Acta Numerica (2005), pp. 194 c Cambridge University Press, 2005DOI: 10.1017/S0962492904000224 Printed in the United KingdomComputation of geometricpartial dierential equationsand mean curvature owKlaus DeckelnickInstitut f ur Analysis und Numerik,Otto-von-Guericke-Universit at Magdeburg, Universit atsplatz 2,D39106 Magdeburg, GermanyE-mail: Klaus.Deckelnick@Mathematik.Uni-Magdeburg.deGerhard DziukAbteilung f ur Angewandte Mathematik,Albert-Ludwigs-Universit at Freiburg i. Br., Hermann-Herder-Strae 10,D79104 Freiburg i.Br., GermanyE-mail: gerd.dziuk@mathematik.uni-freiburg.deCharles M. ElliottDepartment of Mathematics,University of Sussex, Mantell Building,Falmer, Brighton, BN1 9RF, UKE-mail: C.M.Elliott@sussex.ac.ukThis review concerns the computation of curvature-dependent interface mo-tion governed by geometric partial dierential equations. The canonical prob-lem of mean curvature ow is that of nding a surface which evolves so that, atevery point on the surface, the normal velocity is given by the mean curvature.In recent years the interest in geometric PDEs involving curvature has bur-geoned. Examples of applications are, amongst others, the motion of grainboundaries in alloys, phase transitions and image processing. The methodsof analysis, discretization and numerical analysis depend on how the surfaceis represented. The simplest approach is when the surface is a graph overa base domain. This is an example of a sharp interface approach which, inthe general parametric approach, involves seeking a parametrization of thesurface over a base surface, such as a sphere. On the other hand an interfacecan be represented implicitly as a level surface of a function, and this ideagives rise to the so-called level set method. Another implicit approach is thephase eld method, which approximates the interface by a zero level set of a2 K. Deckelnick, G. Dziuk and C. M. Elliottphase eld satisfying a PDE depending on a new parameter. Each approachhas its own advantages and disadvantages. In the article we describe themathematical formulations of these approaches and their discretizations. Al-gorithms are set out for each approach, convergence results are given and aresupported by computational results and numerous graphical gures. Besidesmean curvature ow, the topics of anisotropy and the higher order geometricPDEs for Willmore ow and surface diusion are covered.CONTENTS1 Introduction 1402 Some geometric analysis 1503 Denition and elementary properties ofmean curvature ow 1554 Parametric mean curvature ow 1575 Mean curvature ow of graphs 1666 Mean curvature ow of level sets 1747 Phase eld approach to mean curvature ow 1818 Anisotropic mean curvature ow 1899 Fourth order ows 208Appendix 223References 2251. IntroductionA geometric evolution equation denes the motion of a hypersurface byprescribing the normal velocity of the surface in terms of geometric quantit-ies. As well as being of striking mathematical interest, geometric evolutionproblems occur in a wide variety of scientic and technological applications.A traditional source of problems is materials science, where the understand-ing of the strength and properties of materials requires the mathematicalmodelling of the morphology of microstructure. Evolving surfaces mightbe grain boundaries, which separate diering orientations of the same crys-talline phase, or solidliquid interfaces exhibiting dendritic structures inunder-cooled solidication. On the other hand newer applications are as-sociated with image processing. For example, in order to identify a darkshape in a light background in a two-dimensional image a so-called snakecontour is evolved so that it wraps around the shape.In this article we survey numerical methods for the evolution of surfaceswhose normal velocity is strongly dependent on the mean curvature of thesurface. The objective is to nd a family (t)t[0,T] of closed compact andorientable hypersurfaces in Rn+1whose evolution is dened by specifyingthe velocity V of (t) in the normal direction . An example of a generalComputation of geometric PDEs and mean curvature flow 3geometric evolution equation isV = f(x, , H) on (t), (1.1)where f depends on the application and the x dependence might arise fromevaluating on the surface (t) eld variables which satisfy their own systemof nonlinear partial dierential equations in Rn+1away from the surface. Itis important to note that, in order to specify the evolution of the surface, itis sucient to dene the normal velocity.The prototype problem is motion by mean curvature, for whichV = H on (t), (1.2)where H is the sum of the n principal curvatures of (t). We call H the meancurvature rather than the arithmetic mean of the principal curvatures. Oursign convention is that H is positive for spheres, with being the outwardnormal. It is well known that, starting from an initial surface 0, thisequation is a gradient ow for the area functional,E() =_1 dA. (1.3)In applications the area functional is an interfacial energy with a constantenergy density 1. Equation (1.2) may be viewed as an analogue for surfacesof the parabolic heat equationutu = 0.On the other hand, another geometric equation isV = (t)H on (t), (1.4)where (t) is the LaplaceBeltrami or surface Laplacian operator on (t).This can be viewed as an analogue of the spatially fourth order parabolicequationut + 2u = 0.1.1. ApproachesIn order to solve a surface evolution equation analytically or numerically,we need a description of (t). Each choice of description leads to a partic-ular nonlinear partial dierential equation dening the evolution. Thus thecomputational method depends strongly on the way we choose to describethe surface. For this article we shall focus on four possible approaches.Parametric approach. The hypersurfaces (t) are given as(t) = X(, t)(M),where M is a suitable reference manifold (xing the topological type of (t))and X : M[0, T) Rn+1has to be determined. Here X(p, t), for p M, is4 K. Deckelnick, G. Dziuk and C. M. ElliottFigure 1.1. A dumbbell-shaped two-dimensional surfaceparametrized over the unit sphere.Figure 1.2. A lemniscate, parametrized over the unit circle.the position vector at time t of a point on (t). If we are interested in closedcurves in the plane then M can be the unit circle S1, whereas if (t) is atwo-dimensional surface then M could be the unit sphere S2(see Figures 1.1and 1.2). Geometrical quantities are easily expressed as derivatives of theparametrization so that evolution laws such as (1.2) may be translated intononlinear parabolic systems of PDEs for the vector X. With this approachthere is no notion of the surface being the boundary of an open set andhaving an inside and outside, so self-intersection is perfectly natural forsmooth parametrizations and is not necessarily associated with singularities.For example in the plane a gure of eight curve can be smoothly mappedonto the unit circle one to one (Figure 1.2). At the crossing point the curvehas two smoothly evaluated normals and curvatures which depend on theparametrization. A parametrized curve evolving by mean curvature canevolve smoothly from this conguration.Graphs. We assume that (t) can be written in the form(t) = (x, u(x, t)) [ x ,where Rnand the height function u : [0, T) R has to befound. We shall see that the law (1.2) leads to a nonlinear parabolic PDEfor u. Clearly, the assumption that (t) is a graph is rather restrictive;however, techniques developed for this case have turned out to be very usefulin understanding more general situations. Since the height is a smoothfunction we can view (t) as dividing R into two sets, namely theregions above and below the graph.Computation of geometric PDEs and mean curvature flow 5Figure 1.3. Level lines of a level set function (right) for thegure of eight curve (left).Figure 1.4. Graph of a level set function for the gure ofeight curve, cut at the zero level. Negative part left andpositive part (graphically enlarged) right.Level set method. We look for (t) as the zero level set of an auxiliaryfunction u : Rn+1[0, ) R, that is,(t) = x Rn+1[ u(x, t) = 0.The law (1.2) now translates into a nonlinear, degenerate and singular PDEfor u. Clearly intrinsic to this approach is the notion of (t) being a dividingsurface between the two regions where the level set function is positive andnegative. Thus we have the notion of inside and outside. In order to describea gure of eight by a level set function it is necessary to have the level setfunction positive and negative, as shown in Figures 1.3 and 1.4.Phase eld approach. The phase eld approach is based on an approxima-tion of the sharp interface by a diuse interface

(t) = x Rn+1[ 1 +C u

(x, t) 1 Cof width O(), across which the phase eld function u

has a transitionfrom approximately one bulk negative value 1 to approximately a secondpositive bulk value +1. The zero level set of the phase eld function ap-proximates the surface. Just as in the level set method there is the notion6 K. Deckelnick, G. Dziuk and C. M. Elliottof a material interface separating an inside and outside and in the basic im-plementation interface self-intersection and topological change are handledautomatically. The bulk values of the phase eld function correspond tothe minima of a homogeneous energy function with two equal double wells.Interfacial energy is assigned to the diuse interface via the gradient of thephase eld function. For motion by mean curvature the evolution is denedas a semilinear parabolic equation of reactiondiusion or GinzburgLandautype. Frequently in applications mathematical models are derived which,from the beginning, involve diuse interfaces and phase eld functions.CommentsConceptually the graph formulation is the simplest and most ecient. Itinvolves solving a scalar nonlinear parabolic equation in n space dimensionsand directly computes the surface. However, there are many circumstanceswhere the surface is not a graph. Furthermore, even if the initial surface is agraph it is possible that over the course of the evolution that property mightbe lost, despite the surface evolving smoothly. This would lead to gradientblow-up of the solution of the graph equation. There is the possibility thatthe solution of a numerical discretization exists globally and appears to bestable even though there is no solution to the continuous equation.The parametric approach is also direct. It is conceptually more advancedthan the graph approach and one has to solve for an n-dimensional sur-face a system of n + 1 parabolic equations. If the surface is a graph thenthe parametric approach is less ecient than solving for the height of thesurface. On the other hand it is more widely applicable. In the case of aclosed curve one can use periodic boundary conditions on the unit intervalin order to solve over the circle. A closed two-dimensional surface can beapproximated by a polyhedral surface. A parametrized surface does notsee an inside or outside. From the point of view of dierential geometrythis may not be an issue. However, when the surface separates two phases,or two materials, or two colours, there are signicant issues. For example,consider using two colours in Figure 1.2 in order to dene the curve as theinterface between the coloured regions. Black may be used to colour the in-side of both loops and white to colour the the rest of the plane, but if blackis used inside just one loop then the other loop is lost. Thus, in order touse the parametric approach with this initial condition, one either thinks ofa parametrization which traverses the curve without a crossing, but with asingle self-intersection, or regards them as being two separate closed curveswhich touch at one point. These choices lead to diering evolutions formean curvature ow.Contrary to the parametric approach, the level set method has the capab-ility of tracking topological changes (like pinching-o or merging) of (t) inan automatic way. In the basic implementation of the method topologicalComputation of geometric PDEs and mean curvature flow 7change is nothing special and is observed in post-processing the computa-tional output. This is because, in principle, zero level sets of continuousfunctions can exhibit these features. However, from the mathematical pointof view there are issues of existence of solutions of the degenerate partialdierential equations that the level set approach generates. In the case ofmotion by mean curvature there is the notion of a viscosity solution whichyields a unique evolution from any continuous function. The example of thelemniscate discussed in the context of the parametric approach introducesa new idea in the level set approach of fattening of the interface. The levelset for this example develops an interior whose boundary yields both of thedescribed solutions. Self-intersection, merger and pinch-o can all be simu-lated by this approach. This advantage, however, needs to be oset againstthe fact that the problem now becomes (n + 1)-dimensional in space.The phase eld approach can also handle topological change, self-inter-section, merger and pinch-o without doing anything special. It is the oneapproach which in its conception involves an approximation. The fact thatit involves a new parameter is both an advantage and a disadvantage. Theparabolic equations are in principle easy to solve but possess a certain com-putational stiness due to the thickness of the diuse interface. However,in many applications phase eld models arise naturally and the parameterallows us to resolve singularities in a way which may be viewed as beingphysically motivated. From both the mathematical and physical points ofview it is widely applicable in a rational way, whereas the use of the levelset method is frequently ad hoc.In general, the choice of one or the other approach will depend on whetherone expects topological changes in the ow.1.2. ApplicationsIn what follows we list some problems in which a law of the form (1.1) orgeneralizations of it arise.Grain boundary motionGrain boundaries in alloys are interfaces which separate bulk crystallineregions of the same phase but with diering orientations. Associated withthe grain boundary is a surface energy which gives rise to a thermodynamicrestoring force. For a constant surface energy density this is simply thesurface tension force proportional to the mean curvature and the resultingevolution law is just (1.2). Frequently there is also a driving force causingmotion of the grain boundary.Surface growthThe growth of thin lms on substrates is technologically important. Forexample, epitaxy is a method for growing single crystals by the deposition8 K. Deckelnick, G. Dziuk and C. M. Elliottof atoms and molecules onto a growing lm surface. There are numerousphysical mechanisms operating at diering time and length scales whichaect the growth process. A simple model would have a driving force rep-resenting the deposition ux of atoms onto the surface which might be inthe normal direction or in a xed vertical direction parallel to a beam ofarriving atoms.Image processingOne of the most important problems in image processing is to automaticallydetect contours of objects. We essentially follow the exposition of Aubertand Kornprobst (2002). Suppose that M Rn+1(n = 1 or 2) is a givenobject and let I(x) = \M(x) be the characteristic function of M. Thefunctiong(x) = 11 +[I(x)[2,where I is a mollication of I, will be small near the contour of M. It istherefore natural to look for minimizers of the functionalJ() =_g(x) dAwhere is a curve in R2or a surface in R3. The corresponding L2-gradientow leads to the following evolution law: nd curves/surfaces (movingsnakes) (t) such thatV = (g ) = g H g on (t).Here, t plays the role of an articial time; clearly this law ts into theframework (1.1).Stefan problem for undercooled solidicationConsider a container Rn+1(n = 1 or 2) lled with an undercooledliquid. Solidication of the liquid follows the nucleation of initial solid seedwith characteristic diameter larger than the critical radius. The seed willthen grow into the liquid. A mathematical model for this situation is theStefan problem with kinetic undercooling, in which the solidliquid interfaceis described by a curve/surface (t) and has to be determined togetherwith the temperature distribution. Here the interior of (t) is the solidregion S(t) and the exterior is the liquid region L(t). Using a suitablenon-dimensionalization the problem then reads: for a given initial phaseboundary 0 and initial temperature distribution 0 = 0(x) (x ), ndthe non-dimensional temperature = (x, t) and the phase boundary (t)(t > 0), such that the heat equation is satised in the bulk, that is,t = 0 in (t),Computation of geometric PDEs and mean curvature flow 9together with the initial value (, 0) = 0 in . On the moving boundarythe following two conditions are satised:V = 1l__ on (t), (1.5) +V()V +H = 0 on (t). (1.6)Here, [/] denotes the jump in the normal derivative of the temper-ature eld across the interface and l is the constant measuring the latentheat of solidication. Equation (1.6) is the GibbsThomson law; V, arenon-dimensional positive constants measuring the strength of the kineticundercooling and surface tension which depress the temperature on thesolidliquid interface from the scaled equilibrium zero melting temperat-ure. Furthermore, H is an anisotropic mean curvature associated with asurface energy density, (), depending on the orientation of the normal.There may also be anisotropy, (), in the kinetic undercooling. Note that(1.6) can be rewritten asV ()V = H 1 on (t).If we consider as being given, this equation again ts into our generalframework (1.1) provided we allow for a coecient in front of V and ageneralized notion of mean curvature.Figure 1.5 from Schmidt (1996) shows a simulation in which the freeboundary was described by the parametric approach resulting in a sharpinterface model. One can see the free boundary forming a dendrite. ForFigure 1.5. Evolution of a dendrite with sixfold anisotropy.Time-steps of the free boundary (left) and adapted grid forthe temperature at one time-step (right).10 K. Deckelnick, G. Dziuk and C. M. Elliottresults concerning three-dimensional dendrites and more information aboutthe algorithm we refer to Schmidt (1996).Figure 1.6 from Fried (1999) illustrates a possible eect of using a levelset method for the free boundary in this problem. Dendrites may seem tomerge. But if a smaller time-step is used the dendrites stay apart. Formore information about a level set algorithm for dendritic growth we referto Fried (1999, 2004).Surface diusion and Willmore owThe following laws do not t into (1.1), but we list them as examples of im-portant geometric evolution equations in which the normal velocity dependson higher derivatives of mean curvature.The surface diusion equationV = H (1.7)models the diusion of mass within the bounding surface of a solid body.At the atomistic level atoms on the surface move along the surface owing toa driving force consisting of a chemical potential dierence. For a surfacewith constant surface energy density the appropriate chemical potential inthis setting is the mean curvature H. This leads to the ux lawV = divj,where is the mass density and j is the mass ux in the surface, with theconstitutive ux law (Herring 1951, Mullins 1957)j = DH.Here, D is the diusion constant. From these equations we obtain the law(1.7) after an appropriate non-dimensionalization. In order to model theFigure 1.6. A possible eect of the use of a level set method.Growing dendrites: merging (left) for large time-step sizeand staying apart (right) for smaller time-step size.Computation of geometric PDEs and mean curvature flow 11underlying structure of the solid body bounded by , anisotropic surfacediusion is important, that is,V = H, (1.8)with H denoting the anisotropic mean curvature of the surface as it isintroduced in (8.15).A similar evolution law is Willmore ow,V = H +H[[2 12H3on (t), (1.9)which arises as the L2-gradient ow for the classical bending energy E() =12_H2dA. Apart from applications in mechanics and membrane physicsthis ow has recently been used for surface restoration and inpainting.1.3. Outline of articleThis article is organized as follows. In Section 2 we present some usefulgeometric analysis, in particular the notion of mean curvature. The basicmean curvature ow is dened in Section 3 and some elementary propertiesare described. The next four sections consider in turn basic approaches fornumerical approximation. In Section 4 we consider the parametric approach.We start with the classical curve shortening ow and present a semidiscretenumerical scheme as well as error estimates. Next, we show how to applythe above ideas to the approximation of higher-dimensional surfaces. Acrucial point is to construct numerical schemes which reect the intrinsicnature of the ow. Section 5 is concerned with graphs. We prove an errorbound for a semidiscrete nite element scheme thereby showing the virtueof working with geometric quantities. A fully discrete scheme along withstability issues is discussed afterwards. In Section 6 we introduce the levelset equation as a way of handling topological changes. We briey discuss theframework of viscosity solutions which allows a satisfactory existence anduniqueness theory. For numerical purposes it is convenient to regularize thelevel set equation. We collect some properties of the regularized problemand clarify its formal similarity to the graph setting. The approximation ofmean curvature ow by phase eld methods is considered in Section 7. Evenbefore numerical discretization there is the notion of approximation of asharp interface by a diuse interface of width O(). The phase eld approachdepends on the notion of a diuse interfacial energy composed of quadraticgradient and homogeneous free energy terms involving a phase eld function.The choice of double well energy potential is discussed. We recall someanalytical results as well as a convergence analysis for a discretization inspace by linear nite elements. We nish this section by discussing thediscretization in time together with the question of stability. In Section 8we introduce the concept of the anisotropy together with its relevant12 K. Deckelnick, G. Dziuk and C. M. Elliottproperties and subsequently generalize the ideas of the previous sections tothis setting. Finally, Section 9 is concerned with fourth order ows: wepresent discretization techniques for both surface diusion and Willmoreow.For the convenience of the reader we have included a long list of references,which are related to the subject of these notes, but not all of which are citedin the text.2. Some geometric analysisThe aim of this section is to collect some useful denitions and results fromdierential geometry. We refer to Gilbarg and Trudinger (1998) and Giga(2002) for a more detailed exposition of this material.2.1. HypersurfacesA subset Rn+1is called a C2-hypersurface if for each point x0 thereexists an open set U Rn+1containing x0 and a function u C2(U) suchthatU = x U [ u(x) = 0, and u(x) ,= 0 for all x U . (2.1)The tangent space Tx is then the n-dimensional linear subspace of Rn+1that is orthogonal to u(x). It is independent of the particular choice offunction u which is used to describe . A C2-hypersurface Rn+1iscalled orientable if there exists a vectoreld C1(, Rn+1) (i.e., C1in an open neighbourhood of ) such that (x) Tx and [(x)[ = 1 forall x . In what follows, we shall assume that Rn+1is an orientedC2-hypersurface.We dene the tangential gradient of a function f, which is dierentiablein an open neighbourhood of byf(x) = f(x) f(x) (x) (x), x .Here denotes the usual gradient in Rn+1. Note also that f(x) is theorthogonal projection of f(x) onto Tx. It is straightforward to show thatf only depends on the values of f on . We use the notationf(x) = (D1f(x), . . . , Dn+1f(x)) (2.2)for the n + 1 components of the tangential gradient. Obviouslyf(x) (x) = 0, x .If f is twice dierentiable in an open neighbourhood of , then we deneComputation of geometric PDEs and mean curvature flow 13the LaplaceBeltrami operator of f asf(x) = f(x) =n+1

i=1DiDif(x), x . (2.3)2.2. Oriented distance functionA useful level set representation of a hypersurface can be obtained with thehelp of the distance function. Let be as above and assume in addition that is compact. The JordanBrouwer decomposition theorem then impliesthat there exists an open bounded set Rn+1such that = . Weassume that the unit normal eld to points away from and dene theoriented (signed) distance function d byd(x) =___dist(x, ), x Rn+1 0, x dist(x, ), x .It is well known that d is globally Lipschitz-continuous and that there exists > 0 such thatd C2(), where = x Rn+1[ [d(x)[ < . (2.4)Every point x can be uniquely written asx = a(x) +d(x)(a(x)), x , (2.5)where a(x) . Furthermore, d(x) = (a(x)), x , which implies inparticular that[d(x)[ 1 in . (2.6)Figure 2.1. Graph (right) of the oriented distance functionfor the curve (left).14 K. Deckelnick, G. Dziuk and C. M. Elliott2.3. Mean curvatureLet us next turn to the notion of mean curvature. By assumption, is C1in a neighbourhood of so that we may introduce the matrixHjk(x) = Djk(x), j, k = 1, . . . , n + 1, x . (2.7)It is not dicult to show that (Hjk(x)) is symmetric. Furthermore,n+1

k=1Hjk(x)k(x) =n+1

k=1Djk(x)k(x) = 12Dj[[2(x) = 0,since [[ = 1 on . Thus, (Hjk(x)) has one eigenvalue which is equal tozero with corresponding eigenvector (x). The remaining n eigenvalues1(x), . . . , n(x) are called the principal curvatures of at the point x. Wenow dene the mean curvature of at x as the trace of the matrix (Hjk(x)),that is,H(x) =n+1

j=1Hjj(x) =n

j=1j(x). (2.8)Note that (2.8) diers from the more common denition H = 1n

n+1j=1 Hjj.From (2.7) we derive the following expression for mean curvature,H(x) = (x), x , (2.9)where f = n+1j=1 Djfj denotes the tangential divergence of a vectoreldf. In particular we see that H > 0 if = Snand the unit normal eld ischosen to point away from Sn, i.e., (x) = x.While the sign of H depends on the choice of the normal , the meancurvature vector H is an invariant. A useful formula for this quantity canbe obtained by choosing f(x) = xj, j 1, . . . , n+1 in (2.3) and observingthat Dixj = ij ji. We then deduce with the help of (2.9) thatxj = n+1

i=1Di(ji) = ( )j j = Hj,so thatx = H on . (2.10)Let us next x a point x and calculate H( x) for various representa-tions of the surface near x.Level set representation. Suppose that is given as in (2.1) near x. Clearly,we then have(x) = u(x)[u(x)[Computation of geometric PDEs and mean curvature flow 15for x U . If the plus sign applies we obtainH =

u[u[ = u[u[ = 1[u[n+1

i,j=1_ij uxiuxj[u[2_uxixj. (2.11)In the special case that u(x) = d(x), where d is the oriented distance func-tion to , we obtain in view of (2.6)H(x) = d(x), x . (2.12)Graph representation. Suppose thatU = (x, v(x)) [ x ,where Rnis open, x = (x1, . . . , xn) and v C2(). Dening u(x, xn+1)= v(x) xn+1 we see that U is the zero level set of u and the aboveconsiderations imply thatH(x, v(x)) = _ v(x)_1 +[v(x)[2_, (x, v(x)) U , (2.13)where is the gradient in Rnand the unit normal is chosen as = (v,1)1+|v|2.Parametric representation. Suppose that there exists an open set V Rnand a mapping X C2(V, Rn+1) such thatU = X(V ), rank DX() = n for all V.The vectors X1(), . . . , Xn() then form a basis of Tx at x = X(). Wedene the metric on bygij() = Xi() Xj(), i, j = 1, . . . , nand let gijbe the components of the inverse matrix of (gij). We then havethe following formulae for the tangential gradient of a function f (denedin a neighbourhood of ) and the mean curvature vector H:f =n

i,j=1gij(f X)jXi, (2.14)H = 1gn

i,j=1i_gijg Xj_ (2.15)where g = det(gij).16 K. Deckelnick, G. Dziuk and C. M. Elliott2.4. Integration by partsLet us assume in this section that is in addition compact. The formulafor integration by parts on is (cf. Gilbarg and Trudinger (1998))_Dif dA =_fHi dA i = 1, . . . , n + 1, (2.16)where dA denotes the area element on and f is continuously dierentiablein a neighbourhood of . Applying (2.16) with h = fDig, summing fromi = 1, . . . , n+1 and taking into account that i = 0, we obtain Greensformula,_f g dA = _fg dA. (2.17)In particular, we deduce from (2.10)_H dA =_x dA, (2.18)where is continuously dierentiable in a neighbourhood of with valuesin Rn+1and x = n+1i=1 xi i. This relation will be veryimportant for the numerical treatment of mean curvature ow. The aboveformulae can be generalized to surfaces with boundaries by including anappropriate integral over .2.5. Moving surfacesIn this section we shall be concerned with surfaces that evolve in time. Afamily ((t))t(0,T) is called a C2,1-family of hypersurfaces if, for each point(x0, t0) Rn+1(0, T) with x0 (t0), there exists an open set U Rn+1, > 0 and a function u C2,1(U (t0, t0 +)) such thatU (t) = x U [ u(x, t) = 0 and u(x, t) ,= 0, x U (t). (2.19)Suppose in addition that each (t) is oriented by a unit normal eld (, t) C1((t), Rn+1) and that C0(

0 0 depending on X and T such that for every 0 < h h0there exists a unique solution Xh(, t) = Nj=1Xj(t)j() of the dierencescheme (4.9), (4.10) andmax[0,T]|X Xh|L2(I) +__ T0|X Xh|2L2(I) dt_1/2 ch, (4.12)max[0,T]|XtXht|L2(I) +__ T0|Xt Xht|2L2(I) dt_1/2 ch, (4.13)where c depends on X and T.This algorithm can be generalized without changes to curves evolving inhigher codimension, i.e., X : I [0, T] Rmand m > 2. The curve solving(4.3) has a velocity only in the normal direction. It is also possible to useComputation of geometric PDEs and mean curvature flow 23the parametric equationXt = X[X[2instead, which denes the same curve evolving in the normal direction with anormal velocity being given by the curvature. However, the parametrizationis dierent, with the points on the curve now having a tangential velocity.A nite element error analysis for the motion of a closed curve is given inDeckelnick and Dziuk (1994), while error bounds for the evolution of a curveattached to a xed boundary with a normal contact condition are proved inDeckelnick and Elliott (1998).In order to obtain a practical method we still have to discretize in time.Choose a time-step > 0 and let tm = m, m = 0, . . . , M, M [T ]. Welet Xmh Sh denote the approximation to X(, tm). On the basis of (4.7)we suggest the following scheme:1_I(Xm+1h Xmh ) h[Xmh[ +_IXm+1h h[Xmh[ = 0 for all h Sh. (4.14)Calculations similar to those above yield a time discrete analogue of (4.9),which we formulate as the following algorithm.Algorithm 1. (Curve shortening ow)(1) Let X0j = X0(j) (j = 0, . . . , N).(2) Compute Xm+1j (j = 0, . . . , N) from the tridiagonal systems12(qmj +qmj+1)(Xm+1j Xmj )_Xm+1j+1 Xm+1jqmj+1Xm+1j Xm+1j1qmj_ = 0.(3) If minj=1,...,N+1qm+1j > 0 then replace m by m+ 1 and goto 2.Thus, in each time-step a positive denite and symmetric linear systemhas to be solved for each component of Xm+1h . Each of these linear systemsis of tridiagonal form with two additional entries reecting the periodicitycondition. The system decouples with respect to the dimension of the spacein which the curve moves. For practical purposes a redistribution of nodesaccording to arc length on the curve is sometimes convenient.Let us go back to the more precise notation X(, t) = X((cos , sin ), t).For later purposes it is convenient to look at (4.14) from a slightly dierentangle. We introduce the polygon mh = Xmh (I) along with the spaceSmh = h : mh R2[ h is ane on each face of mh. (4.15)Thus, if h Smh , then h is the restriction of an ane function on R2oneach face of the polygon and thereforeh() = h( Xmh ()), I,24 K. Deckelnick, G. Dziuk and C. M. ElliottFigure 4.1. Curve shortening ow applied to a star-shapedcurve. Time-steps 0, 100, 200, 300, 500, 700, 5000, 7000(time-step size = 8.5586 105), 480 nodes.Figure 4.2. Curve shortening ow applied to a curve with aself-intersection. A singularity (cusp) appears. The eectis that the algorithm jumps across the singularity. SeeFigure 4.3 for a magnied image. Time-steps 0, 1000, 2000,2500, 3000, 5000, 6000, 7000 (time-step size = 8.5586 105),480 nodes.Figure 4.3. Close-up of Figure 4.2. Time-steps 3498 and 3499and 3505. The parametric theory breaks down.Computation of geometric PDEs and mean curvature flow 25belongs to Sh. Recalling (2.14) we havemhh = 1[ Xmh[h Xmh[ Xmh[, p = Xmh (),where (u v)ij = uivj (u, v R2) and mhh is given piecewise on eachface of mh . Let us dene Xm+1h Smh by Xm+1h (p) = Xm+1h (), p = Xmh ().Observing thatmhXm+1h mhh[ Xmh[ =Xm+1h h[ Xmh[for all h Shwe can rewrite (4.14) as1_mh(Xm+1h id)h dA+_mhmhXm+1h mhh dA = 0 for all h Smh .(4.16)Note that the dot between the matrices mhXm+1h and mhh is the stand-ard scalar product in R4. The key point about the formulation (4.16) isthat m+1h is now parametrized with the help of the polygon mh from theprevious time-step, so that the reference manifold M is no longer needed.We can interpret the second integral on the left-hand side of (4.16) as anapproximation to_(tm+1)(tm+1)x (tm+1)dA,which equals _(tm+1)H dA by (2.17) and (2.10). Here, H is just theusual curvature of the curve (tm+1), but of course it is now natural to alsouse (4.16) for approximating surfaces evolving by mean curvature. We willdiscuss this issue in the next section.4.2. Mean curvature ow of surfacesIn this section we shall use a higher-dimensional version of (4.16) in orderto approximate parametric surfaces (t) = X(M, t), which satisfy (4.1). Tobegin, we need an analogue of the polygons used in the previous section.Figure 4.4. Polyhedral surfaces: successively rened gridsapproximating a half sphere. Macro triangulation (left)and triangulation levels 1, 5 and 7.26 K. Deckelnick, G. Dziuk and C. M. ElliottFigure 4.5. First row: Parametric mean curvature ow of adumbbell-shaped surface. Development of a singularity.Second row: Axially symmetric level set computation of thesame ow going beyond the topological change of the surface.Denition 2. We call a set Rn+1a polyhedral surface if =_TThT,where the triangulation Th consists of closed, nondegenerate, n-dimensionalsimplices. The intersection of two adjacent simplices is an (n k)-dimen-sional subsimplex of these simplices (k 1, . . . , n).Our aim is to construct polyhedral surfaces 0h, . . . , Mh (without bound-ary) in such a way that mh is an approximation to (tm). These surfacesare obtained with the help of the following algorithm. We start the com-putations with an initial polyhedral 0h which approximates the initial sur-face 0. In practice there are several ways to construct the initial discretesurface. One way is to map triangulations of charts onto the continuoussurface and to glue them together. A much better way is to construct amacro triangulation, that is, a coarse approximation 0h of 0 such thatComputation of geometric PDEs and mean curvature flow 27Figure 4.6. A thin two-dimensional torus shrinks underparametric mean curvature ow to a circle.Figure 4.7. A thick two-dimensional torus (cut open)shrinks under parametric mean curvature ow to a spheredeveloping a singularity.0h (see (2.4), (2.5)) and then to rene this triangulation in Rn+1andproject the new nodes x orthogonally onto the smooth surface according tox

= xd(x)(x) to obtain the new nodes x

of the next-ner triangulationfor 0h (see Figure 4.4).Algorithm 2. (Mean curvature ow of surfaces)Let 0h be a polyhedral approximation of 0.For m = 0, 1, . . . , M 1 deneSmh = h C0(mh ) [ h|T is ane for each T mh,and nd Xm+1h Smh with1_mh(Xm+1h id)h dA+_mhmhXm+1h mhh dA = 0 for all h Smh(4.17)Generate the new surface m+1h = Xm+1h (mh ), and if it is a polyhedralsurface then goto to the next m.28 K. Deckelnick, G. Dziuk and C. M. ElliottThis algorithm is based on a nite element method for partial dierentialequations on surfaces, developed in Dziuk (1988). Let us have a look at theimplementation of the above algorithm. Fix m 0, . . . , M1 and denoteby a1, . . . , aN Rn+1the nodes of the polyhedral surface mh . The functionsi : mh R, i = 1, . . . , N are uniquely dened by the requirementsi Smh , i(aj) = ij, i, j = 1, . . . , N.It is not dicult to verify that 1, . . . , N actually form a basis of Smh . Now,stiness and mass matrix are dened bySij =_mhmhi mhj dA, i, j = 1, . . . , NMij =_mhij dA, i, j = 1, . . . , N.Expanding (Xm+1h )k(p) = Nj=1(k)j j(p) (where (Xm+1h )k is the kth com-ponent of Xm+1h ), we nd that (4.17) is equivalent to the linear systemsM(k)+S(k)= b(k), k = 1, . . . , n + 1. (4.18)Here, (k)= ((k)1 , . . . , (k)N ) and b(k) RNis given byb(k)j =_mhxkj dA, j = 1, . . . , N.Since the matrix M + S is symmetric and positive denite, the systems(4.18) can be solved with a conjugate gradient method. The only dierenceto a Cartesian FEM is that the nodes have one more coordinate.5. Mean curvature ow of graphsWe turn our attention to the mean curvature evolution of surfaces (t),which can be written as graphs over some base domain Rn, that is,(t) = (x, u(x, t)) [ x .In order to nd the dierential equation to be satised by the height functionu, we recall (2.13) and (2.21) to see that the mean curvature H and thevelocity V in the direction of = (u,1)1+|u|2 are given byH = _ u_1 +[u[2_, V = ut_1 +[u[2. (5.1)Computation of geometric PDEs and mean curvature flow 29Thus, the evolution law V = H on (t) translates into the nonlinearparabolic partial dierential equationut_1 +[u[2_ u_1 +[u[2_ = 0 in (0, T), (5.2)to which we add the following boundary and initial conditionsu = g on (0, T), (5.3)u(, 0) = u0 in , (5.4)where g : R and u0 : R are given functions. The boundarycondition (5.3) implies that the boundaries of the surfaces (t) are keptxed during the evolution. It would also be possible to replace (5.3) byun = 0 on (0, T), (5.5)in which case the surfaces (t) would meet the boundary of the cylinder R at a right angle.5.1. Analytical resultsThe main diculties for the mathematical analysis are due to the fact thatthe operatorA(u) = _1 +[u[2_ u_1 +[u[2_is not uniformly parabolic and not in divergence form. Only in one spacedimension the equation is in divergence form, since A(u) = (arctan ux)x.Theorem 5.1. Let be a bounded domain in Rnwith C2+andu0 C2,().(a) Suppose that g C2,() and that the compatibility conditionsu0 = g and _1 +[u0[2_ u0_1 +[u0[2_ = 0 on are satised. If has nonnegative mean curvature, the initial-boun-dary value problem (5.2), (5.3), (5.4) has a unique smooth solutionwhich converges to the solution of the minimal surface equation withboundary data g as t .(b) Suppose that the compatibility condition u0n = 0 on holds. Thenthe initial-boundary value problem (5.2), (5.5), (5.4) has a uniquesmooth solution which converges to a constant function as t .Proof. See Lieberman (1986) and also Huisken (1989) for (a); (b) is provedin Huisken (1989).30 K. Deckelnick, G. Dziuk and C. M. ElliottThe assumption that the boundary of the domain has nonnegative meancurvature is a necessary condition. If it is dropped, the gradient of thesolution will become innite on the boundary: see Oliker and Uraltseva(1993). The main tool in the proof of the previous theorem is the derivationof an evolution equation for the surface element. Our numerical algorithmswill be based on a variational formulation of (5.2), (5.3). To derive it, divide(5.2) byQ = _1 +[u[2, (5.6)multiply by a test function H10() and integrate. Integration by partsimplies_utQ +_u Q = 0, H10(), 0 < t < T. (5.7)It is straightforward to derive from (5.7) the decrease in area.Lemma 5.2. Suppose that u is a smooth solution of (5.2). Then_u2tQ + ddt_Q = 0. (5.8)Proof. Since u(, t) = g on (0, T) we have ut(, t) = 0 on for0 < t < T. The relation (5.8) now follows by inserting = ut(, t) in (5.7)and observing that Qt = uutQ .Recalling that V = utQ we may rewrite the relation (5.8) in the moregeometric form of Lemma 3.1.5.2. Spatial discretizationLet Th be an admissible nondegenerate triangulation of the domain withmesh size bounded by h, simplices S and h = SThS the correspondingdiscrete domain. We assume that vertices on h are contained in . Thespace of nite elements of order s N is chosen to beXh = vh C0(h) [ vh is a polynomial of order s on each S Th. (5.9)The subspace containing functions with zero boundary values will be de-noted by Xh0.We assume that for s N, p [1, ] there exists an interpolation operatorIh : Hs+1,p() Xh which satises Ihv Xh0 for v Hs+1,p() H10(),as well as|v Ihv|Lp(h) +h|(v Ihv)|Lp(h) chs+1|v|Hs+1,p() (5.10)for all v Hs+1,p(). For dimensions n < p(s + 1), we can, for instance,choose the usual Lagrange interpolation operator; in higher dimensions aComputation of geometric PDEs and mean curvature flow 31possible choice is the Clement operator. For what follows we choose piece-wise linear nite elements: s = 1.We now use (5.7) in order to dene a semidiscrete approximation to thesolution of (5.2)(5.4) as follows: nd uh(, t) Xh with uh(, t) Ihg Xh0and uh(, 0) = uh0 = Ihu0 such that_huhthQh+_huh hQh= 0, for all h Xh0 (5.11)and all t (0, T). Here, we have abbreviated Qh = _1 +[uh[2. Thefollowing lemma establishes the global existence of the discrete solution.Lemma 5.3. The semidiscrete problem has a unique solution uh whichexists globally in time.Proof. We denote by ai, i = 1, . . . , N the nodes of the triangulation Th andby i the corresponding nodal basis functions. We assume that a1, . . . , aN1are the interior nodes, while aN1+1, . . . , aN lie on h. We expand uh(.t) =

N1i=1i(t)i +

Ni=N1+1g(ai)i and the relation (5.11) then amounts to anonlinear system of ODEs for = (1, . . . , N1). Existence of a uniquelocal solution follows from standard ODE theory, while the analogue of(5.8) implies a uniform bound on uh and therefore on since Xh is nite-dimensional. This allows us to continue the solution for all times.In order to prove error estimates for the semidiscrete problem we needto make regularity assumptions on the solution of the continuous problem.Let us suppose that u satises_ T0|ut|2H1,() dt +_ T0|ut|2H2() dt N (5.12)for some N > 0 (see Deckelnick and Dziuk (1999) for sucient conditionswhich imply (5.12)). In the following we shall assume that we have a solutionof this kind until the time T. We shall formulate our error estimates interms of geometric quantities, more specically in terms of the normals = (u,1)Q , h = (uh,1)Qhand the normal velocities V = utQ, Vh = uhtQhreecting the form of the a priori estimate (5.8).Theorem 5.4. Let u be a solution of the continuous problem (5.2)(5.4),which satises (5.12). Then_ T0_h(V Vh)2Qh + sup(0,T)_h[ h[2Qh ch2.The constant c depends on N.32 K. Deckelnick, G. Dziuk and C. M. ElliottProof. Let us give the proof of this theorem for polygonal domains, =h. The proof shows how important it is to work with the geometric quant-ities. The dierence of the discrete weak form (5.11) and the correspondingcontinuous weak form of equation (5.2) reads__utQ uhtQh_h +__uQ uhQh_ h = 0 (5.13)for all discrete test functions h Xh0. As a test function we chooseh = Ihutuht = (utuht) (utIhut).We observe that_utQ uhtQh_(utuht) = (V Vh)(V QVhQh) (5.14)= (V Vh)2Qh + (V Vh)V (QQh) (V Vh)2Qh[V Vh[[V [Q[1Q 1Qh[Qh 12(V Vh)2Qh 12[ut[2[ h[2Qh.Here we have used the fact that1Q 1Qh [ h[. (5.15)For the gradient term in (5.13) we exploit the fact that the last componentof the vector QhQh is zero, and get_uQ uhQh_ (utuht) = ( h) (utuht, 0) (5.16)= ( h) (QhQh)t.With the elementary relation( h) = ( h) h = 12[ h[2,the right-hand side in (5.16) can be estimated as follows:( h) (QhQh)t= ( h) (tQhtQh +QthQht)= 12[ h[2(Qt +Qht) + ( h) ( h)tQh + ( h) t(QQh)= 12([ h[2Qh)t + 12[ h[2Qt + ( h) t(QQh) 12([ h[2Qh)t 12[Qt[[ h[2[t[Q[ h[2Qh,Computation of geometric PDEs and mean curvature flow 33where again we have used (5.15). With this estimate, (5.14) and (5.16) theerror relation (5.13) implies the bound12_(V Vh)2Qh + 12ddt_[ h[2Qh 12_|ut|2L() + 3|ut|2L()__[ h[2Qh+_[V Vh[ [utIhut[ +_[ h[ [(utIhut)[.We estimate the interpolation terms with the help of (5.10), that is,_[V Vh[ [utIhut[ +_[ h[ [(utIhut)[ c|ut|H2()_h2__(V Vh)2_12+h__[ h[2_12_ _(V Vh)2Qh +_[ h[2Qh + c|ut|2H2()h2for every > 0, since Qh 1. After a suitable choice of we arrive at12_(V Vh)2Qh + ddt_[ h[2Qh c_1 +|ut|2H1,()__[ h[2Qh +c|ut|2H2()h2.A Gronwall argument and the choice uh(, 0) = Ihu0 then nally proves thetheorem.Remark 1. It is possible to show that in the two-dimensional case theabove error bounds imply that sup[0,T]Qh C uniformly in h. As aconsequence the error estimate can be written down with the help of theusual norms, namely_ T0|utuh,t|2L2(h) dt + sup(0,T)|(u uh)|2L2(h) ch2.5.3. Time discretizationLet us choose a time-step > 0 and let tm = m, m = 0, . . . , M, M [T ]as well as vm= v(, m) for m = 0, . . . , M. Based on (5.11) we suggest thefollowing algorithm.Algorithm 3. (Mean curvature ow of graphs) Let u0h = Ihu0. Form = 0, . . . , M 1, compute um+1h Xh such that um+1h Ihg Xh0 and,34 K. Deckelnick, G. Dziuk and C. M. Elliottfor every h Xh0,1_hum+1h hQmh+_hum+1h hQmh= 1_humh hQmh. (5.17)with Qmh = _1 +[umh[2.The above scheme is semi-implicit in time and has the property that ineach time-step a linear Laplace-type equation with stiness matrix weightedby Qmh has to be solved. In order to analyse its stability we go back to thebasic energy norms introduced in (5.8).Theorem 5.5. The solution umh , 0 m M of (5.17) satises, for everym 1, . . . , M,m1

k=0_h[Vkh[2Qkh +_hQmh _hQ0h (5.18)where Vkh = (uk+1h ukh) Qkhis the discrete normal velocity.Proof. We choose h = uk+1h ukh as a test function in (5.17) for m = kand get1_h(uk+1h ukh)2Qkh+_huk+1h (uk+1h ukh)Qkh= 0. (5.19)Let us use the notation kh = (ukh,1)Qkh. The integrand in the second termcan be rewritten asuk+1h (uk+1h ukh)Qkh= (Qk+1h )21Qkh uk+1hQk+1h

ukhQkhQk+1h= (Qk+1h )2Qkh+ 12[k+1h kh[2Qk+1h Qk+1h= 12[k+1h kh[2Qk+1h +Qk+1h Qkh + (Qk+1h Qkh)2Qkh.We insert this result into (5.19), sum over k = 0, . . . , m1 and obtain theequationm1

k=0_h[Vkh[2Qkh +m1

k=0_h(Qk+1h Qkh)2Qkh+ 12m1

k=0_h[k+1h kh[2Qk+1h+_hQmh =_hQ0hwhich implies the stability estimate (5.18).Computation of geometric PDEs and mean curvature flow 35Let us emphasize that our scheme is unconditionally stable even thoughthe nonlinear expressions are treated explicitly. Other schemes, such asfully explicit and fully implicit variants are discussed in Dziuk (1999a). Itis natural to follow the ideas of the semidiscrete case in order to analyse theabove algorithm. For the analysis of the fully discrete scheme we need thefollowing regularity assumptions:supt(0,T)_|u(, t)|H2,() +|ut(, t)|H1,()_+_ T0_|ut|2H2() +|utt|2_ds N. (5.20)This leads to the following result.Theorem 5.6. Assume that there exists a solution of (5.2)(5.4) on [0, T],which satises (5.20) and let umh , (m = 1, . . . , M = [T ]) be the solution ofAlgorithm 3. Then there exists a 0 > 0 such that, for all 0 < 0,M1

m=0_h(VmVmh )2Qmh c(2+h2), (5.21)supm=0,...,M_h[mmh [2Qmh c(2+h2). (5.22)Proof. This is a special case of the results obtained in Deckelnick and Dziuk(2002a).For computational tests we refer to the anisotropic case; see Table 8.2.Here we give some test results for the usual norms. Error estimates inthese norms for the two-dimensional case are contained in Deckelnick andDziuk (2000). For the tests we have solved the partial dierential equationTable 5.1. Absolute errors in L((0, T); L2()),L((0, T); H1()) and experimental orders ofconvergence (EOC) for the test problem.h E1 EOC E2 EOC2.0 1.1932 0.9428 1.0 0.6649 0.84 0.9453 0.000.7368 0.2878 2.74 0.5873 1.560.4203 0.1067 1.77 0.2919 1.250.2219 0.04211 1.46 0.1375 1.180.1137 0.01775 1.29 0.06536 1.110.05754 0.007986 1.17 0.03168 1.0636 K. Deckelnick, G. Dziuk and C. M. Elliott(5.2) with a given additional right-hand side. We have chosen u(x, t) =sin ([x[2t) sin (1 t) and calculated a right-hand side from this func-tion. The computational domain was = x R2[ [x[ < 1 and we used theboundary condition u = 0 on . The time interval was [0, 4] and as time-step size we have chosen = 0.125h. For two successive grids with grid sizesh1 and h2 we computed the absolute errors E(hj), (j = 1, 2) between dis-crete solution and exact solution for certain norms. The experimental orderof convergence was then dened by EOC = log (E(h1)/E(h2))/ log (h1/h2).In Table 5.1 the errors in the norms E1 = sup0mM |um umh| withM = T and E2 = sup0mM |(um umh )| are shown. The resultsconrm the theoretical estimates. Note that the L((0, T), L2())-errorbehaves linearly in the grid size h because we have chosen the time-stepproportional to the spatial grid size.6. Mean curvature ow of level setsIf we want to compute topological changes of free boundaries then it isnecessary to leave the parametric world, because this xes the topologicaltype of the interface. One method to do this is to dene the interface as thelevel set of a scalar function:(t) = _x Rn+1[ u(x, t) = 0_.Let us assume for the moment that u C2,1(Rn+1 (0, T)) with u ,= 0in a neighbourhood of t(0,T) (t) t. Recalling (2.11) and (2.20), therelation V = H on (t) would hold ifutn+1

i,j=1_ij uxiuxj[u[2_uxixj = 0 (6.1)in a neighbourhood of t(0,T) (t) t. This partial dierential equationis highly nonlinear, degenerate parabolic and not dened where the gradientof u vanishes. Therefore, standard methods for parabolic equations fail, butit is possible to develop an existence and uniqueness theory for (6.1) withinthe framework of viscosity solutions. The corresponding notion involves apointwise relation and the analysis relies mainly on the maximum principle.It is therefore not straightforward to use nite element methods, whichare typically L2-methods and normally do not allow a maximum principle.This diculty will be reected in the numerical analysis. An example ofthe evolution of level sets under mean curvature ow is shown in Figure 6.1(Deckelnick and Dziuk 2001).Crandall, Ishii and Lions (1992) give a concise introduction to the theoryof viscosity solutions, while Giga (2002) describes in detail the applicationof level set techniques to a large class of geometric evolution equations.Computation of geometric PDEs and mean curvature flow 37Figure 6.1. Evolution of level lines under meancurvature ow.Detailed descriptions of computational techniques for level set methodsalong with a host of applications can be found in the monographs by Sethian(1999) and Osher and Fedkiw (2003).6.1. Analytical resultsStarting from (6.1), we are interested in the following problem:utn+1

i,j=1_ij uxiuxj[u[2_uxixj = 0 in Rn+1(0, ) (6.2)u(, 0) = u0 in Rn+1. (6.3)An existence and uniqueness theory for (6.2), (6.3) can be carried out withinthe framework of viscosity solutions.Denition 3. A function u C0(Rn+1[0, )) is called a viscosity sub-solution of (6.2) provided that for each C(Rn+2), if u has a localmaximum at (x0, t0) Rn+1(0, ), thentn+1

i,j=1_ij xixj[[2_xixj 0 at (x0, t0), if (x0, t0) ,= 0,(6.4)tn+1

i,j=1(ij pipj)xixj 0 at (x0, t0) for some [p[ 1,if (x0, t0) = 0.38 K. Deckelnick, G. Dziuk and C. M. ElliottFigure 6.2. Evolution of a lemniscate under level setmean curvature ow: the zero level.A viscosity supersolution is dened analogously: maximum is replaced byminimum and by . A viscosity solution of (6.2) is a function u C0(Rn+1[0, )) that is both a subsolution and a supersolution.We shall assume that the initial function u0 is smooth and satisesu0(x) = 1 for [x[ S (6.5)for some S > 0. The following existence and uniqueness theorem is a specialcase of results proved independently by Evans and Spruck (1991) and Chen,Giga and Goto (1991).Theorem 6.1. Assume u0 : Rn+1 R satises (6.5). Then there existsa unique viscosity solution of (6.2), (6.3), such thatu(x, t) = 1 for [x[ +t Rfor some R > 0 depending only on S.The level set approach can now be described as follows: given a compacthypersurface 0, choose a continuous function u0 : Rn+1 R such that0 = x Rn+1[ u0(x) = 0. If u : Rn+1 [0, ) R is the uniqueviscosity solution of (6.2), (6.3), we then call(t) = x Rn+1[ u(x, t) = 0, t 0Computation of geometric PDEs and mean curvature flow 39Figure 6.3. Evolution of the oriented distance functionof a lemniscate: level lines.a generalized solution of the mean curvature ow problem. We remark thatEvans and Spruck (1991) and Chen, Giga and Goto (1991) also establishedthat the sets (t) = x Rn+1[ u(x, t) = 0, t > 0 are independent ofthe particular choice of u0 which has 0 as its zero level set, so that thegeneralized evolution ((t))t0 is well dened for a given 0. As (t) existsfor all times, it provides a notion of solution beyond singularities in the ow.For this reason, the level set approach has also become very important inthe numerical approximation of mean curvature ow and related problems.Note however that it is possible that the set (t) may develop an interior fort > 0, even if 0 had none, a phenomenon which is referred to as fattening.The level set solution has been investigated further in several papers: inparticular we mention Evans and Spruck (1992a, 1992b, 1995) and Soner(1993).6.2. RegularizationEvans and Spruck (1991) proved that the (smooth) solutions u

ofu

t n+1

i,j=1_ij u

xiu

xj

2+[u

[2_u

xixj = 0 in Rn+1(0, ), (6.6)u

(, 0) = u0 in Rn+1(6.7)40 K. Deckelnick, G. Dziuk and C. M. ElliottFigure 6.4. Evolution of the oriented distancefunction of a lemniscate: graph.converge locally uniformly as 0 to the unique viscosity solution of (6.2),(6.3). For numerical purposes it is important to know the asymptotic errorbetween the viscosity solution and the solution of the regularized problemquantitatively as 0. In Deckelnick (2000) there is a proof of the followingtheorem together with several a priori estimates and their dependence onthe regularization parameter .Theorem 6.2. For every (0, 12), 0 < T < there is a constantC = C(u0, T, ) such thatsup0tT|u u

|L(Rn+1) Cfor all > 0.If one wants to calculate approximations to the viscosity solution u of(6.2), (6.3) then, according to Theorem 6.2, it is sucient to solve theregularized problem (6.6), (6.7), which we have to study for computationalpurposes, on a bounded domain. For simplicity we choose = BS(0) withS > R = R(S), where R is the radius from Theorem 6.1, and considerutn+1

i,j=1_ij uxiuxj

2+[u

[2_uxixj = 0 in (0, ), (6.8)u

= 1 on (0, ), (6.9)u

(, 0) = u0 in . (6.10)Computation of geometric PDEs and mean curvature flow 41An application of the parabolic comparison theorem yields the followingcorollary of Theorem 6.2.Corollary 6.3. For every (0, 12), 0 < T < there is a constantC = C(u0, T, ) such that|u u

|L((0,T)) C. (6.11)We are now in position to look at the regularized level set mean curvatureow problem as a problem for graphs. If we scaleU = u

(6.12)then U is a solution of the mean curvature ow problem for graphs (see(5.2)), that is,Ut_1 +[U[2 U_1 +[U[2= 0 in (0, T). (6.13)This is a theoretical observation and implies that we can apply techniquesdeveloped for the mean curvature ow of graphs to the mean curvature owof level sets. But for computations we shall not use (6.13) but the unscaledversion for u

.6.3. The approximation of viscosity solutionsNumerical schemes based on the level set approach were rst introduced inOsher and Sethian (1988); see also Sethian (1990). Chen, Giga, Hitaka andHonma (1994) proposed a nite dierence scheme for which they proved sta-bility with respect to the L-norm. Walkington (1996) used a nite elementapproach on the dual mesh to construct a discretization that is stable bothwith respect to L and to W1,1. Evans (1993) analysed a scheme basedon the solution of the usual heat equation, continually re-initialized aftershort time-steps, and which was proposed in Merriman, Bence and Osher(1994). Crandall and Lions (1996) constructed a nite dierence schemethat is both monotone and consistent, and obtained the rst convergenceresult for an approximation of (6.2), (6.3). An error analysis for this schemecan be found in Deckelnick (2000).Here we want to consider a dierent nite element scheme which exploitsthe above-described formal similarity to the graph case. This will also allowus to carry out some basic numerical analysis. In the following we use theabbreviations

(v) = (v, )Q

(v) , Q

(v) = _

2+[v[2, V

(v) = vtQ

(v).Our results for the mean curvature ow of a graph can directly be trans-formed into a convergence result for the regularized level set problem.42 K. Deckelnick, G. Dziuk and C. M. ElliottTheorem 6.4. Let u

be the solution of (6.8), (6.10) and let uh be thesolution of the semidiscrete problem uh(, t) Xh with uh(, t) 1 Xh0,uh(, 0) = uh0 = Ihu0 and_huhthQ

(uh) +_huh hQ

(uh) = 0 (6.14)for all t (0, T) and all discrete test functions h Xh0. Then_ T0_h(V

(u

) V

(uh))2Q

(uh) c

h2,sup(0,T)_h[

(u

)

(uh)[2Q

(uh) c

h2.We omit the proof as it is based on the scaling argument (6.12). Un-fortunately, the constants c

contain a term that depends exponentially on1

, which is due to an application of Gronwalls lemma. Numerical tests,however, suggest that the resulting bound overestimates the error.In two space dimensions we can prove that the computed solutions uhconverge in L to the viscosity solution. The proof is contained in Deckel-nick and Dziuk (2001).Theorem 6.5. Let u be the viscosity solution of (6.2), (6.3) and let uhbe the solution of the problem (6.14) with R2as in Corollary 6.3. Thenthere exists a function h = h() 0 as 0 such thatlim0|u uh()|L((0,T)) = 0.Finally, the fully discrete numerical scheme for (regularized) isotropicmean curvature ow of level sets is now a straightforward adaption of Al-gorithm 3.Algorithm 4. (Mean curvature ow of level sets) Let u0h = Ihu0.For m = 0, . . . , M 1, compute um+1h Xh such that um+1h 1 Xh0 and,for every h Xh0,1_hum+1h hQ

(umh) +_hum+1h hQ

(umh) = 1_humhhQ

(umh). (6.15)For this scheme we have the following convergence result.Theorem 6.6. Let u

be the solution of (6.8)(6.10) and let umh, (m =1, . . . , M) be the solution from Algorithm 4. Then there exists a 0 > 0Computation of geometric PDEs and mean curvature flow 43such that, for all 0 < 0,M1

m=0_h(V

(um

) Vmh)2Q

(umh) c

(2+h2), (6.16)supm=0,...,M_h[

(um

)

(umh)[2Q

(umh) c

(2+h2), (6.17)with M = [T ]. Here Vmh = (um+1h umh)/( Q

(umh)) is the regularizeddiscrete normal velocity.This result implies the convergence of the fully discrete regularized solu-tion to the viscosity solution.Theorem 6.7. Let u be the viscosity solution from Theorem 6.1 and let be the domain from Corollary 6.3 in R2. Let uh denote the time-interpolated solution of the fully discrete scheme (6.15). Then there existfunctions h = h() 0 and = () 0 as 0 such thatlim0|u uh()()|L(h(0,T)) = 0.7. Phase eld approach to mean curvature ow7.1. IntroductionThe phase eld approach to interface evolution is based on physical modelsfor problems involving phase transitions. In this section is a boundeddomain in Rn+1and (t) is a hypersurface moving through . In the case oftwo phases one has the notion of an order parameter or phase eld function : (0, T) R which indicates the phase of a material by associatingwith the phases the minima of a C2double well bulk energy function W() :R R. For simplicity we suppose that W(r) = W(r) and the minima ofW() are at 1. The canonical example isW(r) = 14(r21)2. (7.1)Consider the gradient energy functionalE() =__

2[[2+ W()

_dx, (7.2)where is a small parameter. Steepest descent or gradient ow for thisfunctional leads to the parabolic AllenCahn equation (Allen and Cahn1979)t + 1

W

() = 0 in (0, T) (7.3)44 K. Deckelnick, G. Dziuk and C. M. Elliottwith Neumann boundary conditions. In order to understand the behaviourof this evolution equation for an initial function 0 : R, observe thatthe ow of the ordinary dierential equation t = W

()

2 drives positivevalues of 0 to 1 and negative values to 1. On the other hand the Laplacianterm in the equation (7.3) has a smoothing eect which will diuse largegradients of the solution. Thus, on the basis of these heuristics, after ashort time the solution of (7.3) will develop a structure consisting of bulkregions in which is smooth and takes the values 1, and separating theseregions there will be interfacial transition layers across which changesrapidly from one bulk value to the other. These transition layers are due tothe interaction between the regularizing eect of the gradient energy termand the ow associated with the bi-stable potential term W

. It turns outthat the motion of these interfacial transition layers approximates meancurvature ow.We can argue informally to support this in the following way. Let, fort (0, T), (t) be a smoothly evolving closed and compact hypersurfacesatisfying V = H. Suppose that (t) is the boundary of an open set(t) and denote by d(, t) the signed distance function to (t). Weconsider the semilinear parabolic operatorP(v) = vtv + 1

W

(v).A calculation yields for v(x, t) = _d(x,t)

_, where : R R, thatP(v) = (dtd)

_d

_ 1

_

_d

_W

__d

___.Hence it is natural to dene = (z) to be the unique solution of

(z) +W

((z)) = 0, z R, (7.4)(z) 1, z , (0) = 0,

(z) > 0. (7.5)If W is given by (7.1) we have that (z) = tanh( z2) and thereforeP(v) = (dtd)

_d

_.Recalling (2.12) and (2.20) we obtain dt d = V H = 0 on (t), sothat the smoothness of d implies[dtd[ C[d[in a neighbourhood U of 0 0 such that dist((t), ) for t [0, T];(iii) [dt d[ D0[d[ for [d[ , t [0, T], where d(, t) is the signeddistance function to (t);(iv) V = H on (t) for t [0, T].Let be suciently small such that 12 (1 + 2e2D0T)1and let = be the unique solution of (7.12) with g = 0 and initial data 0 = _d(,0)_.Then, for all t [0, T],d(x, t) 12(1 + 2e2D0t) (x, t) = 1,d(x, t) 12(1 + 2e2D0t) (x, t) = 1.Proof. See Chen and Elliott (1994).Computation of geometric PDEs and mean curvature flow 47A consequence of this theorem is that the diuse interfacial region_(x, t) : [(x, t)[ < 1_is sharply dened with nite width bounded by c(t) and that both thezero level set of (, t) and (t) are in a narrow strip of width c(t). Herec(t) = 12(1 + e2D0t); but in practice it is observed that this is pessimisticand the growth of the interface width is not usually an issue. A morerened analysis by Nochetto, Paolini and Verdi (1994) revealed in the caseof a smooth evolution of the forced mean curvature ow that the Hausdordistance between the zero level set of and the interface of the ow (7.8)is of order O(2). Furthermore, there is convergence to the unique viscositysolution of the level set formulation (Nochetto and Verdi 1996a).7.3. Discretization of the AllenCahn equationWe use the same notation for the discrete spaces as in Section 5.2. We willidentify any function Xh with the vector jNj=1 of its nodal values,so that = Nj=1 jj. By (, ) we denote the L2() inner product.For computational convenience we use a discrete inner product (, )h onC0() dened by(, )h =_Ih() dx for all , C0(), (7.13)where Ih is the usual Lagrange interpolation operator for Xh. Furthermore,let = T/M > 0 be the uniform time-step and tm = m. For any mMm=0,we set m= 1(m+1 m). The fully discrete approximation usingexplicit ( = 0) and implicit ( = 1) time-stepping reads as follows.Algorithm 5. (AllenCahn equation) Let 0= Ih0. For m = 0, . . . ,M 1, nd m+1 Xh, 1 m M 1, such that, for all Xh,(m, )h + (m+, ) 12(W

(m+), )h = cW

(g, )h. (7.14)For initial data we choose the nite element interpolant of the transitionlayer prole0= Ih_d0(x)

_,where d0 is the signed distance function to the initial interface.The explicit scheme requires the usual time-step constraint for parabolicequations, Ch2, (7.15)where the constant C depends on the mesh and the L norm of the initialdata through the magnitude of [W

[. On the other hand the implicit scheme48 K. Deckelnick, G. Dziuk and C. M. Elliottrequires a time-step constraint in order for the nonlinear equations deningm+1to have a unique solution. This constraint is 2, (7.16)where is the minimum value of W

. See Elliott and Stuart (1993) andChen, Elliott, Gardiner and Zhao (1998).The analysis of convergence to mean curvature ow requires considera-tion of the three approximation parameters , h, tending to zero. Standarda priori nite element error analysis for xed would lead to, for the dier-ence between the nite element solution and the solution of the AllenCahnequation, optimal order error bounds in terms of the mesh sizes , h butwith constants depending on the Gronwall-induced factor exp(T

2). Fengand Prohl (2003) have improved the nite element error analysis of theAllenCahn equation using the special structure of the solution. Indeed,they exploit spectral estimates of Chen (1994) which lead to error boundswhose constants show just polynomial growth in 1

. They specically con-sider the implicit scheme without numerical integration. As a consequencethey derive an error bound of order 2between the zero level set of thesolution of the AllenCahn equation and the limiting surface.7.4. Discretization of the double obstacle phase eld modelWe use the nite element setting of Section 7.3. Let /h= Xh : [[ 1. The double obstacle version of Algorithm 5 is as follows.Algorithm 6. (Double obstacle phase eld) Let 0= Ih0. For m =0, . . . , M 1, nd m+1 /hsuch that, for all /h,(m, m+1)h + (m+, m+1) (7.17) 12(m++cWgm+, m+1)h 0.For initial data we choose the nite element interpolant of the transitionlayer prole. The explicit scheme is a discrete obstacle variational inequalityassociated with the mass matrix. Without mass lumping the solution of thisnonlinear algebraic problem would require quadratic programming or linearcomplementarity methods. However, with the mass lumping quadraturerule the explicit scheme is as simple as the explicit scheme for a semilinearparabolic equation. It can be simply written asm+1/2=__1 +

2_I A_m+cW

gm, (7.18)m+1= Tm+1/2. (7.19)Here A = M1K, where M and K are dened byMij = (i, j)h, Kij = (i, j),Computation of geometric PDEs and mean curvature flow 49for 1 i, j N. Furthermore, T : RN RNis the component-wiseprojection onto [1, 1]Ndened by(TV )j = max(1, min(1, Vj)).On the other hand, in linear algebraic form the implicit scheme leads to thediscrete variational inequality: nd m+1 RNsuch that [j[ 1 and__1

2_I+A_m+1

_m+1__m+cW

gm+1_

_m+1_ (7.20)for all RNwith [j[ 1. Because A is symmetric this is equivalent tominimizing a quadratic function subject to bound constraints and can easilybe solved by projected SOR (Elliott and Ockendon 1982). Such a systemcan also be solved by nonlinear multigrid (Kornhuber and Krause 2003).As for the continuous parabolic variational inequality, a discrete compar-ison principle holds for these schemes if the triangulation is acute. Thisprovides the basis for a convergence analysis (Nochetto and Verdi 1996b,1997). For the implicit scheme without numerical integration an O() errorbound for the interface is obtained when = O(h2) = O(4). For the expli-cit scheme without numerical integration in the potential term an O(2) isproved for = O(h2) = O(5).7.5. ImplementationOne expects there to be a relationship between and h in order that the dis-crete phase eld model can approximate the sharp interface motion. Sincethe convergence analysis in the continuous case relies heavily on under-standing the prole of the phase eld function across the transition layer,one would expect that for any the mesh size h should be suciently smallin order to resolve the interface. Indeed the existing convergence analysis de-scribed above indicates that h should tend to zero faster than . In practicethis implies that across the discrete interfacial layer in the normal directionthere should be a sucient number of elements.In the case of the double obstacle potential, at the mth time-step, thenite elements may be divided into three sets:h(m) = j = 1 for each element vertex,h+(m) = j = 1 for each element vertex,1h(m) = Th (h(m) h+(m)).Clearly the approximation to the interface is the zero level set of mwhichlies inside the discrete interfacial region 1h(m). We view 1h(m) as a sharpdiuse interface, as opposed to the interfacial region associated with thesmooth double well, which is not sharply dened. The computationalwork in evolving the interface is then associated with the small number of50 K. Deckelnick, G. Dziuk and C. M. Elliott1 0.5 0 0.5 110.500.510.62 0.67 0.720.0500.05Figure 7.1. Meshes.elements in this region. As observed above, the time-step, , in the phaseeld calculations is substantially smaller than the mesh size, h. Thus, ina numerical simulation one would expect that, for nite normal velocity ofthe interface, the sharp diuse interface should only move by at most theaddition or subtraction of a single layer of elements. In the case of theexplicit scheme this can be made precise. For nodes in h+(m) (or h(m))whose nearest neighbours are also in h+(m) (or h(m)), we ndm+1/2j = 1 +

2(1 +cWgm(aj))which, provided [gm(aj)[ 1cW

, implies that m+1j = 1. It follows thatthe sharp diuse interface can not move more than one element per time-step. It also implies that it is only necessary to compute m+1on the closureof the transition layer. This can be exploited in a number of ways.The two-dimensional dynamic mesh algorithm (Nochetto, Paolini andVerdi 1996) is based on the explicit scheme and carries a mesh only inthe sharp diuse interface; it adds and removes triangles where necessary.The mask method (Elliott and Gardiner 1996) keeps an underlying xedmesh and computes in the sharp diuse interface only. It is possible to storenodal values only in this region.An amalgam of the above is an adaptive procedure which uses a nemesh within the diuse interface and a coarse mesh outside. In Figure 7.1a typical mesh is shown for a phase eld calculation of anisotropic meancurvature ow. The global mesh is shown together with a zoom. Thisapproach requires a ne mesh slightly larger than the diuse interface. Asthe interface region moves the mesh is rened and coarsened appropriately.Computation of geometric PDEs and mean curvature flow 510 0.5 100.510 0.5 100.510 0.5 100.51Figure 7.2. Topological change.0 0.5 100.510 0.5 100.51Figure 7.3. Diuse interfaces with topological change.Sharp diuse interface front trackingUsing the double obstacle phase eld method and only computing within asharp diuse interface as described above can be viewed as a front trackingmethod, which has the advantage of being able to handle topological change.In Figure 7.2 the interfaces at various times are displayed for a forced meancurvature ow starting from initial circles. Eventually the circles intersect.Meshes associated with these computations are shown in Figure 7.3.8. Anisotropic mean curvature ow8.1. The concept of anisotropyIn free boundary problems such as phase transition problems it is oftennecessary to treat interfaces which are driven by anisotropic curvature. Thisis induced by modelling an anisotropic surface energy, which generalizes areain the isotropic case to weighted area in the anisotropic case. Anisotropic52 K. Deckelnick, G. Dziuk and C. M. Elliottsurface energy has the formE() =_() dA, (8.1)where is a surface with normal and is a given anisotropy function. For(p) = [p[ this energy is the area of . For our purposes it will be necessaryto restrict the admissible anisotropies to a certain class.Denition 4. An anisotropy function : Rn+1R is called admissible if(1) C3(Rn+1 0), (p) > 0 for p Rn+1 0;(2) is positively homogeneous of degree one, i.e.,(p) = [[(p) for all ,= 0, p ,= 0; (8.2)(3) there exists 0 > 0 such thatD2(p)q q 0[q[2for all p, q Rn+1, [p[ = 1, p q = 0. (8.3)It is not dicult to verify that (8.2) impliesD(p) p = (p), D2(p)p q = 0, (8.4)D(p) = [[D(p), D2(p) = 1[[D2(p) (8.5)for all p Rn+1 0, q Rn+1and ,= 0. The convexity assumption (8.3)will be crucial for analysis and numerical methods.Anisotropy is normally visualized by using the Frank diagram T and theWul shape J:T = p Rn+1[ (p) 1,J = q Rn+1[ (q) 1.Figure 8.1. Frank diagram (left) and Wul shape (right)for the regularized l1-anisotropy (p) = 3j=1_2[p[2+p2j.Computation of geometric PDEs and mean curvature flow 53Here is the dual of , which is given by(q) = suppRn+1\{0}p q(p). (8.6)Let us consider some examples. Note that not all of them are admissible.The choice (p) = [p[ is called the isotropic case; in particular we havethat T = J = p Rn+1[ [p[ 1 is the closed unit ball.A typical choice for anisotropy is the discrete lr-norm for 1 r ,(p) = |p|lr =_n+1

k=1[pk[r_1r, 1 r < , (8.7)with the obvious modication for r = .For a given positive denite (n + 1) (n + 1) matrix G, the anisotropyfunction(p) = _Gp p (8.8)models an anisotropy which is dened by a (constant) Riemannian metric.In Figure 8.2 we show the Frank diagram and Wul shape for the anisotropy(p) =_(5.5 + 4.5 sign(p1))p21 +p22 +p23. (8.9)One anisotropy function often used in a physical context is(p) =_1 A_1 |p|4l4|p|4l2__|p|l2 (8.10)where A is a parameter. For A < 0.25 the Frank diagram is convex.For more information on this subject, including anisotropies that maydepend on space, see Bellettini and Paolini (1996).Figure 8.2. Frank diagram T (left) and Wul shape J(right) for the anisotropy (8.9).54 K. Deckelnick, G. Dziuk and C. M. Elliott8.2. Anisotropic distance functionLet be an admissible anisotropy function. We can associate with anonsymmetric metric on Rn+1by setting(x, y) = (x y), x, y Rn+1. (8.11)It is possible to prove that is equivalent to the standard Euclidean metric.Suppose next that Rn+1is a bounded open set with smooth boundary. Using we now dene an anisotropic signed distance function d :Rn+1R byd(x) =___infy (x, y), x Rn+1 ,0, x ,infy (x, y), x .Lemma 8.1. There exists an open neighbourhood U of such that d C2(U) and(d) = 1, (8.12)D2dD(d) = 0. (8.13)8.3. Anisotropic mean curvatureOur goal is to generalize the notion of mean curvature to the anisotropicsetting. Suppose that is an admissible anisotropy function and that Rn+1is an oriented hypersurface with normal . We dene the CahnHomann vector on by(x) = D((x)), x , (8.14)and the anisotropic mean curvature byH(x) = (x), x . (8.15)Note that H = H in the isotropic case (p) = [p[. The following lemmashows that H is a natural generalization of mean curvature as the rstvariation of the area functional with respect to normal variations.Lemma 8.2. Suppose that is compact. For C0 (U) (U a neigh-bourhood of ) dene F

(x) = x + (x)(x), x U as well as

= F

().Then,ddE(

)|=0 =_HdA.Proof. Let d(, ) : Rn+1 R denote the signed distance function to

.Consider g : U (0, 0) R, dened byg(x, ) = (

(x)) = (d(x, )),Computation of geometric PDEs and mean curvature flow 55where acts on the x variables only. Now (2.23), (2.20) and (2.6) implyddE(

)|=0 = dd_

g(, ) dA|=0=_g(, 0) dA_g(, 0)d(, 0)H dA_g(, 0)d(, 0) dA.It is not dicult to see that d(, 0) = (x), x , which also implies thatg(, 0) = D() d(, 0) = D() d(, 0) = .Here we have used the denition of and the fact that d(, 0) = 0 on. Thus,ddE(

)|=0 = _ dA+_()H dA+_g(, 0)dA=_ dA+_g(, 0)dA,where the last identity follows from (2.16). Finally, observing that g(, 0) =pi()dxixj(, 0)dxj(, 0) = 0, and recalling the denition of H, the claimfollows.Let us next calculate H for various descriptions of .Level set representation. Suppose that is given as in (2.1) and orientedby = u|u|. Since pi is homogeneous of degree 0, we have (see also (2.2))H = =n+1

i=1Di_pi_ u[u[__ =n+1

i=1Di_pi(u)_=n+1

i,j=1pipj(u)uxixj n+1

i,k,l=1pipl(u)uxlxkuxk[u[uxi[u[.Recalling (8.4) we therefore deduceH =n+1

i,j=1pipj(u)uxixj. (8.16)Graph representation. If is locally given as the graph of the functionx

v(x

), x

= (x1, . . . , xn) with normal = (x v,1)1+|x v|2, formula (8.16)applied to u(x

, xn+1) = v(x

) xn+1 givesH =n

i,j=1pipj(x v, 1)vxixj. (8.17)56 K. Deckelnick, G. Dziuk and C. M. ElliottLet us next derive an analogue of (2.16) with H replaced by H. Observingthat Dkl = Dlk and recalling that Dkxl = klkl, we obtainHl = Dk_pk()_l = Dk_pk()l_pk()Dkl= Dk_pk()l_pk()Dlk= Dk_pk()l_Dl_()_= Dk_pk()l_Dk_()(klkl)_()lDkk= Dk_pk()l_Dk_()Dkxl_()Hl,where summation over k is from 1 to n + 1. For a smooth test function = (1, . . . , n+1) we multiply the above relation by l, sum over l andintegrate over . Using (2.16) we infer_H = n+1

k,l=1_pk()lDkl +n+1

k,l=1_pk()lHkl+n+1

k,l=1_()DkxlDkln+1

l=1_()Hlland (8.4) yields_H = n+1

k,l=1_pk()lDkl +n+1

k,l=1_()DkxlDkl. (8.18)This relation will be at the heart of the numerical methods in the parametriccase. For additional information on the subject of weighted mean curvatureincluding the crystalline case, see Taylor (1992).8.4. Motion by anisotropic mean curvature with mobilityHaving introduced the notion of anisotropic mean curvature we can nowformulate the following generalization of (3.1):()V = H +g on (t). (8.19)Here, : Sn R is a given positive and smooth function of degree zero.In applications where (t) models a sharp phase-interface, the coecient measures the drag opposing interfacial motion and the function 1 is calledmobility. The function g represents the energy dierence in the bulk phases.A detailed derivation of (8.19) from the force balances and the second law ofthermodynamics can be found in Angenent and Gurtin (1989) and Gurtin(1993). Taylor, Cahn and Handwerker (1992) give an overview of variousmathematical approaches to (8.19).Computation of geometric PDEs and mean curvature flow 57In what follows we shall consider the simpler problem()V = H on (t), (8.20)even though all our techniques can be generalized to (8.19). It can be shown(see Bellettini and Paolini (1996)) that for the choice () = 1() there is anexplicit solution of (8.20) consisting of shrinking boundaries of Wul shapes;the sets(t) = p Rn+1[ (p) = _r(0)22ntsatisfy 1()V = H and are therefore a generalization of the shrinkingcircles from the isotropic case. We also have the following analogue ofLemma 3.1.Lemma 8.3. Let (t) be a family of evolving hypersurfaces satisfying(8.20) on (t), and assume that each (t) is compact. Then_(t)()V2dA+ ddt_(t)() = 0.Proof. In the same way as in the proof of Lemma 8.2, we deriveddt_(t)() =_(t)HV,and the claim follows from the evolution law (8.20).8.5. Anisotropic curve shortening owLet us consider a family (t) of closed curves in R2which move according to(8.20). As in Section 4.1 we describe the evolution by means of a mappingX: R[0, T) R2which satises X(, t) = X( +2, t) for t [0, T), R. The curves (t) = X(, t) will move by (8.20) provided that()Xt = H. (8.21)Using the notation (a1, a2) = (a2, a1) we may write = , where = X|X| is the unit tangent to the curve (t). Equation (8.21) amounts to asystem of partial dierential equations for the vector function X. In order towrite down this system, let H1per(I; R2), I = [0, 2], be a test function,which we can think of as being dened on (t) via (X(, t)) = (). Itfollows from (8.18) that_(t)H = 2

k,l=1_(t)pk()lDkl +2

k,l=1_(t)()DkxlDkl= 2

k,l=1_(t)_pk()l()kl_Dkl,58 K. Deckelnick, G. Dziuk and C. M. Elliottsince Dkxl = kl kl. Using l = l,|X| and recalling that (p) =D(p) p, we obtain after some calculations2

k,l=1_pk()l()kl_Dkl = D() [X[.In conclusion we have_(t)H dA =_ 20D(X ) d,so that we obtain the following weak form of (8.21):_ 20_X[X[_Xt[X[ d+_ 20D(X ) d = 0 for all H1per(I; R2).(8.22)We shall base our numerical scheme on this formulation. The classical formof (8.22) is_X[X[_Xt + 1[X[_D_X__ = 0 in I (0, T). (8.23)For the convenience of the reader we explicitly write down the two equationsof this system:_X[X[_X1t[X[ p2p2(X2, X1)X1 +p2p1(X2, X1)X2 = 0,_X[X[_X2t[X[ p1p1(X2, X1)X2 +p1p2(X2, X1)X1 = 0.It is easy to see that this system can be written as_X[X[_Xta_X[X[_ 1[X[_ X[X[_ = 0,wherea(p) = pp(p) p p, p R2 0.Note that (8.3) implies a(p) 0 > 0 for all p R2, [p[ = 1. Analyticalresults for this problem which generalize the theory for the isotropic case(a = 1) have been obtained by Gage (1993). We shall continue to usethe form (8.22) because this equation only contains rst derivatives of theanisotropy function . Recall the denition of Sh from Section 4.1. Adiscrete solution of (8.22) will be a function Xh : [0, T] Sh, such thatXh(, 0) = Xh0 = IhX0 =N

j=1X0(j)j,Computation of geometric PDEs and mean curvature flow 59and for all discrete test functions h Sh_ 20_ Xh[Xh[_Xht h[Xh[ d +_ 20D(Xh) h d = 0. (8.24)In the same way as in the isotropic case we can writeXh(, t) =N

j=1Xj(t)j()with Xj(t) R2, and nd that the discrete weak equation (8.24) is equival-ent to the following system of 2N ordinary dierential equations:16jqj Xj1 + 13(jqj +j+1qj+1) Xj + 16j+1qj+1 Xj+1+D(Xj+1Xj )D(Xj Xj1) = 0,for j = 1, . . . , N, where X0(t) = XN(t), XN+1 = X1(t), andqj = [Xj Xj1[, j = _Xj Xj1qj_.Furthermore, the initial values are given byXj(0) = X0(j), j = 1, . . . , N.We again use mass lumping, which is equivalent to a quadrature formula.Thus we replace this system by the lumped scheme12(jqj +j+1qj+1) Xj +D(Xj+1Xj )D(Xj Xj1) = 0 (8.25)together with the initial conditions Xj(0) = X0(j) for j = 1, . . . , N. Weare now ready to say what we mean by a discrete solution of anisotropiccurve shortening ow. The above system is equivalent to the one we use inthe following denition of discrete anisotropic curve shortening ow.Denition 5. A solution of the discrete anisotropic curve shortening owfor the initial curve h0 = Xh0([0, 2]) is a polygon h(t) = Xh([0, 2], t),which is parametrized by a piecewise linear mapping Xh(, t) Sh, t [0, T],such that Xh(, 0) = Xh0 and for all h Sh_ 20_ Xh[Xh[_Xht h[Xh[ d +_ 20D(Xh) h d+ 16h2_ 20_ Xh[Xh[_Xht h[Xh[ d = 0. (8.26)60 K. Deckelnick, G. Dziuk and C. M. ElliottFigure 8.3. Anisotropic curve shortening ow with asixfold anisotropy function applied to a circle (left) andto a square (right).Here h is the constant grid size of the uniform grid in [0, 2]. The lastterm of (8.26) is introduced by mass lumping. One could also dene thediscrete curve shortening ow without this quantity, but then the geometricproperty of length shortening would not be true for the discrete problem.Dziuk (1999b) proved the following convergence result for = 1. Itis easily extended to the case of general . We formulate the result forthe geometric quantities normal, length and normal velocity. The errorestimates in standard norms then follow easily.Theorem 8.4. Suppose that : S2 R is a smooth positive function.Let X be a solution of the anisotropic curve shortening ow (8.23) on theinterval [0, T] with X(, 0) = X0, min[0,2][0,T][X[ c0 > 0 and Xt L2((0, T), H2(0, 2)). Then there is an h0 > 0 such that, for all 0 < h h0,there exists a unique solution Xh of the discrete anisotropic curve shorteningow (8.26) on [0, T] with Xh(, 0) = Xh0 = IhX0, and th