Universität Regensburg Mathematik

Universität Regensburg Mathematik Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature John W. Barrett, Harald Garcke and Robert Nürnberg Preprint Nr. 01/2016

Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature John W. Barrett Harald Garcke Robert Nürnberg Abstract We study numerical approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in the principal curvatures of a two-dimensional surface. Beside the well-known Willmore and Helfrich flows we will also consider flows involving the Gaussian curvature of the surface. Boundary conditions for these flows are highly nonlinear, and we use a variational approach to derive weak formulations, which naturally can be discretized with the help of a mixed finite element method. Our approach uses a parametric finite element method, whichcanbeshowntoleadtogoodmeshproperties. Weprovestabilityestimatesfor a semidiscrete(discrete in space, continuous in time) version of the method and show existence and uniqueness results in the fully discrete case. Finally, several numerical results are presented involving convergence tests as well as the first computations with Gaussian curvature and/or free or semi-free boundary conditions. Key words. Willmore flow, parametric finite elements, tangential movement, spontaneous curvature, clamped boundary conditions, Navier boundary conditions, Gaussian curvature energy, line energy. AMS subject classifications. 65M60, 65M12, 35K55, 53C44 1 Introduction Energies involving the principal curvatures of a two-dimensional surface in the three dimensional Euclidean space play an important role in geometry, physics, biology and imaging. The Willmore energy given as the integrated square of the mean curvatures is of great interest in geometry, cf. Willmore (1993). However, also more general functionals involving the principal curvatures appear in the theory of elastic plates and shells, and go back to work of Poisson (1812), Germain (1821) and Kirchhoff (1850). In the theory of biological membranes the work of Helfrich (1973) used generalized curvature functionals, which lead to a huge interest for curvature functionals in the field of biophysics. Boundary Department of Mathematics, Imperial College London, London, SW7 2AZ, UK Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany 1

value problems involving curvature functionals also play an important role in imaging, for example in problems involving surface restoration and image inpainting, cf. Clarenz et al. (2004); Bobenko and Schröder (2005). Analytical and numerical work on static and evolutionary questions in the context of curvature functionals so far have been mainly focused on the case of closed surfaces, and we refer to Simon (1993); Kuwert and Schätzle (2001); Rivière (2008); Marques and Neves (2014) for analytical results, and to Mayer and Simonett (2002); Rusu (2006); Dziuk (2008); Barrett et al. (2008) for numerical results. Much less is known for boundary value problems involving functionals that include curvature quantities. Analytical results often need small data assumptions, use symmetries or consider the graph case. We refer to Nitsche (1993); Bergner et al. (2009); Dall Acqua et al. (2008); Deckelnick and Grunau (2009); Schätzle (2010); Deckelnick et al. (2015) for the static case and to Abels et al. (2016) for an evolution problem for the Willmore energy with boundary conditions. Numerical approaches to problems involving the Willmore energy and boundary conditions are discussed in Peres Hari et al. (2001); Clarenz et al. (2004); Bobenko and Schröder (2005); Deckelnick et al. (2015). In this context we refer to Wang and Du (2008), see also Du (2011), who used a phase field approach to study open membranes numerically. Experimental observations for open membranes are reported in Saitoh et al. (1998). Capovilla and Guven (2004); Tu and Ou-Yang (2003) and Biria et al. (2013) used variational calculus to derive equilibrium equations for a bilayer membrane having an edge, and also provided physical interpretations of the equations obtained. To the knowledge of the authors, no results are available in the literature so far for numerical approaches of evolution problems that involve also the Gaussian curvature and/or free or semi-free boundary conditions. It is the goal of this paper to derive and analyze a finite element approximation of L 2 gradient flows for curvature functionals of Willmore and Helfrich type that allow also for Gaussian curvature and (semi-)free boundary conditions. We are interested in discretizations which allow to treat the highly nonlinear boundary conditions in a variational way, which will then make it possible to derive stability estimates. In order to do so, it is necessary to generalize work of Dziuk (2008) and Barrett et al. (2016) on computational Willmore flow for closed surfaces to the case of open surfaces. Due to the highly nonlinear boundary conditions, this is a nontrivial task. In order to formulate the governing problems in more detail, we parameterize the surfaces over a fixed oriented, compact, smooth reference manifoldυ R 3 with boundary Υ. We now consider a hypersurface Γ with boundary Γ parameterized by x : Υ R 3 with normal ν given by the orientation. Denoting by s = ( s1, s2, s3 ) the surface gradient on Γ we define s χ = ( sj χ i ) 3 i,j=1. We then define the second fundamental tensor for Γ as s ν, and we recall that s ν( z), for any z Γ, is a symmetric linear map that has a zero eigenvalue with eigenvector ν. The remaining two eigenvalues, κ 1,κ 2, are the principal curvatures of Γ at z; see e.g. (Deckelnick et al., 2005, p. 152). Hence s ν( z) induces a linear map S : T z Γ T z Γ on the tangent space T z Γ for any z Γ. The map S is called the 2

Weingarten map or shape operator. The mean curvature κ and the Gaussian curvature K can now be stated as κ = trs = κ 1 +κ 2 and K = det(s) = κ 1 κ 2, (1.1) where we note that unit spheres with outer unit normal have mean curvature κ = 2. It then follows that s ν 2 = κ 2 1 +κ 2 2 = κ 2 2K. The mean curvature vector is given as where s = s. s is the Laplace Beltrami operator on Γ. The Willmore energy is now given as E 0 (Γ) := 1 κ 2 dh 2 = 1 2 2 s id = κ ν =: κ on Γ, (1.2) Γ Γ κ 2 dh 2, (1.3) see e.g. Willmore (1993) for details. Here and throughout H d, d = 1,2, denotes the d- dimensional Hausdorff measure. Realistic models for biological cell membranes lead to energies more general than (1.3). In the original derivation of Helfrich (1973) a possible asymmetry in the membrane, originating e.g. from a different chemical environment, was taken into account. This lead Helfrich to the energy E κ (Γ) = 1 (κ κ) 2 dh 2 = 1 κ κ ν 2 dh 2, (1.4) 2 2 Γ where κ R is the given so-called spontaneous curvature. Similarly to Barrett et al. (2016), we will also consider the energy Γ E κ,β (Γ) := E κ (Γ)+ β 2 (M(Γ) M 0) 2 (1.5a) with M(Γ) = Γ κ dh 2 = Γ κ. ν dh 2 (1.5b) and given constants β R 0, M 0 R. Models employing the energy (1.5a) are often called area-difference elasticity (ADE) models, see Seifert (1997). We note that for present models, choosing β > 0 does not have a physically meaningful interpretation for surfaces with boundary. For open surfaces also contributions taking Gaussian curvature and line energy into account are relevant. We hence consider E(Γ) := E κ,β (Γ)+α G K dh 2 +γh 1 ( Γ), (1.6) for given α G R and γ R 0. Similarly to (1.2), fundamental to many approaches, which numerically approximate evolving curves in a parametric way, is the identity Γ id ss = κ Γ on Γ, (1.7) 3

where κ Γ is the curvature vector on Γ. Here we choose the arclength s of the curve Γ such that ( id s, µ, ν), where µ = ν id s on Γ (1.8) denotes the conormal to Γ on Γ, form a positively oriented orthonormal basis of R 3. Note that µ is a vector that is perpendicular to the unit tangent id s on Γ and lies in the tangent space of Γ. Now (1.7) can be rewritten as id ss = κ Γ = κ µ µ+κ ν ν on Γ, (1.9) where κ µ is the geodesic curvature and κ ν is the normal curvature. It then follows from the Gauß Bonnet theorem, K dh 2 = 2πm(Γ)+ κ µ dh 1, (1.10) Γ where m(γ) Z denotes the Euler characteristic of Γ, that the energy (1.6), is equivalent to [ ] E(Γ) := E κ,β (Γ)+α G κ Γ. µ dh 1 +2πm(Γ) +γh 1 ( Γ). (1.11) Γ It turns out that the first variation of the energy is given by, compare (A.49) in the appendix, Nitsche (1993) and Barrett et al. (2016), s κ ( 1 2 (κ κ)2 +β(m(γ) M 0 )κ)κ +(κ κ +β(m(γ) M 0 )) s ν 2, and the gradient flow dynamics hence moves a point on the surface Γ with a normal velocity which is the negative of the above expression. The gradient flow is hence given as a family () t [0,T] of evolving surfaces with boundary that are parameterized by x(,t) : Υ R 3 for which V = s κ +( 1 2 (κ κ)2 +βaκ)κ (κ κ +βa) s ν 2 (1.12) Γ holds, where Here A = M() M 0. (1.13) V( z,t) := x t ( q,t) z = x( q,t) (1.14) defines the velocity of, and V := V. ν is the normal velocity of the evolving hypersurface. The flow (1.12) is of fourth order (taking into account that κ involves two derivatives of the parameterization). In this paper, we consider four different types of boundary conditions on. The boundary can either move freely, or move along the boundary of a fixed domain Ω, or it will be fixed, = Γ(0). For the latter case two types of boundary conditions arise: clamped and Navier. As noted earlier, the flow (1.12) is a highly nonlinear fourth order parabolic partial differential equation for the parameterization x. Hence, if the boundary of is fixed, two boundary conditions are needed in order to yield a wellposed problem. If the boundary can move, however, then an additional boundary 4

condition is needed to close the system. Similarly to Barrett et al. (2012, Remark 2.1), we may write locally near the boundary as a graph over a time-dependent domain D(t). The fact that D(t) can move shows the need for three boundary conditions to obtain a well-posed problem. In the free boundary case, the three necessary boundary conditions are given by ( s κ). µ+γκ ν α G τ s = 0 on, (1.15a) 1 2 (κ κ)2 βaκ +γκ µ α G K = 0 on, κ κ +βa+α G κ ν = 0 on, (1.15b) (1.15c) where τ denotes the torsion of the curve, see the appendix for a derivation. We note that in the case β = γ = α G = 0 the condition (1.15b) collapses to (1.15c), and so we conjecture that for this choice of parameters the evolution problem is not well posed. For the partially free case, when Ω for all t [0,T], where Ω is the boundary of a fixed open domain Ω R 3, we let Ω be given by a function F C 1 (R 3 ) such that Ω = { z R 3 : F( z) = 0} and F( z) = 1 z Ω, and we denote the normal to Ω on Ω by n Ω = F. The necessary boundary conditions are then Ω (1.16a) [( s κ). µ+γκ ν α G τ s ]( µ. n Ω ) [ 1 2 (κ κ)2 βaκ +γκ µ α G K]( ν. n Ω ) = 0 κ κ +βa+α G κ ν = 0 on, see the appendix for a derivation. Clamped boundary conditions are given by on, (1.16b) (1.16c) = Γ(0) and µ(t) = ζ(t) on Γ(0), (1.17) where ζ C([0,T],C( Γ(0),S d 1 )). Similarly, Navier boundary conditions are given by = Γ(0) and κ = κ βa α G κ ν on Γ(0), (1.18) see the appendix for a derivation. Of course, for the two fixed boundary conditions, when = Γ(0) for t 0, the line energy contributions in (1.11) play no role. Similarly, for clamped boundary conditions, (1.17), the last integral in (1.11) is fully determined by the data, and so Gaussian curvature plays no role in this case. In some cases, in particular in applications for biomembranes, cf. Tu(2013), the surface area of Γ needs to stay constant during the evolution. In this case one can consider E λ (Γ) = E(Γ)+λH 2 (Γ) (1.19) has to be considered. Here, λ is a Lagrange multiplier for the area constraint, which can be interpreted as a surface tension. In this case (1.12) is replaced by V = s κ +( 1 2 (κ κ)2 +βaκ)κ (κ κ +βa) s ν 2 +λκ, (1.20) 5

and (1.15b) is replaced by 1 2 (κ κ)2 βaκ +γκ µ α G K = λ on. (1.21) In Section 2 we will derive a weak formulation for the continuous problem. This will be the basis for the semidiscrete finite element approximation introduced in Section 3 for which we can show a stability result. In Section 4 we formulate a fully discrete finite element approximation for which we can show that a unique solution exists. After a discussion on how to solve the fully discrete linear algebra problem in Section 5, we present in Section 6 several numerical results, many of them for situations for which no computations were available beforehand. 2 Weak formulations/formal calculus of PDE constrained optimization On recalling (1.14), we define the following time derivative that follows the parameterization x(,t) of. Let t φ = φ t + V. φ φ H 1 (G T ), (2.1) where we have defined the space-time surface G T := {t}. (2.2) t [0,T] Here we stress that this definition is well-defined, even though φ t and φ do not make sense separately for a function φ H 1 (G T ). For later use we note that d dt ψ,φ = t ψ,φ +ψ, t φ + ψφ, s. V ψ,φ H 1 (G T ), (2.3) see Lemma 5.2 in Dziuk and Elliott (2013). Here, denotes the L 2 inner product on. It immediately follows from (2.3) that d dt H2 () = s. V,1 = sid, s V. (2.4) In this section we would like to derive a weak formulation for the L 2 gradient flow of E(). To this end, we need to consider variations of the energy with respect to = x(υ,t). For any given χ [H 1 ()] 3 and for any ε (0,ε 0 ) for some ε 0 R >0, let Γ ε (t) := { Ψ( z,ε) : z }, where Ψ( z,0) = z and Ψ ( z,0) = χ( z) z. ε (2.5) 6

We note that in the case of a fixed boundary, we choose variations χ [H0()] 1 3, and so Γ ε (t) = = Γ(0). The first variation of H 2 () with respect to in the direction χ [H 1 ()] 3 is given by [ ] δ δγ H2 () ( χ) = d dε H2 (Γ ε (t)) ε=0 [ 1 = lim ε 0 ε H 2 (Γ ε (t)) H 2 () ] = sid, s χ, (2.6) see e.g. the proof of Lemma 1 in Dziuk (2008). For later use we note that generalized variants of (2.6) also hold. Namely, we have that [ ] δ δγ w,1 ( χ) = d dε w ε,1 Γε(t) ε=0= w sid, s χ w L (), (2.7) where w ε L (Γ ε (t)), for any w L (), is defined by w ε ( Ψ( z,ε)) = w( z) z, (2.8) and similarly for w [L ()] 3. This definition of w ε yields that ε 0 w = 0, where 0 ε w( z) = d dε w ε( Ψ( z,ε)) ε=0 z. (2.9) Of course, (2.7) is the first variation analogue of (2.3) with w = ψφ and ε 0ψ = 0 ε φ = 0. Similarly, it holds that [ ] δ δγ w, ν ( χ) = d dε w ε, ν ε Γε(t) ε=0= ( w. ν) sid, s χ + w, ε 0 ν w [L ()] 3, (2.10) where 0 ε w = 0 and ν ε (t) denotes the unit normal on Γ ε (t). In this regard, we note the following result concerning the variation of ν, with respect to, in the direction χ [H 1 ()] 3 : 0 ε ν = [ s χ] T ν on t ν = [ s V] T ν on, (2.11) see Schmidt and Schulz (2010, Lemma 9). Finally, we note that for η [H 1 ()] 3 it holds that [ δ ] sid, s η ( χ) = d sid, s η ε ε=0 = s. η, s. χ δγ dε Γ ε(t) 3 ] + [( ν) l ( ν) m s ( η) m, s ( χ) l ( s ) m ( η) l,( s ) l ( χ) m l,m=1 = s η, s χ + s. η, s. χ ( s η) T,D( χ)( sid) T, (2.12) 7

where 0 ε η = 0, see Lemma 2 and the proof of Lemma 3 in Dziuk (2008). Here D( χ) := s χ+( s χ) T, (2.13) and we note that our notation is such that s χ = ( Γ χ) T, with Γ χ = ( si χ j ) 3 i,j=1 defined as in Dziuk (2008). It follows from (2.12) that d sid, s η = s η, sv + s. η, s. dt V ( s η) T,D( V)( sid) T η { ξ H 1 (G T ) : t ξ = 0}. (2.14) For closed surfaces, in the seminal work Dziuk (2008), the author introduced a stable semidiscrete finite element approximation of Willmore flow, which is based on the discrete analogue of the identity 1 d κ, κ 2 dt = fγ, V, where fγ, χ = s κ, s χ + s. κ, s. χ ( s κ) T,D( χ)( sid) T κ 2 sid, s χ χ [H 1 ()] 3. (2.15) + 1 2 In the recent paper Barrett et al. (2015a) the present authors were able to extend (2.15), and the corresponding semidiscrete approximation, to the case of nonzero β and κ in (1.5a). The approximation is based on a suitable weak formulation, which can be obtained by considering the first variation of (1.5a) subject to the side constraint, the weak formulation of (1.2), κ, η + sid, s η = 0 η [H 1 ()] 3. (2.16) To this end, one defines the Lagrangian L(, κ, y) = 1 2 κ κ ν 2,1 + β 2 ( ) 2 κ, ν M 0 κ, y sid, s y (2.17) with y [H 1 ()] 3 being a Lagrange multiplier for (2.16). Then, on using ideas from the formal calculus of PDE constrained optimization, see e.g. Tröltzsch (2010), one can compute the direction of steepest descent f Γ of E κ,β (), under the constraint (2.16). In particular, we formally require that [ ] δ δγ L 1 ( χ) = lim [ L(Γε L(, κ, y)] (t), κ, y) = fγ, χ, (2.18a) ε 0 ε [ ] δ δ κ L ( ξ) ] 1 = lim [ L(, κ ε 0 ε +εξ, y) L(, κ, y) = 0, (2.18b) [ ] δ δ y L ] 1 ( η) = lim [ L(, κ, y ε 0 ε +ε η) L(, κ, y) = 0. (2.18c) 8

On recalling (2.7) (2.12), this yields that 1 d E 2 dt κ,β() = fγ, V, where fγ, χ = s y, s χ + s. y, s. χ ( s y) T,D( χ)( sid) T 1 [ κ κ ν 2 2( y. κ)] 2 sid, s χ +(βa κ) κ,[ s χ] T ν βa ( κ. ν) sid, s χ = 0 χ [H 1 ()] 3, (2.19a) κ +(βa κ) ν y, ξ = 0 ξ [H 1 ()] 3, (2.19b) κ, η + sid, s η = 0 η [H 1 ()] 3, (2.19c) where A(t) = κ, ν M 0. (2.19d) Clearly, (2.19b) implies that κ+(βa κ) ν = y. In the case κ = β = 0, this collapses to y = κ, andso (2.19a)collapses to(2.15). Inthecontext ofthenumerical approximationof the L 2 gradient flow of E κ,β (), (2.19a d) gives rise to the following weak formulation, where we recall (1.14). Given Γ(0), for all t (0,T] find = x(υ,t), with V(t) [H 1 ()] 3, and y(t) [H 1 ()] 3 such that (2.19a) holds with κ = y (βa κ) ν and A(t) = κ, ν M 0, and such that V fγ, χ y, η + = 0 χ [H 1 ()] 3, (2.20a) = (βa κ) ν, η η [H 1 ()] 3. (2.20b) s id, s η Under discretization, (2.20a,b) does not have good mesh properties. That is because the discretizations will exhibit mesh movements that are almost exclusively in the normal direction, which in general leads to bad meshes. To see this, we note that (2.20a,b) is the weak formulation of V = [ s κ +( 1 2 (κ κ)2 +βaκ)κ s ν 2 (κ κ +βa) ] ν, (2.21) which agrees with Barrett et al. (2008, (1.12)). A derivation of (2.21) is given in the appendix. In order to overcome the undesirable mesh effects for a discretization of (2.20a,b), the authors in Barrett et al. (2016) replaced the side constraint (2.16) with the more general side constraint Q θ κ, η + sid, s η = 0 η [H 1 ()] 3, (2.22) where θ [0,1] is a fixed parameter, and where Q θ is defined by Q θ = θid+(1 θ) ν ν on. (2.23) On recalling(1.2), we note that on the continuous level(2.22) trivially holds independently of the choice of θ [0,1]. However, on the discrete level (2.22), for θ < 1, leads to an 9

induced tangential motion and good meshes, in general. See e.g. (3.39) in Section 3 below for more details. From now on we consider open surfaces. Then, similarly to (2.22), we consider the side constraint Q θ κ, η + sid, s η = m, η η [H 1 ()] 3, (2.24) which again holds trivially on the continuous level for m being the conormal µ. Here, denotes the L 2 inner product on. We stress that in the clamped case, = Γ(0) and m = µ = ζ in (2.24) are fixed given data, recall (1.17). For the other three types of boundary conditions, (2.24) weakly defines the conormal µ(t) to on. In the discrete setting, the discrete analogue of (2.24) will weakly define a discrete conormal m h (t), which will ingeneral bedifferent fromthetrueconormal µ h (t) to on, defined via the discrete analogue of (1.8), where is a discrete approximation of. Similarly to (2.16), and for later use, we introduce the weak formulation of (1.7): Find κ Γ [H 1 ( )] 3 such that κ Γ, η + ids, η s = 0 η [H 1 ( )] 3. (2.25) Similarly to (2.7) it holds that [ ] δ δγ w,1 ( χ) = d dε w ε,1 Γε(t) ε=0= wid s, χ s where 0 ε w = 0, and where w L ( ), χ [H 1 Γ ()]3, (2.26) H 1 Γ () := {η H1 () : η H 1 ( )}. (2.27) Moreover, similarly to (2.12), we note that for η [H Γ 1 ()]3 it holds that [ δ ] ids, η s ( χ) = P Γ η s, χ s δγ, (2.28) where 0 ε η = 0, and where For notational convenience, we also define P Γ = Id id s id s on. (2.29) [H 1 F ()]3 = { η [H 1 Γ ()]3 : η. F = 0 on } (2.30) and [H Γ 1 ()]3 free boundary conditions, X() = [H F 1 ()]3 semi-free boundary conditions, [H0()] 1 3 fixed boundary conditions. (2.31) 10

We first consider the three types of boundary conditions that do not involve fixing the conormal on, i.e. free, semi-free and Navier. We recall the energy (1.11) and the fact that m() is a topological invariant, which does not change its value under continuous deformations of the surface. We hence define the Lagrangian omitting the term m() as follows. Let L(, κ, m, κ Γ, y, z) = 1 2 κ κ ν 2,1 ( ) 2 + β κ, ν 2 M 0 +γh 1 ( ) ] +α G [ κ Γ, m κ Γ, z ids, z s Q θ κ, y sid, s y + m, y, (2.32) where y [H 1 ()] 3 and z [H 1 ( )] 3 are Lagrangemultipliers for (2.24) and (2.25), respectively. We now want to compute the direction of steepest descent f Γ of E(), where the curvature vector, κ, and the conormal m = µ, satisfy (2.24), and the curve curvature vector, κ Γ, satisfies (2.25). This means that f Γ needs to fulfill [ ] fγ δ, χ = δγ E() ( χ) χ X(). (2.33) Using (2.7) (2.12) and (2.18a c), with L replaced by L, as well as [ ] δ δ m L 1 ( ϕ) = lim [L(, κ, m+ε ϕ, κ ε 0 ε Γ, y, z) L(, κ, m, κ Γ, y, z)] = 0, (2.34) δ and similarly for δ κ Γ L = 0, which yields that z = m, and δ L = 0, one computes, on δ z noting (2.26) and (2.28), that fγ, χ = s y, s χ + s. y, s. χ ( s y) T,D( χ)( sid) T 1 [ κ κ ν 2 2Q 2 θ y. κ] sid, s χ βa ( κ. ν) sid, s χ (βa κ) κ, ε 0 ν + ε 0 [Q θ κ], y γ ids, χ s [ +α G κ Γ. m, id s. χ s + ] P Γ m s, χ s χ X(), (2.35a) κ +(βa κ) ν Q θ y, ξ = 0 ξ [H 1 ()] 3, (2.35b) α G κ Γ + y, ϕ = 0 ϕ [H 1 ( )] 3, (2.35c) with (2.24) and (2.25). As 0 ε κ = 0, we have that 0 ε [Q θ κ] = (1 θ) [ ( κ. 0 ε ν) ν +( κ. ν) 0 ε ν ]. (2.36) 11

We observe that (2.35b,c) imply that Q θ y = κ +(βa κ) ν on and y = α G κ Γ on. (2.37) If θ = 0, then it follows from (2.37), together with (1.2), (1.7) and (1.9), that κ = κ βa α G κ ν holds on. However, if θ (0,1], then the two conditions in (2.37) are incompatible if α G 0, unless the geodesic curvature κ µ vanishes on, since the first condition in (2.37) yields that y = (κ + βa κ) ν. Hence for general boundaries and α G 0 we need to take θ = 0, at least locally at the boundary. Therefore we need to consider a variable θ L (). The calculation (2.35a c) remains valid provided that 0 ε θ = 0. We will make this more rigorous on the discrete level, see (3.18) below. It follows from (2.35a), (2.36) and (2.11) that fγ, χ = s y, s χ + s. y, s. χ ( s y) T,D( χ)( sid) T [ κ κ ν 2 2 y.q θ κ] sid, s χ βa ( κ. ν) sid, s χ 1 2 +(βa κ) κ,( s χ) T ν (1 θ) κ.([ s χ] T ν) ν, y (1 θ) ( κ. ν)[ s χ] T ν, y γ ids, χ s +α G [ κ Γ. m, id s. χ s see Barrett et al. (2016) for a similar computation. + P Γ m s, χ s ] χ X(); (2.38) For the case of clamped boundary conditions, (1.17), the unknown m in (2.24) and (2.32) is replaced by the given data ζ. Then there is no variation in m, so that we no longer obtain (2.35c) and, of course, the terms involving in (2.38) play no role as χ [H 1 0 ()]3. Hence in this case it is not necessary to take θ = 0 in the vicinity of = Γ(0). If the surface area of has to be preserved during the evolution, cf. (1.19) (1.21), the right hand side of (2.38) has an additional term λ s id, s χ, on recalling (2.6). 3 Semidiscrete finite element approximation The parametric finite element spaces are defined as follows, see also Barrett et al. (2008). Let Υ h R 3 be a two-dimensional polyhedral surface, i.e. a union of nondegenerate triangles with no hanging vertices (see Deckelnick et al. (2005, p. 164)), approximating the reference manifold Υ. In particular, let Υ h = J j=1 oh j, where {oh j} J j=1 is a family of mutuallydisjoint opentriangles. Thenlet V h (Υ h ) := { χ C(Υ h,r 3 ) : χ o h j is linear, j = 1,...,J}. We consider a family of parameterizations X h (,t) V h (Υ h ) with X h (Υ h,t) = and with Γ h (0) an approximation of Γ(0). In particular, let = J j=1 σh j (t), 12

where {σ h j(t)} J j=1 is a family of mutually disjoint open triangles with vertices { q h k (t)}k k=1. Then let V h () := { χ [C()] 3 : χ σ h j is linear, j = 1,...,J} =: [W h ()] 3 [H 1 ()] 3, where W h () H 1 () is the space of scalar continuous piecewise linear functions on, with {χ h k (,t)}k k=1 denoting the standard basis of Wh (), i.e. χ h k( q h l (t),t) = δ kl k,l {1,...,K}, t [0,T]. (3.1) For later purposes, we also introduce π h (t) : C() W h (), the standard interpolation operator at the nodes { q k h(t)}k k=1, and similarly πh (t) : [C()] 3 V h (). Let V h 0 (Γh (t)) := V h () [H0 1 (Γh (t))] 3 (3.2a) and V h ( ) := { ψ [C( Γ h (0))] 3 : χ V h () χ Γ h (0)= ψ}. For later use, we introduce the decomposition (3.2b) V h () = V h Γ (Γh (t)) V h 0 (Γh (t)), (3.3) where we note that V h Γ (Γh (t)) is clearly isomorphic to V h ( ). We also introduce V h F (Γh (t)) := { χ V h () : ( χ. F)( q h k (t)) = 0 qh k (t) Γh (t)}. (3.4) In order to treat all four boundary conditions in a compact way, we also define V h () free boundary conditions, X() = V h F (Γh (t)) semi-free boundary conditions, V h 0 (Γh (t)) fixed boundary conditions, (3.5) where fixed boundary conditions can be either clamped or Navier. We denote the L 2 inner products on and by, and, Γ (t), h respectively. In addition, for piecewise continuous functions, with possible jumps across the edges of {σj h}j j=1, we also introduce the mass lumped inner product η,φ h := 1 3 J H 2 (σj) h j=1 3 (η.φ)(( q j h k ) ), k=1 where { q j h k } 3 k=1 are the vertices of σh j, and where we define η(( qh j k ) ) := lim η( p). We σj h p qh j k naturally extend this definition to vector and tensor functions. We also define the mass lumped inner product, h in the obvious way. 13

Following Dziuk and Elliott (2013, (5.23)), we define the discrete material velocity for z by K [ ] d V h ( z,t) := dt qh k (t) χ h k ( z,t). (3.6) Then, similarly to (2.1), we define,h t φ = φ t + V h. φ k=1 φ H 1 (G h T ), where Gh T := t [0,T] On differentiating (3.1) with respect to t, it immediately follows that {t}. (3.7),h t χ h k = 0 k {1,...,K}, (3.8) see also Dziuk and Elliott (2013, Lem. 5.5). It follows directly from (3.8) that,h t φ(,t) = K χ h k (,t) d dt φ k(t) on (3.9) k=1 for φ(,t) = K k=1 φ k(t)χ h k (,t) Wh (). For later use, we also introduce the finite element spaces W(G h T ) := {χ C(Gh T ) : χ(,t) Wh () t [0,T]}, W T (G h T) := {χ W(G h T) :,h t χ C(G h T)}. We recall from Dziuk and Elliott (2013, Lem. 5.6) that d φ dh 2 =,h t φ+φ s. V dt h dh 2 φ H 1 (σ h σj h(t) σj h(t) j (t)),j {1,...,J}, (3.10) which immediately implies that d dt η,φ =,h t η,φ +η,,h t φ +ηφ, s. V h Similarly, we recall from Barrett et al. (2015b, Lem. 3.1) that η,φ W T (G h T). (3.11) d dt η,φh =,h t η,φ h +η,,h t φ h +ηφ, s. V h h η,φ W T (G h T ). Moreover, it holds that (3.12) d dt η,φh =,h t η,φ h +η,,h t φ h +ηφ, id s. V h s h η,φ W T (G h T). (3.13) 14

We also note the discrete version of (2.14), d sid, s η dt = s η, sv h + ( s η) T,D( V h )( sid) T as well as the corresponding version for, d ids, η s = P h dt Γ h Γ η s, Vs h (t) s. η, s. V h η { ξ [W T (G h T )]3 :,h t η { ξ [W T (G h T )]3 :,h t which follows similarly to (2.28). Here, similarly to (2.29), we have defined ξ = 0}, (3.14) ξ = 0}, (3.15) P h Γ = Id id s id s on. (3.16) For later use, we introduce the vertex normal function ω h (,t) V h () with ω h ( q h k(t),t) := 1 H 2 (Λ h k (t)) j Θ h k H 2 (σ h j(t)) ν h σ h j (t), where for k = 1,...,K we define Θ h k := {j : qh k (t) σh j (t)} and set Λh k (t) := j Θ hσh k j (t). Here we note that z,w ν h h = z,w ω h h z V h (), w W h (). (3.17) In addition, for a given parameter θ [0,1] we introduce θ h W h () such that { θ h ( q k h (t),t) = 1 q k h(t) Γh (t), θ q k h(t) for clamped boundary conditions, Γh (t), { θ h ( q k h (t),t) = 0 q k h(t) Γh (t), θ q k h(t) for all other boundary conditions. (3.18) Γh (t), Then we introduce Q h θ h [W h ()] 3 3 by setting, for k {1,...,K}, Q h θ h ( q h k(t),t) = θ h ( q h k(t),t)id+(1 θ h ( q h k(t),t)) ωh ( q h k (t),t) ωh ( q h k (t),t) ω h ( q h k (t),t) 2, (3.19) wherehereandthroughoutweassumethat ω h ( q k h(t),t) 0fork = 1,...,K andt [0,T]. Only in pathological cases could this assumption be violated, and in practice this never occurred. We note that Q h θ h z, v h = h z,q h θ v h and h Q h θ z, ω h = z, ω h h (3.20) h 15

for all z, v V h (). Moreover, in the case of clamped boundary conditions, we let ζ h (t) V h ( Γ h (0)) be a suitable approximation of ζ(t) on Γ(0). On recalling from the introduction that at present β > 0 does not make sense from a modelling point of view for open surfaces, we set β = 0 from now on for simplicity. Mathematically the case β > 0 may be considered, and the resulting terms can then treated as in the closed surface case, see Barrett et al. (2016) for details. Similarly to the continuous setting, recall (1.11), (2.24), (2.25), we consider the first variation of the discrete energy E h () := 1 2 κ h κ ν h 2,1 h +α G subject to the side constraints h Q h θ κ h, η + h sid, s η [ κ h Γ, m h h +2πm(Γh (t)) ] +γh 1 ( ) (3.21) = m h, η h η V h () (3.22) and κ h Γ, χ h + ids, χ s = 0 χ V h ( ). (3.23) Of course, for clamped boundary conditions we set m h = ζ h, whereas for the other three boundary conditions m h (t) V h ( ) is an unknown. When taking variations of (3.22), we need to compute variations of the discrete vertex normal ω h. To this end, for any given χ X() we introduce Γ h ε(t) as in (2.5) and ε 0,h defined by (2.9), both with replaced by. We then observe that it follows from (3.17) with w = 1 and the discrete analogue of (2.10) that z, 0,h ε ω h h = z, 0,h ε ν h h h ( z.( ν + h ω h )) sid, s χ z V h (), χ X(). (3.24) An immediate consequence is that h h h z,,h t ω h = z,,h Γ h t ν h ( z.( ν (t) Γ (t)+ h ω h )) sid, s V h h In addition, we note that for all ξ, η V h () with 0,h ε where 0,h ε π h [( ξ. ω h ω h )( η. )] ω h [ = π h G h ( ξ, η). ω h ] ε 0,h ω h [ ( G h ( ξ, η) = π h 1 ( ξ. ω ω h h ) η+( η. ω h ) ξ 2 ( η. ωh )( ξ. ω h ) 2 ω h 2 z V h (). (3.25) ξ = 0,h ε η = 0 it holds that on, (3.26) ω h )]. (3.27) It follows that G h ( ξ, η). ω h = 0 ξ, η V h (). (3.28) 16

Considering at first all the boundary conditions that do not involve fixing the conormal, i.e. free, semi-free and Navier, we have the discrete analogue of (2.32) and define the Lagrangian L h (, κ h, m h, κ h Γ, Y h, Z h ) = 1 2 κ h κ ν h 2,1 h +γh1 ( ) [ κ h +α G Γ, m h h κ h Γ, Z h h ids, ] Z s Q h θ κ h, h Y h h sid, s Y h + m h, h Y h, (3.29) where κ h V h () and κ h Γ V h Γ() satisfy (3.22) and (3.23), respectively, with Y h V h () and Z h V h Γ (Γh (t)) being the corresponding Lagrange multipliers. Similarly to (2.35a c) with (2.24), (2.25), on recalling the formal calculus of PDE constrained optimization, we obtain an L 2 gradient flow of E h () subject to the side constraint (3.22) by setting [ δ δγ h L h ]( χ) = [ δ δ Q h, θ V h, χ h for χ X(), L h ]( ξ) = 0 for ξ V h (), [ δ κ h δy h Lh ]( η) = 0 for η V h (), [ L h ]( ϕ) = 0 δ m h for ϕ V h ( Γ h δ (t)), [ L h ]( φ) = 0 for φ V h ( ), yielding Z h = m h, and δ κ h Γ δ [ δz h Lh ]( φ) = 0 for φ V h ( ). Here we have defined δ Q h, θ ( q h k (t),t) = { Id Q h θ h q h k (t), q h k (t) Γh (t). (3.30) h h Hereweconsider [ δ L h ]( χ) = Q h, δγ h θ V h, χ inplaceof[ δ L h ]( χ) = V h, χ Γ h δγ (t) h in order to allow implicit tangential motion of vertices. In particular, we will show in Theorem 3.1, below, that for θ = 0 good meshes are enforced via the equation (3.31d). But these meshes can only be realized, if the motion of the vertices is not constrained to be in normal direction only. On the other hand, we must not allow an implicit tangential motion at the boundary nodes of, as we wish to reparameterize and not change the shape of via this tangential motion. Hence, for the boundary nodes we replace θ with 1 in the definition (3.30). Overall this gives rise to the following semidiscrete finite element approximation, where we note that ε 0,h θ h = 0. Given Γ h (0), for all t (0,T] find, with V h X(), κ h (t) V h (), Y h (t) V h (), κ h Γ (t) V h ( ) and m h (t) V h ( ) such 17

that h Q h, θ V h, χ + sy h, s χ ( s Y h ) T,D( χ)( s id) T + 1 2 s. Y h, s. χ +γ ids, χ s [ κ h κ ν h 2 2Y ] h.q hθ κ h h sid, s χ +κ κ h,[ s χ] T ν h h (1 θ h )( G h ( Y h, κ h ). ν h ) sid, s χ + (1 θ h ) G h ( Y h h, κ h ),[ s χ] T ν h [ α G κ h Γ. m h, id h s. χ s + ] P h Γ h Γ m h s, χ s = 0 χ X(), (3.31a) (t) κ h κ ν h Q h θ Y h, h ξ = 0 ξ V h (), (3.31b) h α G κ h Γ + Y h h, ϕ = 0 ϕ V h ( ), (3.31c) h Q h θ κ h, η + h sid, s η = m h, η h η V h (), (3.31d) h κ h Γ, φ + ids, φ s = 0 φ V h ( ), (3.31e) where G h ( Y h, κ h ) V h () is defined as in (3.27). Of course, in the case of fixed boundary conditions we have X() = V h 0 (Γh (t)), and so the terms involving in (3.31a) drop out. In addition, for fixed boundary conditions (3.31e) is invariant in time. Moreover, in the case of clamped boundary conditions, (1.17), m h (t) = ζ h (t) on Γ h (0) is fixed, and so the Lagrangian (3.29) simplifies. The semidiscrete finite element approximation is then given by (3.31a,b,d), with m h in (3.31d) replaced by ζ h. For later use we also observe that combining (3.31b) and (3.31d), on recalling (3.20), yields that Q h h θ Y h,q h h θ η + h sid, s η = m h, η h κ ω h, η h h h η V h (). (3.32) In deriving (3.31a d) from the six variationsof L h mentioned above, we have made use of the obvious discrete variants of (2.7) (2.12), (2.26), (2.28) and recalled (3.24), (3.26) and (3.28), which requires (3.16). We note that (3.31b,c) and (3.17) imply that π h [Q h θ h Y h ] = κ h κ ω h and Y h = α G κ h Γ, (3.33) which is the discrete analogue of (2.37). In order to be able to consider surface area conserving variants of (3.31a d), we introduce a Lagrange multiplier λ h (t) R for the constraint d dt H2 () = s. V h,1 = sid, s V h = 0, (3.34) 18

where we recall (2.4). Now, on writing (3.31a) as h Q h, θ V h, χ = sy h, s χ + f h, χ we consider Q h, θ V h, χ h = where ( λ h (t) = sy h, s [ Π h 0 κ h ] + m h, V ) h h / h χ X(), h sy h, s χ + f h, χ λ h sid, s χ χ X(), (3.35) f h, Π h h 0 κ h + Π h Γ h 0 V h h V h,q h θ κ h (t) h Q h θ κ h, h Π h h 0 κh, (3.36) with Π h 0 : V h () V h 0 (Γh (t)) being the projection onto V h 0 (Γh (t)). Here we note that Q h θ κ h, h Π h 0 κ h = Q h h Γ h θ Π h 0 κ h, h Π h 0 κ h 0, (3.37) (t) h with strict inequality for θ (0,1] unless Π h 0 κh = 0, and for θ = 0 unless κ h ( q k h(t),t). ωh ( q k h(t),t) = 0 for all qh k (t) Γh (t) \. In order to motivate (3.36) we note, on recalling (3.31d), (3.30) and (3.20) that sid, s Π h0 κ h = Q h Γ h θ κ h, h Π h (t) h 0 κh (3.38a) and Q h, θ V h, Π h h 0 κh = Π h h Γ h 0 V h,q h θ κ h (t) h = Π h 0 V h V h,q h θ h κ h h + m h, V h h sid, s V h. (3.38b) Hence (3.35) with χ = Π h 0 κ h and (3.36), (3.38a,b) yield that (3.34) is satisfied. Of course, in the case of fixed boundary conditions, the terms involving V h V h 0 (Γh (t)) in (3.36) drop out, and the last term on the right hand side of (3.35) can be equivalently written as λ h Q h θ h κ h, χ h, on noting (3.31d), since χ V h 0 (Γh (t)). The following theorem establishes that (3.31a e) is indeed a weak formulation for the L 2 gradient flow of E h (), recall (3.21), subject to the side constraints (3.22) and (3.23). We will also show that for θ = 0 the scheme produces conformal polyhedral surfaces. Here we recall from Barrett et al. (2008, 4.1) that the surface is a conformal polyhedral surfaces if s id, s η = 0 η { ξ V h 0 (Γh (t)) : ξ( q h k (t)). ωh ( q h k (t),t) = 0,k = 1,...,K }. 19 (3.39)

Note that the definition in Barrett et al. (2008, 4.1) is for closed surfaces, and that it implies that ω h is parallel to the discrete Laplacian of id. Hence (3.39) is a natural generalization of that definition, since we only enforce this constraint at the interior nodes of. We recall from Barrett et al. (2008) that conformal polyhedral surfaces exhibit good meshes. In particular, coalescence of vertices cannot occur. Moreover, we recall that the two-dimensional analogue of conformal polyhedral surfaces are equidistributed polygonal curves, see Barrett et al.(2007, 2011). Now introducing the parameter θ [0, 1], we obtain a family of schemes that interpolate between the choices θ = 0 and θ = 1, with the latter meaning that all vertices are transported approximately only in the normal direction. This corresponds to the original approach in Dziuk (2008), and so the choice θ = 1 can be interpreted as a natural generalization of the approximation in Dziuk (2008) to surfaces with boundary. We now present a stability proof for the semidiscrete scheme (3.31a e), where in the case of clamped boundary conditions we assume that ζ h C( Γ h (0),S d 1 ) does not vary in time. Theorem. 3.1. Let θ [0,1] and let {(Γ h, κ h, m h, κ h Γ, Y h )(t)} t [0,T] be a solution to (3.31a e), where in the clamped case we fix m h (t) = ζ h and do not require (3.31c). Then d dt Eh () = Q h, θ V h, V h h 0. (3.40) Moreover, if θ = 0 then is a conformal polyhedral surface for all t (0,T]. Proof. First we consider the cases where the boundary is not clamped. Taking the time derivative of (3.31d) with,h t η = 0, yields that h h,h t (Q h θ κ h ), η + (Q h h θ κ h. η) h sid, s V h + sv h, s η = + s. V h, s. η h,h t m h, η + ( s η) T,D( V h )( s id) T m h. η, id s. V h s, (3.41) where we have noted (3.12), (3.13) and (3.14). Similarly, taking the time derivative of (3.31e) with,h t φ = 0 yields, on noting (3.13) and (3.15), that h,h t κ h Γ, φ + κ h Γ h Γ. φ, id s. h Vs h + P h (t) Γ h Γφ s, Vs h = 0. (3.42) (t) Choosing χ = V h X() in (3.31a), η = Y h V h () in (3.41) and combining 20

yields, on noting the discrete variant of (2.11), that Q h, θ V h, V h h + 1 [ κ h κ ν h 2 2 h Y h.q h Γ h 2 θ κ h ] (t) h sid, s V h κ + h κ h,,h t ν h +,h t (Q h θ h κ h ), Y h h + (1 θ h )( G h ( Y h, κ h ). ν h ) s id, s V h h (1 θ h ) G h ( Y h, κ h ),[ s V h ] T ν h h +γ ids, V h s (Q h θ h κ h. Y h ) s id, s V h ] [ α G κ h Γ. mh, id s. V h s h + P h Γ h Γ mh s, Vs h (t) =,h t m h, Y h h + m h. Y h, id s. h V h Γ h s. (3.43) (t) Choosing φ = m h in (3.42), it follows from (3.43), on recalling (3.17) and (3.33), that Q h, θ V h, V h h h + 1 κ h κ ν h 2 Γ h 2 sid, s V h (t) + +,h t (Q h θ κ h ), h Y h h (1 θ h ) G h ( Y h h, κ h ),[ sv h ] T ν h +γ h κ κ h,,h t ν h (1 θ h )( G h ( Y h, κ h ). ν h ) s id, s V h ids, V s h [ h h = α G κ h Γ,,h t m h +,h Γ h t κ h Γ, mh + (t) h κ h Γ. mh, id s. V h s h h d = α G κ h dt Γ, m h h. (3.44) We have from (3.20), (3.33) and (3.17) that,h t (Q h θ κ h ), h Y h κ h =,h = 1 2 + κ h,,h t ν h h κ h κ ν h,,h t ν h h t κ h,q h h θ Y h κ h,h t (Q h θ κ h ) Q h h θ,h h t κ h, Y h h,h t κ h κ ν h 2,1 h +,h t (Q h θ κ h ) Q h h θ,h h t κ h, Y h h. (3.45) Combining (3.44) and (3.45), on noting (3.13), (3.21),,h t θ h = 0 (which follows from (3.9) and (3.18)) and the invariance of m() under continuous deformations, yields that Q h, θ V h, V h h + d dt Eh ()+P = 0, ] 21

where P := (1 θ h ) κ h.,h t ω h Y, h. ω h ω h 2 h + (1 θ h ) Y h.,h t ω h, κh. ω h ω h 2 2 (1 θ h )( κ h. ω h )( Y h. ω h ), ωh.,h t ω h ω h 4 (1 θ h )( G h ( Y h h, κ h ). ν h ) sid, s V h + (1 θ h ) G h ( Y h h, κ h ),[ sv h ] T ν h. (3.46) It remains to show that P as defined in (3.46) vanishes. To see this, we observe that it follows from (3.28), (3.27), the discrete variant of (2.11) and (3.25) that P = h h (1 θ h )( G h ( Y h h, κ h ),,h t ω h + (1 θ h )( G h ( Y h h, κ h ).( ω h ν h ) sid, s V h (1 θ h )( G h ( Y h h, κ h ),,h t ν h = 0. (3.47) This proves the desired result (3.40) when the boundary is not clamped. In the case of clamped boundary conditions we have that m h (t) = ζ and V h V h 0(), and so the right hand side of (3.41) is zero, which means that we do not need (3.42). Hence the right hand side of (3.44) is zero, and so the desired result (3.40) follows for clamped boundary conditions. If θ = 0 then it immediately follows from (3.31d) that (3.39) holds. Hence is a conformal polyhedral surface. Remark. 3.1. It is clear from the above proof that on replacing Q h, θ V h, χ h in (3.31a) with Q h, ς V h, χ h, for ς [0,1], we obtain a slightly different family of schemes that are also stable. I.e. solutions to these schemes satisfy d dt Eh () = Q h, ς V h, V h h in place of (3.40). In view of the desired tangential motion for θ = 0, it would be natural to choose ς = 0 in this case, or at least to choose ς [0,1), in order to allow for nonzero tangential motion in (3.31a). In fact, in practice we observe that for ς = 1 the corresponding fully discrete finite element approximation yields unsatisfactory results. Moreover, the proof of the following theorem demonstrates that in order to satisfy the conservation property (3.34), it is desirable to keep the left hand side of (3.35) as stated, i.e. to choose ς = θ. Theorem. 3.2. Let θ [0,1] and let {(Γ h, κ h, m, κ h Γ, Y h,λ h )(t)} t [0,T] be a solution to (3.35), (3.31b e) and (3.36), where in the clamped case we fix m h (t) = ζ h and do not 22

require (3.31c). Then it holds that as well as d dt Eh () = Q h, θ V h, V h h 0, (3.48) d dt H2 () = 0. (3.49) Moreover, if θ = 0 then is a conformal polyhedral surface for all t (0,T]. Proof. Choosing χ = Π h 0 κh in (3.35) yields, on noting (3.36), (3.37) and (3.38a) that (3.34) holds, which yields the desired result (3.49). The stability result (3.48) directly follows from the proof of Theorem 3.1. In particular, choosing χ = V h in (3.35), on noting (3.34), yields that Q h, θ V h, V h h = sy h, sv h + f h, h V h. Combining this with (3.41) yields that (3.43) holds, and the rest of the proof proceeds as that of Theorem 3.1. Finally, as in the proof of Theorem 3.1, for θ = 0 it follows from (3.31d) that is a conformal polyhedral surface. Remark. 3.2. Similarly to Remark 3.3 in Barrett et al. (2016), we can also consider a natural alternative to the scheme (3.31a e), which does not use the normalization of the discrete vertex normal ω h as in (3.19). In particular, on letting Q h θ h [W h ()] 3 3 be defined by Q h θ h ( q h k (t),t) = θh ( q h k (t),t)id+(1 θh ( q h k (t),t)) ωh ( q h k (t),t) ωh ( q h k (t),t) for all k {1,...,K}, and on replacing Q h θ h and Q h, θ in (3.31a e) by Q h θ h and Q h, θ, respectively, as well as adjusting the terms involving (1 θ h ) in (3.31a), we obtain a new scheme that can be shown to satisfy all the properties of (3.31a e). In fact, in the case of closed surfaces this new scheme collapses to the scheme (3.41a c) from Barrett et al. (2016) in the case β = 0. However, in the interest of consistency and continuity, we concentrate on the scheme (3.31a e) in this paper, as we did in Barrett et al. (2016). 4 Fully discrete finite element approximation In this section we consider a fully discrete variant of the scheme (3.35), (3.31b e) and (3.36) from Section 3. To this end, let 0 = t 0 < t 1 <... < t M 1 < t M = T be a partitioning of [0,T] into possibly variable time steps τ m := t m+1 t m, m = 0,...,M 1. Let Γ m be a polyhedral surface, approximating Γ h (t m ), m = 0,...,M, with boundary Γ m. Following Dziuk(1991), wenowparameterize thenewsurfaceγ m+1 over Γ m. Hence, we introduce the following finite element spaces. Let Γ m = J j=1 σm j, where {σm j }J j=1 23

is a family of mutually disjoint open triangles with vertices { q k m}k k=1. Then for m = 0,...,M 1, let V h (Γ m ) := { χ [C(Γ m )] 3 : χ σ m j is linear j = 1,...,J} =: [W h (Γ m )] 3 [H 1 (Γ m )] 3, for m = 0,...,M 1. We denote the standard basis of W h (Γ m ) by {χ m k }K k=1. In addition, similarly to (3.2a,b), we also introduce V h 0(Γ m ) and V h ( Γ m ). We also introduce π m : C(Γ m ) W h (Γ m ), the standard interpolation operator at the nodes { q k m}k k=1, and similarly π m : [C(Γ m )] 3 V h (Γ m ). Throughout this paper, we will parameterize the new closed surface Γ m+1 over Γ m, with the help of a parameterization X m+1 V h (Γ m ), i.e. Γ m+1 = X m+1 (Γ m ). Similarly to (3.5), let V h (Γ m ) free boundary conditions, X(Γ m ) = V h F (Γm ) semi-free boundary conditions, (4.1) V h 0(Γ m ) fixed boundary conditions, where V h F (Γm ) := { χ V h (Γ m ) : ( χ. F)( q m k ) = 0 qm k Γm }. (4.2) We also introduce the L 2 inner products, Γ m and, Γ m, as well as their mass lumped inner variants, h Γ and m, h Γm. Similarly to (3.17), we note that z,w ν m h Γ m = z,w ω m h Γ m z V h (Γ m ), w W h (Γ m ), where ω m := K k=1 χm k ωm k V h (Γ m ), and where for k = 1,...,K we let Θ m k := {j : qm k σj m} and set Λm k := j Θ mσm k j and ω k m := 1 H 2 (Λ j Θ H 2 (σ m k ) m j m) νm j. k We make the following very mild assumption. (A) We assume for m = 0,...,M 1 that H 2 (σj m 0 { ω k m}k k=1, for all m = 0,...,M 1. ) > 0 for all j = 1,...,J and that In addition, and similarly to (3.18) and (3.19), we first introduce θ m W h (Γ m ), and then Q m θ m [Wh (Γ m )] 3 3 by setting Q m θ m( qm k ) = θm ( q k m) Id+(1 θ m ( q k m)) ωm k 2 ω k m ωm k for k = 1,...,K. We also define Q m, θ [W h (Γ m )] 3 3 similarly to (3.30) in terms of Q m θ m. Similarly to (3.27) and (3.16), we let [ ( G m ( ξ, η) = π m 1 ω m 2 ( ξ. ω m ) η+( η. ω m ) ξ 2 ( η. ωm )( ξ. ω m ) ω m 2 ω m )] (4.3) and P m Γ = Id id s id s on Γ m. (4.4) Given Γ 0 and κ 0, Y 0 V h (Γ 0 ), m 0 V h ( Γ 0 ), let κ 0 Γ V h ( Γ 0 ) be such that κ 0 Γ, η h + ids, η Γ 0 s = 0 η V h ( Γ 0 ). (4.5) Γ 0 24

On recalling (3.32) and (3.17), we consider the following fully discrete approximation of (3.35), (3.31b e) and (3.36). For m = 0,...,M 1, find (δx m+1, Y m+1 ) X(Γ m ) V h (Γ m ), with X m+1 = id Γ m +δx m+1, and ( κ m+1 Γ, mm+1 ) [V h ( Γ m )] 2 such that Q m, θ X m+1 id, χ τ m h Γ m sy m+1, s χ +γ Γ m s. Y m, s. χ +α G m m+1 s, χ s = Γ m Γ m ( sy m ) T,D( χ)( sid) T [ κ 1 m κ ν m 2 2 ] Y m.q mθ 2 m κm sid, s χ + (1 θ m )( G m ( Y h m, κ m ). ν m ) sid, s χ Γ m κ κ m,[ s χ] T ν m h Γ m Γ m h Γ m X m+1 s, χ s Γ m (1 θ m ) G m ( Y m, κ m ),[ s χ] T ν m h Γ m +α G κ m Γ. mm, id h s. χ s +α G (Id+P m Γ ) m m Γ m s, χ s Γ m λ m sid, s χ χ X(Γ m ), (4.6a) Γ m Q m h θ Y m+1,q m m θ m η + sx m+1, s η = m m+1, η h κ ω m, η h Γ m Γ m Γ m Γ m η V h (Γ m ), (4.6b) α G κ m+1 Γ + Y h m+1, ϕ = 0 ϕ V h ( Γ m ), (4.6c) Γ m κ m+1 Γ, η h + X m+1 Γ m s, η s = 0 η V Γ h ( Γ m ) (4.6d) m and set κ m+1 = π m [Q m θ Y m+1 ]+κ ω m and Γ m+1 = X m+1 (Γ m ). We note that for α m G = 0 the scheme simplifies, as we no longer need (4.6d). In addition, for clamped conditions, we replace m m+1 in (4.6b) with ζ m, an approximation of ζ(t m ), and do not require (4.6c). Finally, for m 1 we note that here and throughout, as no confusion can arise, we denote by κ m the function z V h (Γ m ), defined by z( q k m) = κm ( q m 1 k ), k = 1 K, where κ m V(Γ m 1 ) is given, and similarly for Y m, m m and κ m Γ. Of course, (4.6a d) with λ m = 0 corresponds to a fully discrete approximation of (3.31a c,e), (3.32). For a fully discrete approximation of surface area preserving Willmore flow, on recalling (3.36), we let λ m = ( sy m, s [ Π m 0 κm ] + Γ m f m, Π m 0 κm h Γ m ( Π m 0 Id) id X m 1,Q m θ τ m κm m h Γ m + m m, id, X h m 1 τ m Γ m / Q m θ m κm, Π h m 0 κm, (4.7) Γ m 25

where for convenience we have re-written (4.6a) as X m+1 id h, χ Q m, θ τ m Γ m h sy m+1, s χ = f, χ m λ m sid, s χ Γ m Γ m Γ m χ X(Γ m ), (4.8) and where Π m 0 : V h (Γ m ) V h 0(Γ m ) is the projection onto V h 0(Γ m ). We also define X 1 = X 0 = id Γ 0. Similarly to (3.37), we note that the denominator in (4.7) is always nonzero for θ (0,1] unless Π m 0 κm = 0, and for θ = 0 unless κ m ( q k m). ωm ( q k m ) = 0 for all q k m Γm \ Γ 0. 4.1 Fixed cases In the case of fixed boundary, the scheme (4.6a d) simplifies dramatically. First of all, we note that the equation (4.6d) is not needed, since κ m+1 Γ = κ 0 Γ is fixed given data, recall (4.5), and the terms involving Γ m in (4.6a) disappear. Moreover, in the case of Navier boundary conditions, it is also possible to eliminate the unknown m m+1 from the finite element approximation (4.6a d). Overall, for Navier boundary conditions we obtain: Given Γ 0 and κ 0, Y 0 V h (Γ 0 ), let κ 0 Γ V h ( Γ 0 ) be defined by (4.5). Then, for m = 0,...,M 1 find (δx m+1, Y m+1 ) V h 0(Γ m ) V h (Γ m ), with X m+1 = id Γ m +δx m+1 and Y m+1 Γ 0= α G κ 0 Γ, such that X Q m, m+1 h id θ, χ sy m+1, s χ = s. τ Y m, s. χ m Γ m Γ m Γ m ( sy m ) T,D( χ)( sid) T κ κ m,[ s χ] T ν m h Γ m Γ m [ κ 1 m κ ν m 2 2 ] h Y m.q mθ 2 m κm sid, s χ Γ m + (1 θ m )( G m ( Y h m, κ m ). ν m ) sid, s χ (1 θ m ) G m ( h Y m, κ m ),[ s χ] T ν m Γ m Γ m λ m sid, s χ χ V h 0(Γ m ), (4.9a) Γ m Q m h θ Y m+1,q m m θ m η + sx m+1, s η = κ ω m, η h Γ m Γ m Γ η V h m 0 (Γm ), (4.9b) and set κ m+1 = π m [Q m θ m Y m+1 ]+κ ω m and Γ m+1 = X m+1 (Γ m ). For clamped boundary conditions, on the other hand, we let ζ m V h ( Γ m ) be an approximation to ζ(t m ) [C( Γ(t m ))] 3, and then consider: Given Γ 0 and κ 0, Y 0 V h (Γ 0 ), for m = 0,...,M 1 find (δx m+1, Y m+1 ) V h 0 (Γm ) V h (Γ m ), with X m+1 = id Γ m +δx m+1, such that (4.9a) holds as well as h ζ, η m Q m θ m Y m+1,q m θ m η h Γ m + s X m+1, s η Γ m = Γ 0 κ ω m, η h Γ m η V h (Γ m ). (4.10) Then set κ m+1 = π m [Q m θ m Y m+1 ]+κ ω m and Γ m+1 = X m+1 (Γ m ) as before. 26

4.2 Implicit treatment of area conservation In practice it can be advantageous to consider an implicit Lagrange multiplier λ m+1 in order to obtain a better discrete surface area conservation. In particular, we replace (4.8) with Q m, θ X m+1 id, χ τ m h Γ m h sy m+1, s χ = f, χ m λ m+1 sx m+1, s χ Γ m Γ m Γ m χ X(Γ m ) (4.11) andrequirethecoupledsolution( X m+1, Y m+1, κ m+1 ) [V h (Γ m )] 3 andλ m+1 Rtosatisfy the nonlinear system (4.11), (4.6b d) as well as an adapted variant of (4.7), where the superscript m is replaced by m + 1 in all occurrences of κ m, Y m and λ m. In addition, id X m 1 τ m X in (4.7) is replaced by m+1 id τ m. In practice this nonlinear system can be solved with a fixed point iteration as follows. Let λ m+1,0 = λ m and κ m+1,0 = κ m. Then, for i 0, find a solution ( X m+1,i+1, Y m+1,i+1, κ m+1,i+1 Γ, m m+1,i+1 ) X(Γ m ) V h (Γ m ) [V h ( Γ m )] 2 to the linear system (4.11), (4.6b d), where any superscript m + 1 on left hand sides is replaced by m + 1,i + 1, and by m + 1,i on the right hand side of (4.11). Then let κ m+1,i+1 = π m [Q m θ Y m+1,i+1 ] + κ ω m be defined as usual, and compute λ m+1,i+1 as the m unique solution to λ m+1,i+1 = + ( sy m+1,i+1, s [ Π m 0 κm+1,i+1 ] f m, h Π m Γ m 0 κm+1,i+1 Γ m ( Π m 0 Id) X m+1,i+1 id,q m θ τ m κm+1,i+1 m h Γ m + / Q m θ m κm+1,i+1, Π h m 0 κm+1,i+1 Γ m m m+1,i+1 X, m+1,i+1 id h τ m Γ m andcontinuetheiterationuntil λ m+1,i+1 λ m+1,i < 10 8. Inpracticethisiterationalways converged in fewer than ten steps, and at little extra computational cost compared to the linear scheme (4.6a d), since the linear system (4.8), (4.6b d) can be easily factorized with the help of sparse factorization packages such as UMFPACK, see Davis (2004). 4.3 Existence and uniqueness Theorem. 4.1. Let the assumptions (A) hold, and let θ [0,1]. Then there exists a unique solution (δx m+1, Y m+1, κ m+1 Γ, mm+1 ) X(Γ m ) V h (Γ m ) [V h ( Γ m )] 2 to (4.6a d) in all the situations where the boundary Γ m is not clamped. In the case of clamped boundary conditions, there exists a unique solution (δx m+1, Y m+1 ) V h 0 (Γm ) V h (Γ m ) to (4.9a), (4.10). Proof. We first consider the three situations where the surface is not clamped at the boundary. As this system is linear, existence follows from uniqueness. To investigate the 27