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Contributors
NumbersinParenthesesindicatethepagesonwhichtheauthor’scontributionsbegin.
EberhardB€ansch(555),AppliedMathematicsIII,DepartmentMathematik,Friedrich-
Alexander-Universit€atErlangen-N€urnberg, Erlangen,Germany
John W.Barrett (275),Department of Mathematics,Imperial CollegeLondon,
London, UnitedKingdom
S€orenBartels (221),Abteilung f€urAngewandte Mathematik,Albert-Ludwigs-
Universit€atFreiburg, Freiburgim Breisgau, Germany
Andrea Bonito (1),Department ofMathematics, TexasA&M University, College
Station, TX,United States
Alan Demlow (1),Department ofMathematics, TexasA&M University, College
Station, TX,United States
Qiang Du(425),Department of AppliedPhysicsand AppliedMathematics and the
Data Science Institute,Columbia University, New York, NY,UnitedStates
XiaobingFeng (425),Department ofMathematics, TheUniversity ofTennessee,
Knoxville, TN,UnitedStates
HaraldGarcke(275),Fakult€atf€urMathematik,Universit€atRegensburg,Regensburg,
Germany
BehrendHeeren(621),InstituteforNumericalSimulation,UniversityofBonn,Bonn,
Germany
Michael Neilan(105),Department of Mathematics,University ofPittsburgh,
Pittsburgh, PA,United States
Ricardo H. Nochetto (1),Department ofMathematics and InstituteforPhysical
ScienceandTechnology,UniversityofMaryland,CollegePark,MD,UnitesStates
Robert Nu€rnberg(275),Department of Mathematics,Imperial CollegeLondon,
London, UnitedKingdom
Martin Rumpf (621),InstituteforNumerical Simulation, University ofBonn,Bonn,
Germany
Abner J.Salgado(105),Department of Mathematics,University ofTennessee,
Knoxville, TN,UnitedStates
Robert I. Saye(509),Mathematics Group, Lawrence Berkeley NationalLaboratory,
Berkeley, CA,UnitedStates
Alfred Schmidt(555),Center forIndustrial Mathematics and MAPEXCenter for
Materials andProcesses, Universit€atBremen,Bremen, Germany
xiii
xiv Contributors
JamesA.Sethian (509),Mathematics Group, Lawrence Berkeley National
Laboratory; Department ofMathematics, Universityof California,Berkeley,
Berkeley,CA,United States
MaxWardetzky(621),InstituteofNumericalandAppliedMathematics,Universityof
G€ottingen, G€ottingen,Germany
BenediktWirth (621),Institute forAnalysisand Numerics,Universityof M€unster,
M€unster,Germany
WujunZhang(105),DepartmentofMathematics,RutgersUniversity,Piscataway,NJ,
UnitedStates
Preface
This and the following volumes are devoted to the numerical approximation
of geometric partial differential equations (GPDEs). Before describing the
objectivesandcontentsofthisproject,itisappropriatetoexplainthemeaning
ofGPDEsandthereasonswhytheydeservetwovolumesoftheHandbookof
Numerical Analysis. GPDEs are governing equations of natural, social and
economic phenomena where geometry plays a prominent to dominant role.
Examples abound from interfaces and free boundaries in fluids and solids—
such as modelling of surface tension and bending effects involving second
fundamental forms—to the development of defects in director and line fields
in liquid crystal modelling, motion, merging and splitting of droplets within
incompressible fluids, total variation minimization in imaging science and
shape morphing and extrapolation, just to name a few. These problems pos-
sess an intrinsic mathematical beauty and pose a formidable challenge both
in analysis and computation. Besides their overwhelming mathematical rich-
ness, GPDEs are ubiquitous in many scientific, engineering and industrial
applications, such as fluid and solid mechanics, materials science, biology,
chemistry, astrophysics, plasma physics, imaging and computer animation.
The last three decades have witnessed the development of powerful algo-
rithmsandcorrespondingnumericalanalysisforthedescriptionandcomputa-
tionofinterfaces.Thelevelsetandphasefieldmethodshavejoinedthemore
traditional front tracking techniques and, together with the advent of ever
more powerful and versatile computers, have allowed for the simulation and
understanding of rather complex phenomena involving interfaces. It is fair
to assert that, besides theory and experimentation, mathematical modelling
and computation have established themselves as the third pillar of scientific
inquiry. A well-designed computational model can replace a very expensive
or even unrealizable experimental setting and give new insight into the
theoretical developments of a specific discipline. Because of their technical
complexity and practical relevance, GPDEs are a chief example.
Thepurposeofthistwo-volumecontributionistoprovideamissingrefer-
encebookthatportraysthestateoftheartonbasicalgorithmsforGPDEsand
their analysis, along with their impacts in a wide variety of areas of science
and engineering. Since this field has grown tremendously over the last few
years, the selection of topics and authors is a formidable task. Our intention
is to present different and complementary approaches ranging from funda-
mental numerical analysis of basic algorithms to scientific computation and
xv
xvi Preface
excitingapplications.Thechoiceofauthorsreflectstheirexpertiseonvarious
aspects of this ambitious and multifaceted project.
This first volume consists of eight chapters which encompass numerical
analysis of GPDEs, algorithm design, analysis and simulation of interfaces
and shape morphing. A brief description follows.
l Numerical analysis of GPDEs. Chapter 1, by Bonito, Demlow and
Nochetto, reviews and extends three popular finite element methods to
approximatetheLaplace–Beltramioperatoronacodimensiononesurface;
they are the parametric, trace and narrow band methods. The discussion
centres around the relationship between the (minimal) regularity of the
surface and the manner it is represented and approximated. Chapter 2,
byNeilan,SalgadoandZhang,describesandanalyzes severalapproxima-
tion techniques for the fully nonlinear Monge–Ampe`re equation. Key
notions such as stability, consistency and continuous dependence on data
are developed in the max norm and used to derive rates of convergence
to the viscosity solution.
l Geometrically nonlinear plates and rods. Chapter 3, by Bartels, discusses
geometrically nonlinear bending models for plates and rods that
allow large deformations. Plate deformations are constrained to be isome-
tries, whereas rods are elastic and self-avoiding. Recent finite element
algorithms are reviewed along with their numerical analysis.
l Interfacesandfreeboundaries.Thisisthefocusofthenextfourchapters.
Chapter 4, by Barrett, Garcke and N€urnberg, collects their work on the
parametricfiniteelementapproximationofcurvaturedriveninterfaceevo-
lutions. Finite element methods for surface geometric equations, coupling
of surface geometric equations with bulk equations, and two phase flows
are presented and analyzed. Chapter 5, by Du and Feng, overviews the
phase-field modelling and corresponding diffuse interface approximation.
The main numerical analysis techniques are described and applied.
Chapter6,bySayeandSethian,reviewsthelevel-setmethodstartingfrom
basic ideas and techniques and culminating with complex and intricate
interfacedynamics.Adescriptionoftheinherentchallengestomultiphase
andsharp-interfacephysicsisprovidedalongwithnumericalalgorithmsto
overcome them. These three chapters present variational front tracking,
phasefieldandlevelsetmethods,threecompetingtechniquesforinterface
evolution. Chapter 7, by B€ansch and Schmidt, discusses different aspects
of phase transitions in materials science and fluids along with efficient
algorithms for their approximation.
l Riemmanian calculus and applications. Chapter 8, by Heeren, Rumpf,
Wardetzky and Wirth, develops a Riemannian calculus on the space of
discrete triangular shells along with its discretization. This leads to
adequate algorithms for shape morphing, shape extrapolation, parallel
transport, smooth interpolation and other geometric processes of interest.
Computer-animated video sequences and movies are presented.
Preface xvii
Acknowledgements
A.B. is partially supported by NSF grant DMS-1817691. R.H.N. is partially
supported by NSF grants DMS-1411808 and DMS-1908267.
Andrea Bonito
Ricardo H. Nochetto
Chapter 1
Finite element methods for
the Laplace–Beltrami operator
Andrea Bonitoa,*, Alan Demlowa and Ricardo H. Nochettob
aDepartmentofMathematics,TexasA&MUniversity,CollegeStation,TX,UnitedStates
bDepartmentofMathematicsandInstituteforPhysicalScienceandTechnology,University
ofMaryland,CollegePark,MD,UnitesStates
*Correspondingauthor:e-mail:[email protected]
Chapter Outline
1 Introduction 2 4 Parametricfinite
2 Calculusonsurfaces 7 elementmethod 40
2.1 Parametricsurfaces 7 4.1 FEMonLipschitzparametric
2.2 Differentialoperators 9 surfaces 41
2.3 Signeddistance 4.2 Geometricconsistency 46
function 14 4.3 Apriorierroranalysis 51
2.4 Curvatures 16 4.4 Aposteriorierroranalysis 61
2.5 Surfaceregularityand 5 Tracemethod 66
propertiesofthedistance 5.1 Preliminaries 69
function 18 5.2 Apriorierrorestimates 76
2.6 Divergencetheoremon 5.3 Aposteriorierrorestimates 81
surfaces 20 6 Narrowbandmethod 87
3 Perturbationtheory 22 6.1 ThenarrowbandFEM 88
3.1 Perturbationtheory 6.2 PDEgeometricconsistency 90
forC1,αsurfaces 22 6.3 Propertiesofthenarrow
3.2 Perturbationtheory bandFEM 95
forC2surfaces 26 6.4 Apriorierrorestimates 99
3.3 H2extensionsfrom Acknowledgements 100
C2surfaces 31 References 101
Abstract
Partialdifferentialequationsposedonsurfacesariseinanumberofapplications.Inthis
surveywedescribethreepopularfiniteelementmethodsforapproximatingsolutionsto
theLaplace–Beltramiproblemposedonann-dimensionalsurfaceγembeddedinn+1:
the parametric, trace, and narrow band methods. The parametric method entails
constructing an approximating polyhedral surface Γ whose faces comprise the finite
HandbookofNumericalAnalysis,Vol.21.https://doi.org/10.1016/bs.hna.2019.06.002
©2020ElsevierB.V.Allrightsreserved. 1
2 HandbookofNumericalAnalysis
element triangulation. The finite element method is then posed over the approximate
surfaceΓinamannerverysimilartostandardFEMonEuclideandomains.Inthetrace
method it is assumed that the given surface γ is embedded in an n + 1-dimensional
domain Ω which has itself been triangulated. An n-dimensional approximate surface
Γ is then constructed roughly speaking by interpolating γ over the triangulation of Ω,
andthefiniteelementspaceoverΓconsistsofthetrace(restriction)ofastandardfinite
elementspaceonΩtoΓ.InthenarrowbandmethodthePDEposedonthesurfaceis
extendedtoatriangulatedn+1-dimensionalbandaboutγ whosewidthisproportional
tothediameterofelementsinthetriangulation.Inallcasesweprovideoptimalapriori
errorestimatesforthelowestorderfiniteelementmethods,andwealsopresentapos-
teriori error estimates for the parametric and trace methods. Our presentation focuses
especially on the relationship between the regularity of the surface γ, which is never
assumed better than of class C2, the manner in which γ is represented in theory and
practice,andthepropertiesoftheresultingmethods.
Keywords: Surface partial differential equations, Laplace–Beltrami operator, Surface
finiteelementmethods,Parametricfiniteelementmethods,Tracefiniteelementmeth-
ods,Narrowbandfinite elementmethods,Aprioriandaposteriorierrorestimates
AMSClassificationCodes:65N15,65N30,35A99,53-01,58J32
1 Introduction
Partialdifferentialequations(PDEs)posedonsurfacesplayanimportant role
inmanydomainsofpureandappliedmathematics,includinggeometry,mod-
ellingofmaterials,fluidflow,andimageandshapeprocessing.Thenumerical
approximation of such surface PDEs is both practically important and the
source of many mathematically rich problems.
We consider a closed, compact and orientable surface γ in n+1 of
co-dimension 1. The Laplace–Beltrami operator (cid:1)Δγ, which acts as a gener-
alization of the standard Euclidean Laplace operator, plays a central role in
both static and time-dependent surface PDE models arising in a wide range
of applications. Because of this a wide variety of numerical methods have
been developed for the Laplace–Beltrami equation
(cid:1)Δγue¼fe,
R
e e
where f is a given forcing function satisfying f ¼0. In this article we first
γ
lay out some important notions from differential geometry. We then describe
three important classes of finite element methods (FEMs) for the Laplace–
Beltrami problem: the parametric method, the trace method, and the narrow
bandmethod.Inallthreecaseswefocusonthesimplestcaseofpiecewiselin-
ear finite element spaces and give an in-depth discussion of the effects of
geometry on error behaviour.
The parametric finite element method was introduced by Dziuk (1988),
with some important related techniques appearing in earlier works on
boundary element methods (Bendali, 1984; N(cid:1)ed(cid:1)elec, 1976). This method
TheLaplace–Beltramioperator Chapter 1 3
is the simplest of the many FEM that have been developed for solving
the Laplace–Beltrami problem. TRhe given PDE is first written in weak
form as: Find ue2H1ðγÞ such that ue¼0 and
γ
Z Z
aðue,veÞ:¼ rγue(cid:3)rγve¼ feve 8ve2H1ðγÞ:
γ γ
Here H1(γ) is the set of functions ve in L (Ω) whose tangential gradient
2
rγve2½L2ðγÞ(cid:4)n+1. The continuous surface γ is approximated by a polyhedral
surface Γ whose faces serve as a finite element mesh, and the finite element
space is made of continuous piecewise linear functions over Γ. The finite
element method then consists of finding U2 such that
Z Z
AðU,VÞ¼ rΓU(cid:3)rΓV¼ FV 8V2,
Γ γ
whereFisasuitableapproximation(lift)offdefinedonΓ.Initsconceptionand
implementation,theresultingmethodisverysimilartocanonicalFEMforsolv-
ingPoisson’sproblemonEuclideandomains.ToquoteDziuk,“…thenumerical
schemeisjustthesameasinaplane-twodimensionalproblem.Theonlydiffer-
ence is that in our case the computer has to memorize three-dimensional nodes
insteadoftwo-dimensionalones”(Dziuk,1988,p.143).Thestrategyunderlying
parametricsurfacefiniteelementmethods—directtranslationofFEMonEuclid-
ean spaces to triangulated surfaces—has subsequently been applied to a variety
of methods. These include higher order standard Lagrange methods (Demlow,
2009), various types of discontinuous Galerkin methods (Antonietti et al.,
2015; Cockburn and Demlow, 2016; Dedner et al., 2013), and mixed methods
in classical, hybridized, and finite element exterior calculus formulations
(Bendali, 1984; Cockburn and Demlow, 2016; Ferroni et al., 2016; Holst and
Stern, 2012). A posteriori error estimation and adaptivity have been studied in
Demlow and Dziuk (2007), Wei et al. (2010), Bonito et al. (2013), Dedner
and Madhavan (2016), Bonito et al. (2016), andBonito and Demlow (2019).
Finally, we refer to the survey article (Dziuk and Elliott, 2013).
In many applications in which surface PDEs are to be solved, a back-
ground volume (bulk) mesh is already present. A paradigm example is two-
phase fluid flow, in which effects on the interface between the two phases
suchassurfacetensionarecoupledwithstandardequationsoffluiddynamics
onthebulk.Inthesecasesitisadvantageoustoutilizethebackgroundvolume
meshtosolvesurfacePDEsinsteadofindependentlymeshingγ.Thisisespe-
ciallythecasewhenγ isevolving,sincethemeshesneededfortheparametric
method typically distort as γ changes and periodic remeshing is thus neces-
sary. The trace and narrow band methods both employ background bulk
meshes in order to solve surface PDEs.
Trace (or cut) FEMs for the Laplace–Beltrami problem were first intro-
duced in Olshanskii et al. (2009). In this method an approximating surface
