Table Of ContentMS&A – Modeling, Simulation and Applications 19
Daniele Antonio Di Pietro
Jérôme Droniou
The Hybrid High-
Order Method for
Polytopal Meshes
Design, Analysis, and Applications
MS&A
Volume 19
Editor-in-Chief
AlfioQuarteroni
SeriesEditors
TomHou
ClaudeLeBris
AnthonyT.Patera
EnriqueZuazua
Moreinformationaboutthisseriesathttp://www.springer.com/series/8377
Daniele Antonio Di Pietro • Jérôme Droniou
The Hybrid High-Order
Method for Polytopal Meshes
Design, Analysis, and Applications
DanieleAntonioDiPietro JérômeDroniou
InstitutMontpelliérainAlexander SchoolofMathematics
Grothendieck MonashUniversity
UniversitédeMontpellier Clayton
Montpellier,France VIC,Australia
ISSN2037-5255 ISSN2037-5263 (electronic)
MS&A
ISBN978-3-030-37202-6 ISBN978-3-030-37203-3 (eBook)
https://doi.org/10.1007/978-3-030-37203-3
©SpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Cover illustration by the Authors: Streamlines for the three-dimensional lid-driven cavity flow on a
Voronoimesh
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Originallyintroducedin[146,158],HybridHigh-Order(HHO)methodsprovidea
frameworkforthe discretisationofmodelsbasedonPartialDifferentialEquations
(PDEs) with features that set them apart from traditional ones. The construction
hinges on discrete unknownsthat are broken polynomialson the mesh and on its
skeleton,fromwhichtwokeyingredientsaredevised:
1. Local reconstructionsobtainedby solving small, embarrassinglyparallel prob-
lemsinsideeachelement,andtypicallyconceivedsothattheircompositionwith
the natural interpolator of sufficiently smooth functions yields a physics- and
problem-dependentprojectoronlocalpolynomialspaces;
2. Stabilisation terms that penalise residuals designed at the element level so
as to ensure stability while preserving the approximation properties of the
reconstruction.
These ingredients are combined to formulate local contributions, which are
then assembled as in standard Finite Element Methods. From this construction,
several appealing features ensue: the support of polytopal meshes and arbitrary
approximation orders in any space dimension, an enhanced compliance with the
physics, and a reduced computational cost thanks to the compact stencil along
with the possibility of locally eliminating a large portion of the unknowns. This
monographprovidesanintroductiontothedesignandthemathematicalaspectsof
HHOmethodsfordiffusiveproblems,alongwithapanelofapplicationstoadvanced
modelsincomputationalmechanics.
The support of polytopal meshes is perhapsthe most defining feature of HHO
methods. Mesh generation and adaptation is often the bottleneck of computer-
assistedmodelling:despitetheenormousprogressinthisfield,traditionalunstruc-
tured meshes suffer from intrinsic drawbacks, and disposing of discretisation
methodsthatdeliverhigh-orderapproximationson polytopalmeshesconstitutesa
veritabletechnologicaljump.Letusconsiderafewexamples.Capturinggeometric
microstructuresinthedomaintraditionallyrequirestheuseofsmallelements,which
cansignificantlyaddtothecomputationalburden.Withpolyhedralmeshes,onthe
other hand, one can incorporate such geometric features into larger agglomerated
v
vi Preface
elements[17,38],thusachievingasignificantcostreductionwithoutcompromising
the accuracy. In the context of conforming Finite Element Methods, local mesh
adaptation requires special strategies to either prevent or treat hanging nodes.
When conforming mesh refinement is performed, one typically faces the choice
ofacceptingadegradationofthemeshqualityorrenouncingthebenefitofnested
meshes. Polyhedral methods, on the other hand, can usually treat hanging nodes
seamlessly and even support innovative strategies such as adaptive coarsening
[38]; see, e.g., the a posteriori-based adaptive HHO algorithm devised in [154]
for electrostatic models. The seamless support of meshes that are nonconforming
in the traditional sense is also crucial for models that feature inner interfaces. In
geosciencesapplications,e.g.,accountingforthepresenceoffracturesorfaultsin
the subsoil is paramount to accurately reproduce the flow patterns. In petroleum
basinmodelling,fracturesaretypicallyincorporatedintothenumericalmodelsby
the mutualsliding of two portionsof a corner-pointgrid along the fracture plane,
resulting in highly nonconforming meshes [192]. Polytopal methods offer a true
advantage in this case, as no special strategy is required to handle this situation;
see,e.g.,[106,107],whereHHO methodsforDarcyflowandpassivetransportin
fracturedporousmediaaredevisedandanalysed,or[55],wherefracturenetworks
aresimulatedusingVirtualElementMethods.
AnotherkeyfeatureofHHO methodsiscompliancewith thephysics,meaning
that they can incorporate fundamental properties of the model into the design.
Let us examine a few examples. In the context of diffusive conservation laws
in divergence form, e.g., HHO schemes can typically be interpreted as enforcing
polynomialmomentsoflocalbalanceswithconservativenumericalapproximations
of the fluxes; see, e.g., [129] and [155, Section 4.3.2.5] concerning scalar linear
diffusionproblems,[68,147]concerninglinearandnonlinearelasticity,[58,73]on
linearand nonlinearporoelasticity,and[70] for the incompressibleNavier–Stokes
equations. Another example of compliance with the physics is provided by the
robustnesswithrespecttothevariationsoftheproblemcoefficients;see,e.g.,[149]
concerninganisotropicheterogeneousdiffusionproblemsinmixedformulation.In
somecases, robustnesscanbeextendedtoobtaina seamlesstreatmentofsingular
limitsofPDEmodels.AfirstexampleisprovidedbythePéclet-robustHHOscheme
of [159], which fully supports locally degenerate diffusion with exact solutions
that exhibit jumps inside the domain; see also [157] for further insight into this
topic. Another example is provided by [69], where the authors propose an HHO
scheme for the Brinkman problem that is fully robust in the Darcy limit. In this
case,anovelapproachtotheanalysiswasalsodevisedinordertoidentifythelocal
(Darcy- or Stokes-dominated) regime inside each element. A third example of a
physics-compliant HHO scheme is provided by [70, 151], where non-dissipative
discretisationsoftheconvectivetermintheincompressibleNavier–Stokesequations
areproposed.
The construction of HHO methods is conceived so as to enable efficient
sequential and parallel implementations in arbitrary space dimension [114]. Both
the reconstruction and the stabilisation terms are devised at the element level,
and the coupling amongneighbouringelementsonly occursvia the commonface
Preface vii
unknowns.Asaresult,elementunknownscanbeeliminatedpriortotheassembly
step bycomputinga localSchurcomplement;see, e.g.,[155, Section4.3.2.4]and
[160, Section 3.3.1]. This procedure is often referred to as “static condensation”
intheFiniteElementsliterature,atermreminiscentofitsoriginsincomputational
mechanics [206, 214]. For the HHO approximation of degree k ≥ 0 of a scalar
three-dimensionaldiffusionproblem,thenumberofdegreesoffreedomafterstatic
condensation is 1(k + 2)(k + 1)N , with N denoting the number of non-
2 faces faces
Dirichletmeshfaces.Forhighpolynomialdegrees,thisisadramaticimprovement
over,e.g., vanilla implementationsof DiscontinuousGalerkinmethods, where the
number of degrees of freedom is 1(k + 3)(k + 2)(k + 1)N , with N denoting
6 el el
the number of mesh elements. The improvement is even more dramatic when
considering more complex models such as those encountered in incompressible
fluidmechanics.Asnoticedin[160]inthecontextofthelinearStokesproblem,itis
possibleinthiscasetodeviseastaticcondensationstrategythat,foranypolynomial
degree, leads to global systems with only one pressure unknown per element; cf.
also [151, Remark 6] for the extension of such a strategy to the fully nonlinear
Navier–Stokesproblem.
To closethisintroductorysection,weprovidea briefhistoricaloverviewofthe
developmentof HHO methods. Hybrid High-Order methods in primal form were
originallyintroducedin[146]inthecontextofquasi-incompressiblelinearelasticity
models, with the first fully referencedpublication[158] dating back to 2014.The
earlystepsinthedevelopmentofthemethodwere,ontheonehand,theidentifica-
tion of equilibratedtractions[147] and, on the other hand,its extensionto locally
variable coefficients [148] and more general, possibly degenerate scalar second-
order models [159]. In parallel, the Mixed High-Order method was introduced
in [149], and its link with primal HHO methods first recognised in [9]. Among
the first applications to engineering problems, we can cite the a posteriori-driven
adaptive algorithm for electrostatics devised in [154]; see also [162]. A keystone
in the understanding of the relations between HHO and other hybrid methods is
[129],wherebridgesarebuiltwithHybridisableDiscontinuousGalerkinandHigh-
Order Mimetic methods. Further progressin this directionis made in [56], where
a unified framework comprising a large number of mixed and hybrid methods is
proposed, and in [163], where HHO methods are bridged with Nonconforming
Virtual Elements, and a stable gradient reconstruction is proposed which enables
theinterpretationofbothmethodsasGradientDiscretisationMethods[178].Atthe
same time, a large effort was undertaken for the application, and corresponding
analysis, of HHO methods to complex models, which are more realistic from
the point of view of scientific and engineering applications. A first remarkable
contribution is the HHO method for the Cahn–Hilliard problem designed and
analysed in [105], which constitutes the first application to nonlinear problems.
Landmarkcontributionsinthisdirectionarethepapers[141,142],whichstudythe
applicationoftheHHOmethodtofullynonlinear,Leray–Lionstypeellipticmodels.
Here, a systematic development of the tools for the design and analysis of HHO
methodsfornonlinearproblemswasundertaken,leadingtokeyfunctionalanalysis
resultsofbroaderapplicability.A first exampleof suchresults isthe development
viii Preface
ofaframeworkforthestudyoftheapproximationpropertiesofprojectorsonlocal
polynomialspaces,whichalsoresultedinachangeofthecanonicalwayofintroduc-
ing HHO methods[155]. Anothervaluable set of results are Sobolevembeddings
and compactness results for bounded sequences of HHO functions, which form
the cornerstone of the convergence analysis by compactness [171, Section 1.2].
TheapplicationofHHOmethodstomorecomplexmodelsofengineeringinterest
subsequentlyfocusedmainlyonincompressiblefluidmechanics,solidmechanics,
andgeosciences.Concerningthefirstapplicativefield,keycontributionsincludethe
HHOmethodfortheStokesproblemdevelopedin[9],laterhackedin[99,161]to
robustlyhandlesmallvaluesoftheviscosityandlargeirrotationalbodyforces,and
theHHOmethodsfortheNavier–Stokesproblemdevelopedandanalysedin[151]
and[70],whereaconservativeformulationbasedonTemam’sdeviceisproposed.
A recent contribution inspired by the HHO literature is [98], where Bingham
pipe flows are considered. We can also cite [8], where the influence of dominant
convection on the order of convergenceis evaluated using the Oseen model. Still
in the contextof fluid dynamics, [69] proposes and analyses an HHO method for
the Brinkmanmodel,thatis fullyrobustin boththe Stokesand Darcylimits. Key
contributions to the application of HHO methods to problems in solid mechanics
include[68],concerningnonlinearelasticmodelsvalidunderthesmalldeformation
assumption, [60] on the application of HHO methods to Kirchhoff–Love plate
bending problems, and, more recently [4], on finite deformations of hyperelastic
materials.ApplicationsoftheHHOtechnologytoproblemsingeosciencesinclude
fluid flows in fractured porous media [106, 107], miscible fluid flows in porous
media[14],aswellaslinearandnonlinearporoelasticity[58,73]possiblyincluding
stochastic coefficients [72]. Finally, more recent methodological developments
include, in particular, the extension of the HHO method to meshes with curved
faces[66],thetreatmentofunfittedinterfaceproblems[89]basedontheCutFEM
technology[90],anditsapplicationtohighlyoscillatoryellipticmodels[115].We
alsopointoutheretheanalysisframeworkformethodsinfullydiscreteformulation
developedin[143],whichshowsthebenefitsofthisapproach,originallyadoptedin
thecontextofHHOmethods,whenappliedtootherrecentpolytopaltechnologies.
OtherPolytopal Methods
Discretisation methods that support polytopal meshes and, possibly, arbitrary
approximation orders have experienced a vigorous development over the last
decade. Novel approaches to the design and analysis have been developed or
rediscoveredborrowingideasfromotherbranchesofmathematicssuchastopology
and geometry, or expanding past their initial limits the original ideas underlying
Finite Element or Finite Volume Methods. A brief historical perspective focusing
ondiffusiveproblemsissketchedinwhatfollows.
Since their introduction, usually attributed to Tihonov and Samarski˘ı [271],
Finite Volume Methods have been extensively used for the discretisation of PDE
Preface ix
modelsexpressedaslinearornonlinearconservationlawsindivergenceform.The
classicalTwo-PointFluxApproximationschemerequires,however,strictconditions
ofmesh-datacomplianceforconsistency;see,e.g.,thereferencemonograph[188].
Severallowest-orderpolytopalmethodshavethereforebeendevelopedintheFinite
Volumes spirit in an attempt to circumvent these conditions. In Multi-Point Flux
Approximation methods [1–3, 7, 181], consistent approximations of the flux are
obtained through local reconstructions involving elements that share a common
node. In Mixed [172] and Hybrid [190] Finite Volume Methods, a similar result
isachievedbyintroducingauxiliaryunknownsatfaces,whichcaninsomecasesbe
eliminated resorting to local interpolation. In the Discrete Duality Finite Volume
Method [137, 166, 208], two simultaneous discrete versions of the conservation
law are solved on primal and dual meshes, using both vertex and cell unknowns.
All of the above-mentionedmethods possess local conservation properties on the
primal mesh and enable an explicit identification of continuous approximations
of the normal trace of the flux. We refer the reader to [156, 171] for a more
comprehensive list and a critical review of Finite Volumes and related methods
on generic polytopal meshes. Over the years, polytopal Finite Volume Methods
have been applied to a variety of linear and nonlinear PDE models including,
e.g., fully nonlinearelliptic problemsin divergenceform [15,169], miscible fluid
flows in porous media [102, 104], the incompressible Navier–Stokes equations
[173,202,203],etc.
WhileclassicalFiniteElementMethodsarerestrictedtocertainelementgeome-
tries,extensionstomoregeneralmeshesarepossibleinsomecases.Anypolytopal
meshcanbesplit,e.g.,intoasimplicialconformingsubmeshoverwhichstandard
FiniteElementMethodscanbeapplied,possiblywithsomemodifications.In[221],
thisideaisappliedtoconstructmixedFiniteElementMethodsonpolytopalmeshes
by solving local problems; see also [275, Section 7] for a discussion on this and
related approaches. Conforming P1 elements on a simplicial submesh are at the
core of the Vertex Approximated Gradient method, see [191] and [178, Section
8.5], which additionally uses barycentric eliminations to prevent the submeshing
from introducing additional unknowns, and mass-lumping to mitigate the issues
of P1 elements on coarse meshes in case of heterogeneities. It is also possible to
constructFiniteElementsoncertaintypesofpolytopalelementswithoutresorting
tosubmeshing.VariousconstructionsofH1-conformingpolygonalFiniteElements
areexplored,e.g.,in[265],wheretheauthorscompareLaplace[111,210],Wach-
spress[276],andmeanvalueshapefunctionsonconvexpolygons.Furtherrelated
worksinclude [266,267]. Similar constructionsare also possible formixedFinite
Elements. In [219], H(div)-conforming Finite Elements on polygons and certain
polyhedraareconstructedusingbarycentriccoordinates.Ageneralconstructionof
scalar and vector Finite Elements families on convex polygonal and polyhedral
elements is proposed in [198], where basis functions are expressed in terms of
barycentriccoordinatesand their gradients;this approachis inspiredby the Finite
ElementsExteriorCalculusformalism[21,24,25].Similarideasaredevelopedin
[109]todeviseH(curl)andH(div)FiniteElementsonpolytopalmesheswiththe
minimumnumberofdegreesoffreedomensuringtheappropriateglobalcontinuity.
