degree V (A) Q (A) τ u_h σh u h σ h H-RT k k ≥ 0 _P d k(A) ⊕ x P d k(A) P d k(A) τn k + 1 k + 1 k + 2 – H-BDM_k k ≥ 2 P d k(A) P d k−1(A) τn k k + 1 k + 2 – H-LDG k k ≥ 1 P d k(A) P d k(A) τm k + 1 k + 1 k + 2 k + 1 H-LDG LS k k ≥ 0 P d k(A) ⊕ x P d k(A) P d k+1(A) τLS k + 2 k + 1 – k + 1 Table 3

The first and second columns indicate the dimension of local spaces V (A) and Q (A) , respectively. The dimension of the local solver is equal to the sum of these two quanti- ties. The third and fourth columns indicate the size of the local matrix to compute the postprocessed variables u 

h and σ

h , respectively.

Method dim V (A) dim Q (A) u^_h σ h H-RT_k (k + 1)(k + 3) (k + 1)(k + 2) / 2 (k + 2)(k + 3) / 2 − H-LDG k (k + 1)(k + 2) (k + 1)(k + 2) / 2 (k + 2)(k + 3) / 2 (k + 1)(k + 3) H-LDG LS k (k + 1)(k + 3) (k + 2)(k + 3) / 2 − 3 k + 3 Table 4

Total number of interior Lagrange multipliers, i.e., di- mension of the stiffness matrix A , as a function of the mesh refinement n and the polynomial degree k of the hybrid method on regular meshes.

kn 4 8 16 32 64

0 40 176 736 3008 12 , 160 1 80 352 1472 6016 24 , 320 2 120 528 2208 9024 36 , 480 3 160 704 2944 12 , 032 48 , 640

whereτF isastrictlypositiveconstant. Byimposingtransmissionconditions[[σˆh]]=0on interiorfacetsF∈Ei

h,we obtain anexplicitexpressionofthe numericaltraces(uˆ_h,σˆh) intermsof(u_h,σh):

 ˆ u_h ˆ σh  =  {u_h}ω {σh}_ω  +  0 αF βF 0  ·  [[u_h]] [[σh]]  (25)

whereω,αF andβF arepositiveconstantsdefinedasfollows: ω= τ1 τ1+τ2, τ2 τ1+τ2 , αF= τ1+¹τ2 and βF = τ1τ2 τ1+τ2. (26)

Here, τ1 and τ2 represent the stabilization parameters associated with the common facet F adjacent to A₁ and A₂, re-spectively.Let us specify thatthe H-LDG method cannotbe considered as an LDG method since the parameterαF isnot null forany finitevalue of τ1,2.In this case, the localspaces Q(A), V(A) and (∂A)must satisfythe following inclusion constraints:

∇Q(A)⊂ V(A), Q(A)|∂A⊂(∂A) and V(A)· n|∂A⊂(∂A). (27)

Severalcombinationsofadmissiblelocalspaceshave beenstudiedintheliterature[10],assummarizedinTable1.Previous reportshave concludedthatboth discretevariablesconvergeoptimally (k+1)inL² assumingthat u_h and σh are approxi-mated locallyby polynomials of degreek(cf. Table1, Variant 2)and thatτF istakentobe of orderone(cf. Table2,line 3).

Localization of degrees of freedom of the local approximation spaces V (A) , Q (A) and (F) used by the H-LDG LS

k method for different polynomial orders

k = 0 , . . . , 3 . Here, A is a triangle ( d = 2 ), and F is an edge of ∂A . For the local space V (A) = RT k(A) , the arrows indicate the value of the normal component ( 3(k + 1) -normal basis functions), and the double dots indicate the value of both components ( k (k + 1) -interior basis functions).

Table 6

Test A – Homogeneous and isotropic test case on regular meshes. H-LDGLS

k H-RT_k H-LDG_k

k n eu_h ECR eσ _h ECR eu _h ECR eσh ECR eu _h ECR eσ _h ECR

4 2 . 10E −1 − 1 . 99E −0 − 1 . 27E −1 − 1 . 96E −0 − 1 . 27E −1 − 1 . 97E −0 − 8 4 . 41E −2 2.25 1 . 01E −0 0.98 3 . 11E −2 2.03 1 . 01E −0 0.96 6 . 63E −1 0.94 1 . 01E −0 0.96 0 16 9 . 83E −3 2.16 5 . 06E −1 1.00 7 . 79E −3 2.00 5 . 05E −1 1.00 3 . 41E −1 0.96 5 . 06E −1 1.00 32 2 . 38E −3 2.04 2 . 52E −1 1.00 1 . 95E −3 2.00 2 . 52E −1 1.00 1 . 73E −1 0.98 2 . 52E −1 1.01 64 5 . 95E −4 2.00 1 . 26E −1 1.00 4 . 90E −4 2.00 1 . 26E −1 1.00 8 . 72E −2 0.99 1 . 26E −1 1.00 4 3 . 64E −2 − 4 . 31E −1 − 2 . 27E −2 − 4 . 13E −1 − 2 . 67E −2 − 4 . 24E −1 − 8 3 . 09E −3 3.56 1 . 14E −1 1.92 3 . 15E −3 2.85 1 . 13E −1 1.87 3 . 38E −3 2.98 1 . 16E −1 1.87 1 16 3 . 38E −4 3.19 2 . 87E −2 1.99 3 . 99E −4 2.98 2 . 86E −2 1.98 4 . 21E −4 3.00 2 . 92E −2 1.99 32 3 . 96E −5 3.09 7 . 04E −3 2.01 4 . 86E −5 3.04 7 . 02E −3 2.02 5 . 15E −5 3.03 7 . 17E −3 2.03 64 4 . 80E −6 3.04 1 . 74E −3 2.00 5 . 98E −6 3.02 1 . 74E −3 2.01 6 . 33E −6 3.02 1 . 77E −3 2.02 4 4 . 59E −3 − 8 . 65E −2 − 5 . 21E −3 − 8 . 58E −2 − 5 . 26E −3 − 8 . 75E −2 − 8 2 . 35E −4 4.29 9 . 62E −3 3.17 2 . 80E −4 4.22 9 . 54E −3 3.17 2 . 80E −4 4.23 9 . 83E −3 3.15 2 16 1 . 29E −5 4.18 1 . 18E −3 3.03 1 . 76E −5 4.00 1 . 17E −3 3.03 1 . 76E −5 3.99 1 . 21E −3 3.02

32 7 . 99E −7 4.01 1 . 51E −4 2.97 1 . 14E −6 3.95 1 . 51E −4 2.96 1 . 15E −6 3.94 1 . 55E −4 2.96 64 5 . 08E −8 3.98 1 . 92E −5 2.98 7 . 33E −8 3.96 1 . 92E −5 2.98 7 . 36E −8 3.97 2 . 01E −5 2.95

3.2. Postprocessingtechniques

Until 1985, hybridization was considered onlyas an implementation trick to overcome the saddle-point problem that ariseinMFEmethods.However,ArnoldandBrezziestablishedin[4]thattheadditionalinformationuˆ_hdefinedonthemesh skeletoncan beexploitedtolocallyconstructanewapproximatedvariableu_h^ convergingfasterthanu_h inL2.Weendthis

Dans le document Simulation numérique des phénomènes d'écoulement et de transport de masse en milieu poreux (Page 76-96)