A fully discrete evolving surface finite element method

Original citation:

Elliott, Charles M. and Dziuk, G.. (2012) A fully discrete evolving surface finite element

method. SIAM Journal of Numerical Analysis, Vol.50 (No.5). pp. 2677-2694. ISSN

1095-7170

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A FULLY DISCRETE EVOLVING SURFACE FINITE

ELEMENT METHOD∗

GERHARD DZIUK† AND CHARLES M. ELLIOTT‡

Abstract. In this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface. In earlier papers using a suitable variational formulation in time dependent Sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces [IMA J. Numer Anal., 25 (2007), pp. 385–407;Math. Comp., to appear]. Optimal orderL2_{(Γ(t)) andH}1_{(Γ(t)) error}

bounds were proved for linear finite elements. In this work we prove optimal order error bounds for a backward Euler time discretization.

Key words. surface finite elements, advection diffusion, moving surface, error analysis

AMS subject classifications.65M60, 65M15, 35K99, 35R01, 35R37, 76R99

DOI.10.1137/110828642

1. Introduction. Partial differential equations on moving surfaces appear in a wide range of applications such as material science [3, 7], fluid flow [10, 12], and biology and biophysics [2, 8, 14, 16]. Numerical approaches include level set methods, surface finite elements, finite volume methods, and diffuse interface methods; see [1, 4, 5, 9, 13, 15, 17] and the references cited therein.

In this paper we focus on

(1.1) ∂•u+u∇_Γ·v− ∇_Γ·(A∇_Γu) = 0

on an evolving compact hypersurface Γ(t)⊂Rm+1,m= 1,2,for timet∈[0, T], T >0. The methods apply also to higher dimensional hypersurfaces. Note that although such a Γ(t) does not have a boundary, the method may be extended to a hypersurface with a boundary on which an appropriate boundary condition is satisfied. Herev_N is the normal velocity of the surface,v_T is an advective tangential velocity field,v=v_N+v_T,

∇Γ is the tangential surface gradient,Ais a given diffusion tensor, and

∂•u=u_t+v_N· ∇u+v_T · ∇_Γu

is the material derivative.

This is an advection and diffusion equation (see [4] for a derivation) arising from conservation of a scalar field uon an evolving hypersurface with a fluxq comprising a diffusive term and an advective tangential velocityv_τ so that

q=−A∇_Γu+v_T ·u.

Note that the term u∇_Γ·v in (1.1) is needed for conservation and arises from the local shrinking or expansion of the surface.

∗_{Received by the editors March 25, 2011; accepted for publication (in revised form) August}

22, 2012; published electronically October 23, 2012. This work was supported by the Deutsche Forschungsgemeinschaft via SFB/TR 71 and by the UK Engineering and Physical Sciences Research Council (EPSRC), grant EP/G010404.

http://www.siam.org/journals/sinum/50-5/82864.html

†_{Abteilung f¨ur Angewandte Mathematik, University of Freiburg, Hermann-Herder-Straße 10,}

D–79104 Freiburg i. Br., Germany ([email protected]).

‡_{Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (c.m.elliott@}

warwick.ac.uk).

2677

Our finite element method is based on the variational form (see [4])

(1.2) d

dt

Γ(t)

uϕ+

Γ(t)A∇Γ

u· ∇_Γϕ=

Γ(t)

u∂•ϕ,

whereϕis an arbitrary test function defined on the space-time surface

GT =

t∈[0,T]

Γ(t)× {t}.

A continuous in time finite element approximation was proposed and analyzed in [4] using piecewise linear finite elements on a triangulated surface interpolating Γ(t) whose vertices move with the velocityv.

In this paper we prove optimal order error bounds for the following fully discrete finite element method:

(1.3)

1

τ

Γn+1

h

U_hn+1φn_h+1−

Γn h

U_hnφn_h

+

Γn+1

h

A−l_∇

ΓhUhn+1· ∇Γhφnh+1=

Γn h

U_hn∂_h•φn_h,

where U_hn andφn_h belong to a piecewise linear finite element space,S_hn, defined on a triangulation Γn_h of Γ(tn),τ >0 is the time step size,A−l is an appropriate extension of the given diffusion tensor onto the discrete surface, and ∂_h•φn_h is a time discrete material derivative.

Using the transport property of the basis functions (see [4] and section 2 of this paper) this approximation can be written as

(Mn+1+τSn+1)αn+1=Mnαn,

which we observe is a backward Euler discretization of the following system of ordinary differential equations:

d dt

M(t)α+S(t)α= 0,

arising from the continuous in time evolving surface finite element method (ESFEM) of [4], whereMn=M(tn) andSn=S(tn),M(t) andS(t) are time dependent mass and stiffness matrices, and αn = (αn₁, . . . , αn_J) where {αn_j}J_j=1 are the coefficients of the piecewise linear nodal basis functions{χn_j}J_j=1 such that

U_hn=

N

j=1

αn_jχn_j.

For the error between the continuous solutionuand the continuous in time spa-tially discrete solution u_h (which is a lift onto Γ(t) of the finite element solution on the triangulated surface) we proved in [4, 6] the bound

sup

t∈(0,T)

u(·, t)−u_h(·, t)2_L2_(Γ(t))+h2 _T

0 ∇Γ

(u(·, t)−u_h(·, t))2_L2_(Γ(t))dt≤ch4,

where the constantcdepends on norms of the continuous solution and the data of the problem. The purpose of this paper is to prove an optimalL2-estimate for the error between the solutions of the continuous and fully discrete problems:

u(·, tn)−un_h2_L2_(Γ(tn₎₎+h2τ n

k=1

∇Γu(·, tk)−ukh

2

L2_(Γ(tk₎₎≤c(τ2+h4).

This is consistent with numerical experiments performed in [4, Table 2 for Example 7.3].

The challenges for convergence and stability analysis of discretization methods for advection-diffusion equations on moving surfaces arise because of the moving ge-ometry. Existing results are for direct approximations based on interpolations of the evolving surface. An optimal order error analysis for the semidiscrete ESFEM based on piecewise linear finite element approximations has been given in [4, 6]. High or-der Runge–Kutta schemes for this semidiscretization have been or-derived and analyzed in [11]. In particular, stability and optimal order error bounds for the ordinary dif-ferential equation systems arising from ESFEM approximation of advection-diffusion equations on moving surfaces are obtained. In [13] Lenz, Nemadjieu, and Rumpf provedL2 and H1 error bounds for their proposed fully discrete time implicit finite volume scheme. The purpose of this paper is to extend the results of [5] to the case when the time discretization is based on a backward Euler scheme.

This paper is organized as follows. In section 2 we formulate the variational prob-lem and the finite eprob-lement method. In section 3 we derive discrete in time transport formulae and provide some estimates necessary for the error analysis. In section 4 we list some approximation properties established in [6]. Finally, in section 5 we prove the error bound.

2. The partial differential equation and finite element method.

2.1. Weak and variational formulations. In the following we assume thatA is a sufficiently smooth symmetric (m+ 1)×(m+ 1) matrix which maps the tangent space of Γ at a point into itself and is positive definite on the tangent space, i.e.,

(2.1) Aξ·ξ≥c₀|ξ|2 ∀ξ∈Rm+1, ξ·ν = 0,

with some constant c₀ > 0. For the definition of a solution we assume that the elements ofAbelong toL∞(G_T).

For each t ∈ [0, T] let Γ(t) be a compact hypersurface oriented by the normal vector fieldν(·, t) and Γ₀ = Γ(0). We assume that there exists a mapG(·, t) : Γ₀ → Γ(t), G ∈ C1([0, T], C2(Γ₀)) such that G(·, t) is a diffeomorphism from Γ₀ to Γ(t), and we set v(G(·, t), t) =G_t(·, t), G(·,0) = Id. We assume that v(·, t) ∈ C2(Γ(t)). The normal velocity of Γ then is defined byv_N =v·νν.

Forϕ, ψ∈H1(Γ) we define the bilinear forms

a(ϕ(·, t), ψ(·, t)) =

Γ(t)A

(·, t)∇_Γϕ(·, t)· ∇_Γψ(·, t),

(2.2)

m(ϕ(·, t), ψ(·, t)) =

Γ(t)

ϕ(·, t)ψ(·, t),

(2.3)

g(v(·, t);ϕ(·, t), ψ(·, t)) =

Γ(t)

ϕ(·, t)ψ(·, t)∇_Γ·v(·, t).

(2.4)

Note that above we have suppressed the explicit dependence ontof the forms on the left-hand side. This dependence should be understood by the dependence on time of the arguments.

Using this notation, the variational form (1.2) becomes

(2.5) d

dtm(u, ϕ) +a(u, ϕ) =m(u, ∂

•_ϕ).

In [4] we proved the existence of a weak solution. Furthermore for the initial data

u₀∈H1(Γ₀), Γ₀= Γ(0), and the elements ofAandv∈C1(G_T), the solution satisfies

sup

(0,T)u 2

L2_(Γ)+ T

0 ∇Γ

u2_L2_(Γ)dt≤cu02L2_(Γ0),

(2.6)

_T

0

∂•u2_L2_(Γ)dt+ sup (0,T)∇Γ

u2_L2_(Γ)≤cu02H1_(Γ0),

(2.7)

wherec=c(A, v,G_T, T). Throughout the paper we usecto denote a generic constant which is independent of the approximation parametershandτbut possibly dependent on the data.

2.2. Surface finite element method. Let N be a positive integer and set

τ = T /N. For each n ∈ {0, . . . , N} set tn = nτ. For a discrete time sequence fn,

n∈ {0, . . . , N}, we use the notation

∂_τfn= 1

τ

fn+1−fn.

2.2.1. Triangulated surface. The smooth evolving surface Γ(t) (∂Γ(t) =∅) is approximated by an evolving surface

Γ_h(t)⊂ N(t), (∂Γ_h(t) =∅),

which for each t is of class C0,1 and is smooth in time. We assume that Γ_h(t) is homeomorphic to Γ(t). Here N(t) is a neighborhood in Rm+1 of Γ(t) such that for eachx∈ N(t) there is a uniquep(x, t)∈Γ(t) which is the normal projection ofxonto Γ(t), i.e.,x−p(x, t) = d(x, t)ν(p(x, t), t), where |d(x, t)| is the distance of xto the hypersurface andν(p(x, t), t) is normal to the surface. In particular form = 2, Γ_h(t) is a triangulated (and hence polyhedral) compact hypersurface consisting of simplices

E(t)∈ T_h(t), which form an admissible triangulation; i.e., each face of a simplex is always a face of an adjoining simplex. We suppose that the maximum diameter of the simplices inT_h(t) is bounded uniformly in time byh. Observe that this is an implicit restriction on the size of T for a given velocity v. Note that for each E(t)⊂Γ_h(t) there is a uniquee(t)⊂Γ(t),e(t) =p(E(t), t), whose edges are the unique projections of the edges ofE(t) onto Γ(t). This induces an exact triangulation of Γ(t) with curved edges.

In this paper we consider triangulated surfaces for which the vertices{X_j(t)}J_j=1 of the simplices sit on Γ(t) so that Γ_h(t) is an interpolation. Furthermore we advect the nodes in the tangential direction with the advective velocityv_T and keep them on the surface using the normal velocityv_N so that

(2.8) dXj

dt (t) =v(Xj(t), t) (j = 1, . . . , J).

2.2.2. Finite element spaces. We use piecewise linear finite elements on the evolving discrete surface Γ_h(t) and G_Th = _t∈[0,T]Γ_h(t)× {t}. For any continuous functions η_h and η defined, respectively, on Γ_h(t) and Γ(t) we define extensions or lifts η_hl and η−l by extending constantly in the direction of the normal ν of the continuous surface so that (η_hl)−l = η_h; see [4, 6]. This has two purposes. First, we use the lift from Γ_h(t) to Γ(t) in order to define extensions of our finite element space which allows an error analysis of the discretization on the smooth surface Γ(t).

Second, since the numerical method is based on integration of finite element functions on Γ_h(t) we define approximations of the data,A, given on Γ(t) usingA−l.

Definition 2.1 (_{finite element spaces}). _{For each t and t}n _{we define the finite}

element spaces

S_h(t) =φ_h(·, t)∈C0(Γ_h(t))|φ_h(·, t)|_E is linear affine for each E(t)∈ T_h(t), S_hl(t) =ϕ_h(·, t) =φ_h(·, t)l|φ_h(·, t)∈S_h(t),

S_hn=S_h(tn), S_hn,l=Sl_h(tn).

Remark 2.2. For eachϕ_h ∈Sl

h

ϕn_h ∈S_hn,l there is a uniqueφ_h ∈S_hφn_h ∈S_hn

such thatϕ_h=φl_h(ϕn_h=φn,l_h ).

We denote by {χ_j(·, t)}J_j=1 the piecewise linear basis functions from S_h(t) such thatχ_j(X_i(t), t) =δ_ij and observe that{χl_j(·, t)}J_j=1 form a basis ofS_hl(t) and

φ_h(·, t) =

J

j=1

φ_j(t)χ_j(·, t)∈S_h(t),

ϕ_h(·, t) =φl_h(·, t) =

J

j=1

φ_j(t)χl_j(·, t)∈S_hl(t).

Settingχn_j =χ_j(·, tn), χn,l_j =χl_j(·, tn) forj ∈ {1, . . . , J}, we use the notation

φn_h=

J

j=1

φn_jχn_j ∈S_hn, ϕn_h =φn,l_h =

J

j=1

φn_jχn,l_j ∈S_hn,l.

We will find it convenient to define, forφn_h andφn_h+1inS_hn andS_hn+1fort∈[tn, tn+1] andα= 0,1,

φn+α h (·, t) =

J

j=1

φn_j+αχ_j(·, t)∈S_h(t),

(2.9)

ϕn+α

h (·, t) = (φnh+α(·, t))l= J

j=1

φn_j+αχl_j(·, t)∈S_hl(t) (2.10)

as well as

φL_h(·, t) = (t

n+1_−t)

τ φ

n h(·) +

(t−tn)

τ φ

n+1

h (·),

ϕL_h(·, t) =(t

n+1_−t)

τ ϕ

n h(·) +

(t−tn)

τ ϕ

n+1

h (·).

2.2.3. Discrete material derivative. Associated with the moving triangulated surface Γ_h(t) we define two material velocitiesV_h on Γ_h and v_h on Γ. First, for X₀

on Γ_h0= Γ_h(0) we setX(t) on the surface Γ_h(t) by

˙

X(t) =

J

j=1

˙

X_j(t)χ_j(X(t), t), X(0) =X₀,

and defineV_h to be the material velocity

(2.11) V_h(x, t) =

J

j=1

˙

X_j(t)χ_j(x, t), x∈Γ_h(t).

It follows that V_h is an interpolation ofv on Γ_h(t) and we use it to define the finite element approximation. On the other hand, we wish to analyze the method on the smooth surface Γ(t), and it is natural to construct a discrete material velocity on Γ(t) which evolves the curvilinear trianglese(t). For eachX(t) on Γ_h(t) there is a unique

Y(t) =p(X(t), t)∈Γ(t),

˙

Y(t) = ∂p

∂t(X(t), t) +Vh(X(t), t)· ∇p(X(t), t),

and we set

(2.12) v_h(p(X(t), t), t) = ˙Y(t).

We observe the following:

– The edges ofe(t) are the projections onto Γ(t) of the edges ofE(t)⊂Γ_h(t) and evolve with the material velocityv_h(·, t).

– The discrete material velocity v_h is notthe interpolation of v in S_hl(t), which actually is given by

I_hv(p(x, t), t) =

J

j=1

˙

X_j(t)χl_j(p(x, t), t) =

J

j=1

˙

X_j(t)χ_j(x, t) =V_h(x, t)

forx∈Γ_h(t).

Given the discrete velocity field V_h ∈(S_h)m+1 and the associated velocity field

v_h on Γ(t), (2.12), we define discrete material derivativeson Γ_h(t) and Γ(t) element by element through the equations

∂_h•φ_h|_E(t)= (φ_ht+V_h· ∇φ_h)|_E(t), ∂_h•ϕ_h|_e(t)= (ϕ_ht+v_h· ∇ϕ_h)|_e(t).

It was shown in [4, 6] that the basis functions satisfy thetransport propertythat

(2.13) ∂_h•χ_j= 0, ∂_h•χl_j= 0,

and thus forφ_h∈S_h, φ_j(t) =φ_h(X_j(t), t),

(2.14) ∂•_hφ_h(·, t) =

J

j=1

˙

φ_j(t)χ_j(·, t)∈S_h(t),

and similarly forϕ_h=φl_h∈Sl_h,

(2.15) ∂_h•ϕ_h= (∂_h•φ_h)l=

J

j=1

˙

φ_jχl_j ∈S_hl.

We also have the estimate [6]

(2.16) |∂•ϕ_h−∂_h•ϕ_h| ≤ch2|∇_Γϕ_h| on Γ.

From (2.15) we see that the material derivative and lifting process commute.

Definition 2.3 (time discrete material derivative). Given φn

h∈Shn andφnh+1 ∈ S_hn+1 set

∂_h•φn_h =

J

j=1

∂_τφn_jχn_j ∈S_hn, ∂_h•ϕn_h =

J

j=1

∂_τφn_jχn,l_j ∈S_hn,l.

We observe that by definition the time discrete transport property of the basis func-tions hold:

∂•_hχn_j = 0, ∂_h•χn,l_j = 0,

and on [tn, tn+1],

∂_h•φn+α

h = 0, ∂h•ϕnh+α= 0

forα= 0,1. From the above we deduce

(2.17) φn+1

h (·, tn) =φnh+τ ∂h•φnh, ϕhn+1(·, tn) =ϕhn+τ ∂h•ϕnh.

2.2.4. Discrete bilinear forms. As discrete analogues of the bilinear forms (2.2), (2.3), and (2.4) we define forφ_h(·, t), W_h(·, t)∈S_h(t)

a_h(φ_h(·, t), W_h(·, t)) =

E(t)∈Th(t)

E(t)A

−l_{(·, t)∇}

Γhφh(·, t)· ∇ΓhWh(·, t), (2.18)

m_h(φ_h(·, t), W_h(·, t)) =

Γh(t)

φ_h(·, t)W_h(·, t),

(2.19)

g_h(V_h(·, t);φ_h(·, t), W_h(·, t)) =

Γh(t)

φ_h(·, t)W_h(·, t)∇_Γh·V_h(·, t).

(2.20)

We keep in mind that the forms explicitly depend onttoo as discussed in section 2.1 for the time continuous forms. We indicate this in the time discrete case by writing forφn_h, W_hn∈S_hn,a_h(φn_h, W_hn), etc.

We will use the notation

|φ_h|_h,n=φ_hL2_(Γh(tn₎₎=m_h(φn_h, φn_h)1/2, φ_h∈S_hn,

|ϕ|_n=ϕ_L2_(Γ(tn₎₎ =m(ϕn, ϕn)1/2, ϕ∈H1(Γ(tn)).

2.3. Fully discrete scheme. We now can write the fully discrete scheme in the following form.

Given U_h0 ∈ S_h0 find U_hn ∈ S_hn, n ∈ {1, . . . , N}, such that for all φn_h ∈ S_hn and

φn_h+1∈Sn_h+1 andn∈ {0, . . . , N−1},

(2.21) ∂_τm_h(U_hn, φn_h) +a_h(U_hn+1, φn_h+1) =m_h(U_hn, ∂_h•φn_h).

Let us discuss the matrix-vector form of this fully discrete scheme. Choosing

φn_h=χn_i in (2.21), it follows that this definition isequivalentto

(2.22) ∂_τm_h(U_hn, χn_i) +a_h(U_hn+1, χn_i+1) = 0

for alli= 1, . . . , J.

SettingM(t),Mn to be the time dependent mass matrices

M(t)_jk=

Γh(t)

χ_j(·, t)χ_k(·, t), Mn=M(tn)

(j, k= 1, . . . , J) andS(t),Sn to be the time dependent stiffness matrices

S(t)_jk=

Γh(t)

A−l_{(·, t)∇}

Γhχj(·, t)· ∇Γhχk(·, t), Sn=S(tn),

we arrive at the following simple version of the fully discrete finite element approxi-mation:

∂_τ(Mnαn) +Sn+1αn+1= 0.

Equivalently,

(2.23) Mn+1+τSn+1αn+1=Mnαn,

where

U_hn+1=

N

j=1

αn_j+1χn_j+1

has to be determined as the solution of the sparse system of linear equations (2.23). Since for eachn the mass matrixMn is uniformly positive definite and the stiffness matrix Sn is positive semidefinite, we get existence and uniqueness of the discrete finite element solution.

2.4. Error bound. In section 4 we will prove the following error estimate for the fully discrete scheme.

Theorem 2.4. _{Let ube a sufficiently smooth solution of (1.1)and assume that}

(2.24) u₀−u0_hL2_(Γ0)≤ch2.

Let uk_h=U_hkl (k= 0, . . . , N) be the lift of the solution of the fully discrete scheme

(2.21). Then for 0< τ ≤τ₀ and0< h≤h₀ we have the error bounds

(2.25) u(·, tn)−un_h2_L2_(Γ(tn₎₎+h2τ n

k=1

∇Γu(·, tk)− ∇_Γuk_hL22_(Γ(tk₎₎≤c(τ2+h4)

for all n∈ {0, . . . , N} with a constant c which is independent of τ andh. Note that

c, h₀, andτ₀ depend on the data of the problem.

3. Transport and Leibniz formulae.

3.1. Continuous in time transport and Leibniz formulae. The following formulae for the differentiation of time dependent surface integrals are calledtransport formulaeand are proved in [4,6]. We use the notation∂•Ato denote the tensor which is obtained by taking the material derivative ofAcomponentwise.

Lemma 3.1 (transport theorems on surfaces). Let M(t) be an evolving surface

with normal velocity v_N. Let v_T be a tangential velocity field on M(t). Let the

boundary ∂M(t)evolve with the velocity v :=v_N +v_T. Assume that f is a function such that all of the following quantities exist. Then

(3.1) d

dt

M(t)

f =

M(t)

∂•f+f∇_Γ·v.

We have forϕ, ψ∈H1(G_T),

(3.2) d

dtm(ϕ, ψ) =m(∂

•_{ϕ, ψ) +m(ϕ, ∂}•_{ψ) +g(v;ϕ, ψ)}

and

(3.3) d

dta(ϕ, ψ) =a(∂

•_{ϕ, ψ) +a(ϕ, ∂}•_{ψ) +b(v;ϕ, ψ)}

with the bilinear form

(3.4) b(v;ϕ, ψ) =

ΓB

(v)∇_Γϕ· ∇_Γψ.

With the deformation tensor D(v)_ij = ₂1n_k=1+1(A_ik(∇_Γ)_kv_j+A_jk(∇_Γ)_kv_i) (i, j= 1, . . . , n+ 1)and the tensor

(3.5) B(v) =∂•A+∇_Γ·vA −2D(v),

we have the formula

(3.6)

d dt

M(t)A∇Γ

f·∇_Γg=

M(t)

A∇Γ∂•f·∇_Γg+A∇_Γf·∇_Γ∂•g

+

M(t)B

(v)∇_Γf·∇_Γg.

Lemma 3.2 (transport theorems on triangulated surfaces). LetΓ_h(t)be an

evolv-ing admissible triangulation with material velocity V_h. Then

(3.7) d

dt

Γh(t) f =

Γh(t)

∂_h•f+f∇_Γh·V_h.

For φ∈S_h(t), W_h∈S_h(t),

d

dtmh(φ, Wh) =mh(∂

•

hφ, Wh) +m_h(φ, ∂_h•W_h) +g_h(V_h;φ, W_h),

(3.8)

d

dtah(φ, Wh) =ah(∂

•

hφ, Wh) +ah(φ, ∂•hWh) +bh(Vh;φ, Wh)

(3.9)

with the bilinear form

(3.10) b_h(V_h;φ, W_h) =

E(t)∈Th(t)

E(t)Bh

(V_h)∇_Γhφ· ∇_ΓhW_h,

where

Bh(Vh) =∂h•A−l+∇Γh·VhA−l−2Dh(Vh),

D_h(V_h)_ij= 1 2

n+1

k=1

A−l

ik (∇Γ)kVhj+Ajk−l(∇Γ)kVhi

, i, j= 1, . . . , n+ 1.

LetΓ(t)be an evolving surface decomposed into curved elementse(t)whose edges move with velocity v_h. Then

(3.11) d

dt

Γ(t)

f =

Γ(t)

∂_h•f +f∇_Γh·v_h.

For ϕ, w, ∂_h•ϕ, ∂_h•w∈H1(Γ(t)),

d

dtm(ϕ, w) =m(∂

•

hϕ, w) +m(ϕ, ∂h•w) +g(vh;ϕ, w),

(3.12)

d

dta(ϕ, w) =a(∂

•

hϕ, w) +a(ϕ, ∂h•w) +b(vh;ϕ, w)

(3.13)

Remark 3.3. From the smoothness of Γ and A and the fact that V_h is the

interpolant of the smooth velocityvwe have that

∇ΓhVhL∞(Γh)+Bh(Vh)L∞(Γh)≤c

uniformly in time.

3.2. Discrete in time transport and Leibniz formulae. In this section we find an adequate form for discrete in time transport.

For W_h(·, t) ∈ S_h(t) and w(·, t) ∈ H1(Γ(t)), (t ∈ [tn, tn+1]) we set (as usual)

W_hn =W_h(·, tn) ∈ S_hn and W_hn+1 =W_h(·, tn+1) ∈S_hn+1. Similarly w(·, tn) = wn ∈ H1(Γ(tn)) and w(·, tn+1) = wn+1 ∈ H1(Γ(tn+1)). For given φn_h+1 ∈ S_hn+1 and its lifted versionϕn_h+1=φ_hn+1,l∈S_hn+1,l we define

Lh(W_h, φn_h+1) =∂_τm_h(W_hn, φn_h)−m_h(W_hn, ∂_h•φn_h),

(3.14)

L(w, ϕn_h+1) =∂_τm(wn, ϕ_hn)−m(wn, ∂_h•ϕn_h).

(3.15)

This is motivated by the fact that according to the fully discrete scheme (2.21) we will have to work with quantities of the form (3.14). We observe that this quantity depends only on W_h and on φn_h+1 – and so the definition of L_h makes sense—since by Definition 2.3 we have

Lh(Wh, φnh+1) =

1

τ

m_h(W_hn+1, φn_h+1)−m_h(W_hn, φn_h)−1

τmh

⎛ ⎝W_hn,

J

j=1

(φn_j+1−φn_j)χn_j

⎞ ⎠

= 1

τ

⎛

⎝m_h(W_hn+1, φn_h+1)−m_h

⎛ ⎝W_hn,

J

j=1

φn_j+1χn_j

⎞ ⎠

⎞ ⎠.

A similar argument holds forL.

In the following lemma we will exploit the time continuous transport formula (3.2). For this we use the definitions (2.9) ofφn+1

h and (2.10) ofϕnh+1.

Lemma 3.4. _{Assume the situation defined above. Then}

Lh(Wh, φnh+1)

(3.16)

= 1

τ

_tn+1

tn

m_h(∂_h•W_h(·, t), φn+1

h (·, t)) +gh(Vh(·, t);Wh(·, t), φnh+1(·, t))dt

and

(3.17)

L(w, ϕn_h+1) = 1

τ

_tn+1

tn

m(∂_h•w(·, t), ϕn+1

h (·, t)) +g(vh(·, t);w(·, t), ϕnh+1(·, t))dt.

Proof. Rewriting (3.14) we find that

Lh(Wh, φnh+1) =

1

τ

m_h(W_h(·, tn+1), φn+1

h (·, tn+1))−mh(Wh(·, tn), φnh+1(·, tn))

,

and using the transport formula (3.7) and the fact that∂_h•φ

h= 0 we obtain the desired

equation (3.16). Equation (3.17) follows by a similar calculation.

Lemma 3.5. _{For Wh(}·_{, t)}∈_Sh(t)andwh(·_{, t)}∈_Sl

h(t),(t∈[tn, tn+1]), we have

∂_τm_h(W_hn, W_hn)−m_h(W_hn, ∂_h•W_hn) = 1

2∂τmh(W

n

h, Whn) +

1 2τ mh(∂

•

hWhn, ∂h•Whn)

+ 1 2τ

m_h(W_hn+1, W_hn+1)−m_h(W_hn+1(·, tn), Wn_h+1(·, tn)),

∂_τm(wn_h, wn_h)−m(wn_h, ∂_h•wn_h) = 1 2∂τm(w

n h, wnh) +

1 2τ m(∂

•

hwnh, ∂h•wnh)

+ 1 2τ

m(wn_h+1, wn_h+1)−m(w_hn+1(·, tn), w_hn+1(·, tn)).

Proof. These identities follow by rearranging the terms and using the equations (2.17).

3.3. Properties of the mass and stiffness matrices. For later use we have to control the time derivative of the mass and stiffness matrices and the related bilinear forms.

Lemma 3.6. For the continuous and the discrete bilinear forms we have the

estimates

|m_h(φn_h+1, φ_hn+1)−m_h(φn+1

h (·, t), φnh+1(·, t))| ≤c τ mh(φnh+1, φnh+1),

(3.18)

|a_h(φn_h+1, φ_hn+1)−a_h(φn+1

h (·, t), φnh+1(·, t))| ≤c τ ah(φnh+1, φnh+1),

(3.19)

|m(ϕn_h+1, ϕ_hn+1)−m(ϕn+1

h (·, t), ϕnh+1(·, t))| ≤c τ m(ϕnh+1, ϕnh+1)

(3.20)

for t∈[tn, tn+1] and

m_h(W_hn+1, W_hn+1)−m_h(W_hn+τ ∂_h•W_hn, W_hn+τ ∂_h•W_hn)

≤cτ m_h(W_hn+1, W_hn+1),

(3.21)

m(wn_h+1, wn_h+1)−m(wn_h+τ ∂_h•w_hn, wn_h+τ ∂_h•wn_h)≤cτ m(wn_h+1, wn_h+1) (3.22)

if τ≤τ₀ with a time step size τ₀ which depends on the data of the problem. We also have for t∈[tn, tn+1]

φn+1

h (·, t)L2(Γh(t))≤c|φnh+1|n+1,h,ϕn_h+1(·, t)L2(Γ(t)) ≤c|ϕnh+1|n+1, (3.23)

∇Γφn_h+1(·, t)L2_(Γh(t)) ≤c|∇_Γφn_h+1|_n+1,h,

(3.24)

∇Γϕn_h+1(·, t)L2_(Γ(t)) ≤c|∇_Γϕn_h+1|_n+1.

Proof. Because ofφn+1

h (·, tn+1) =φhn+1 and∂h•φnh+1= 0 we have

m_h(φn_h+1, φ_hn+1)−m_h(φn+1

h (·, t), φnh+1(·, t))

=m_h(φn+1

h (·, tn+1), φhn+1(·, tn+1))−mh(φnh+1(·, t), φnh+1(·, t))

=

_tn+1

t d dsmh(φ

n+1

h (·, s), φnh+1(·, s))ds=

_tn+1

t

Γh(s)

∇Γh·Vh(·, s)φn_h+1(·, s)2ds

≥ −c

_tn+1

t

m_h(φn+1

h (·, s), φnh+1(·, s))ds.

For the nonnegative functionf(s) =m_h(φn+1

h (·, s), φnh+1(·, s)) this means

f(tn+1)−f(t)≥ −c

tn+1

t

f(s)ds.

A standard Gronwall argument and the assumptionτ ≤τ₀ lead to the estimate

f(tn+1)−f(t)≥ −cτ f˜ (tn+1).

This proves the estimate (3.18). The same argument holds for (3.20).

The estimates (3.21) and (3.22) now follow from (2.17) applied to φ_h = W_h

and ϕ_h =w_h. Finally, we observe that a similar argument may be used to obtain (3.19).

We add two estimates which will be used in the next section.

Lemma 3.7. _{Assume that ∂}•

hφh= 0and ∂•_hψ_h= 0. Then we have the estimates m_h(φ_h(·, tk+1), ψ_h(·, tk+1)−m_h(φ_h(·, tk), ψ_h(·, tk)

(3.25)

≤c

_tk+1

tk

m_h(φ_h(·, t), φ_h(·, t))12m_h(ψ_h(·, t), ψ_h(·, t))12dt,

a_h(φ_h(·, tk+1), φ_h(·, tk+1))−a_h(φ_h(·, tk), φ_h(·, tk)) (3.26)

≤c

tk+1

tk

a_h(φ_h(·, t), φ_h(·, t))dt.

Proof. We use the transport formula (3.7) and have

m_h(φ_h(·, tk+1), ψ_h(·, tk+1))−m_h(φ_h(·, tk), ψ_h(·, tk))

=

_tk+1

tk

g_h(V_h(·, t);φ_h(·, t)ψ_h(·, t))dt≤c

_tk+1

tk

φ_h(·, t)_L2_(Γh(t))ψ_h(·, t)_L2_(Γh(t))dt.

For the second estimate of the lemma we use the transport formula (3.9) and get

a_h(φ_h(·, tk+1), φ_h(·, tk+1))−a_h(φ_h(·, tk), φ_h(·, tk))

=

_tk+1

tk

b_h(V_h(·, t);φ(·, t)φ(·, t))dt≤c

_tk+1

tk ∇Γh(t)

φ_h(·, t)2_L2_(Γh)dt

≤c

tk+1

tk

a_h(φ_h(·, t), φ_h(·, t))dt,

and the lemma is proved.

4. Proof of the error bound.

4.1. Stability. We begin with a stability result which is the fully discrete ana-logue of the continuous estimates (2.6) and (2.7).

Lemma 4.1. The fully discrete solutionUk

h (k= 0, . . . , N)satisfies the following

a priori bounds for τ≤τ₀:

|U_hn|2_h,n+τ c₀ n

j=1

|∇ΓhUhk|2h,k≤c|Uh0|2h,0, (4.1)

|un_h|2_n+τ c₀ n

j=1

|∇Γuk_h|2_k≤c|u0_h|2₀,

(4.2)

τ n−1

k=0

|∂_h•U_hL(·, tk+1−0)|2_h,k+c₀|∇_ΓhU_hn|2_h,n≤c|U_h0|2_h,0+|∇_ΓhU_h0|2_h,0,

(4.3)

τ n−1

k=0

|∂_h•uL_h(·, tk+1−0)|2_k+c₀|∇_Γu_hn|2_n≤c|u0_h|2₀+|∇_Γu0_h|2₀.

The constants depend on the data of the problem including the final time T.

Proof. Using Lemma 3.5, the finite element equation (2.21), withφ_h =U_h, can be written as

1

2∂τmh(U

n h, Uhn) +

1 2τ mh(∂

•

hUhn, ∂h•Uhn) +ah(Uhn+1, Uhn+1)

=−1 2τ

m_h(U_hn+1, U_hn+1)−m_h(Un_h+1, Un_h+1).

We omit one positive term on the left-hand side and estimate the right-hand side with (3.18) to get

1 2τ

|U_hn+1|2_h,n+1− |U_hn|2_h,n+c₀|∇_ΓhU_hn+1|2_h,n+1≤c|U_hn+1|2_h,n+1,

from which we obtain (4.1) by a discrete Gronwall argument.

The bound (4.2) follows by norm and seminorm equivalence for|φn_h|_h,n and|ϕn_h|_n

and|∇_Γhφn_h|_h,n and|∇_Γϕn_h|_n; see Lemma 5.2 in [4].

We derive the next bound using the matrix form of the discretization. Suppose that with vectorsw, v∈RJ we have

Mn+1_{w− M}n_v+τSn+1_{w= 0.}

Then multiplying byw−v and rearranging yields

Mn+1_{(w−v)·(w−v) +}τ

2

Sn+1_{w·w− S}n_v·v+τ

2S

n+1_{(w−v)·(w−v)}

= (Mn− Mn+1)v·(w−v) +τ 2(S

n+1_{− S}n_)v·v.

We choosev_j=U_jk andw_j=U_jk+1 (j= 1, . . . , J) and find withUm= (U₁m, . . . , U_Jm) form=k+ 1, k that

Mk+1_(Uk+1_−Uk_)·(Uk+1_−Uk_{) +}τ

2

Sk+1_Uk+1_·Uk+1_{− S}k_Uk_·Uk

=−τ 2S

k+1_(Uk+1_−Uk_)·(Uk+1_−Uk₎

−(Mk+1− Mk)(Uk+1−Uk)·Uk+τ 2(S

k+1_{− S}k_)Uk_·Uk_.

If we rewrite this equation using the bilinear forms, then we arrive at

τ2m_h(∂_h•U_hL(·, tk+1−0), ∂_h•U_hL(·, tk+1−0)) +τ 2

a_h(U_hk+1, U_hk+1)−a_h(U_hk, U_hk)

=−τ

3

2 ah(∂

•

hUhL(·, tk+1−0), ∂h•UhL(·, tk+1−0))

−τ

m_h(∂_h•U_hL(·, tk+1−0), Uk_h(·, tk+1))−m_h(∂_h•U_hL(·, tk+ 0), Uk_h(·, tk))

+τ 2

a_h(Uk_h(·, tk+1), Uk_h(·, tk+1))−a_h(Uk_h(·, tk), Uk_h(·, tk))

.

After dropping the first term, using (3.25) and (3.26) together with (3.23) and (3.24), the right-hand side of this equation can be estimated from above by

1 2τ

2_m

h(∂h•UhL(·, tk+1−0), ∂h•UhL(·, tk+1−0)) +cτ2mh(Uhk, Uhk) +cτ2|∇ΓUhk+1|2h,k+1.

Summation overkfrom 0 ton−1 and division byτ leads us to the estimate

τ n−1

k=0

m_h(∂_h•U_hL(·, tk+1−0), ∂_h•U_hL(·, tk+1−0) +a_h(U_hn, U_hn)−a_h(U_h0, U_h0)

≤cτ n−1

k=0

m_h(U_hk, U_hk) +cτ n−1

k=0

|∇ΓhUhk+1|2h,k+1.

But this gives the estimate

τ n−1

k=0

|∂_h•U_hL(·, tk+1−0)|2_h,k+1+c₀|∇_ΓhU_hn|2_h,n

≤c|∇_ΓhU_h0|2_h,0+cτ n−1

k=0

|∇ΓhUhk+1|2h,k+1+cτ

n−1

k=0

|U_hk|2_h,k.

We use the a priori estimate (4.1), and the estimate (4.3) is proved.

4.2. Error decomposition and error equation. Setting (R_hu)(·, t) to be the Ritz projection (see (A.6)) andρ(·, t) =u(·, t)−(R_hu)(·, t) it is convenient to introduce the error decomposition

u(·, tn)−un_h=ρn+θn, ρn=ρ(·, tn), θn= (R_hu)(·, tn)−un_h∈S_hl,n.

Lemma 4.2. The finite element solution satisfies the error relation

(4.4) L(θ, ϕn_h+1) +a(θn+1, ϕn_h+1) =F₂(ϕ_hn+1)−F₁(ϕn_h+1) ∀ϕn_h+1∈S_hn+1,l,

where

F₁(ϕn_h+1) =L(u_h, ϕ_hn+1)− L_h(U_h, φn_h+1)

+a(un_h+1, ϕn_h+1)−a_h(U_hn+1, φn_h+1),

F₂(ϕn_h+1) =−L(ρ, ϕn_h+1) +L(u, ϕn_h+1) +a(un+1, ϕn_h+1).

Proof. We begin by rewriting the finite element equation (2.21) on the time interval [tn, tn+1] as

L(u_h, ϕn_h+1) +a(u_hn+1, ϕn_h+1) =F₁(ϕn_h+1)

On the other hand, using the definition (A.6) of the Ritz projection,

L(R_hu, ϕn_h+1) +a(R_hun+1, ϕ_hn+1) =F₂(ϕn_h+1) ∀ϕ_h∈Sl_h.

Taking the difference of these two equations gives the desired result.

Equation (4.4) is the basis of the error bound. We proceed by boundingF₁(ϕn_h+1) andF₂(ϕn_h+1).

Lemma 4.3. _{For every >0we have the estimate}

|F₁(ϕn_h+1)| ≤c()h

4

τ

_tn+1

tn

∂_h•uL_h2_L2_(Γ)+uL_h2_H1_(Γ)dt+c()h4|∇Γun_h+1|2_n+1

+|∇_Γϕn_h+1|2_n+1+c|ϕn_h+1|2_n+1.

Proof. It is convenient to write

F₁(ϕn_h+1) =L(u_h, φ_hn+1)− L_h(U_h, φn_h+1)

+a(un_h+1, ϕn_h+1)−a_h(U_hn+1, φn_h+1)=I+II.

We observe that L_h(U_h, φ_hn+1) = L_h(U_hL, φ_hn+1), L(u_h, ϕn_h+1) = L(uL_h, ϕn_h+1), and

uL_h = (U_hL)l. Using (3.17) and (3.16) and the estimates (A.1) and (A.3) we find

|I| ≤ch

2

τ

_tn+1

tn

∂_h•uL_h(·, t)_L2_(Γ)ϕn+1

h (·, t)L2(Γ)+uLhH1(Γ)ϕnh+1H1(Γ)dt

≤ε|∇_Γϕn_h+1|2_n+1+c(ε)h

4

τ

_tn+1

tn

∂_h•uL_hL22_(Γ)+uL_h2_H1_(Γ)dt+c|ϕn_h+1|2_n+1,

where we have used (3.23) and (3.24). By (A.2) we have that

|II| ≤ch2|∇_Γun_h+1|_n+1|∇_Γϕn_h+1|_n+1≤ε|∇_Γϕn_h+1|2_n+1+c(ε)h4|∇_Γun_h+1|2_n+1.

The result is proved.

Lemma 4.4. For F₂ we have the estimate

|F₂(ϕn_h+1)| ≤ch

4

τ

_tn+1

tn

u2_H2_(Γ)+∂•u2_H2_(Γ)dt

+c()τ

_tn+1

tn u

2

H1_(Γ)+∂•u_H21_(Γ)dt+|∇ϕ_hn+1|2_n+1+c|ϕn_h+1|2_n+1

for arbitrary >0.

Proof. It follows from the variational form (2.5) that

m(u(·, tn+1), ϕn+1

h (·, tn+1))−m(u(·, tn), ϕnh+1(·, tn)) +

_tn+1

tn

a(u(·, t), ϕn+1 h (·, t))dt

=

_tn+1

tn

m(u(·, t), ∂•ϕn+1

h (·, t))dt.

Sinceϕn+1

h (·, tn+1) =ϕnh+1 andϕnh+1(·, tn) =ϕnh+τ ∂h•ϕnh we find that

m(u(·, tn+1), ϕn_h+1)−m(u(·, tn), ϕn_h) +

_tn+1

tn

a(u(·, t), ϕn+1 h (·, t))dt

=τ m(u(·, tn), ∂•_hϕn_h) +

_tn+1

tn

m(u(·, t), ∂•ϕn+1

h (·, t))dt.

Hence

L(u, ϕn_h+1) +1

τ

_tn+1

tn

a(u(·, t), ϕn+1

h (·, t))dt=

1

τ

_tn+1

tn

m(u(·, t), ∂•ϕn+1

h (·, t))dt,

from which we deduce that

F₂(ϕn_h+1) =−L(ρ, ϕn_h+1) +1

τ

_tn+1

tn

a(u(·, tn+1), ϕ_hn+1)−a(u(·, t), ϕn+1 h (·, t))dt

+1

τ

_tn+1

tn

m(u(·, t), ∂•ϕn+1

h (·, t))dt=I+II+III.

It follows from (3.17) that

|I| ≤ c τ

_tn+1

tn

∂_h•ρ(·, t)_L2_(Γ)+ρ(·, t)_L2_(Γ)ϕn+1

h (·, t)L2(Γ)dt

and from applying the bounds (A.4), (A.8), (A.7) and (3.23) that

|I| ≤ch

4

τ

_tn+1

tn

u2_H2_(Γ)+∂•u2_H2_(Γ)dt+c|ϕn_h+1|2_n+1.

Using (3.3) we find that

|II|=1

τ

_tn+1

tn

_tn+1

t

a(∂_h•u(·, s), ϕn+1

h (·, s)) +b(vh(·, s);u(·, s), ϕnh+1(·, s))dsdt

≤c

_tn+1

tn

u(·, t)_H1_(Γ)+∂_h•u(·, t)_H1_(Γ)∇_Γϕn+1

h (·, t)L2(Γ)dt

≤c()τ

_tn+1

tn

u2_H2_(Γ)+∂•u2_H1_(Γ)dt+|∇Γϕn_h+1|2_n+1.

For termIII we use the fact that∂_h•ϕn+1

h = 0 and get with (2.16)

|III| ≤ε|∇_Γϕn_h+1|2_n+1+c(ε)h

4

τ

_tn+1

tn

u(·, t)2_L2_(Γ)dt.

The estimates forI,II, andIII then imply the lemma.

4.3. Proof of Theorem 2.4. We insertϕ_h=θ in (4.4) and get

∂_τm(θn, θn)−m(θn, ∂_h•θn) +a(θn+1, θn+1) =F₂(θn+1)−F₁(θn+1).

We use Lemma 3.5 together with (3.20) and get

1 2τm(θ

n+1_{, θ}n+1₎₋ 1

2τm(θ

n_{, θ}n_{)−c m(θ}n+1_{, θ}n+1_{) +a(θ}n+1_{, θ}n+1₎

≤ |F₂(θn+1)|+|F₁(θn+1)|,

and the coercivity ofathen leads to

1 2τ

|θn+1|2_n+1− |θn|_n2+c₀|∇_Γθn+1|2_n+1

(4.5)

≤c|θn+1|2_n+1+|F₂(θn+1)|+|F₁(θn+1)|.

Using the estimates for F₁ and F₂ from Lemmas 4.3 and 4.4 and summing we find, after a suitable choice ofε >0,

|θn|2_n+c₀τ n

k=1

|∇Γθk|2k ≤ |θ0|20+cτ

n

k=1 |θk|2_k

+ch4

_tn

0

u2_H2_(Γ)+∂•u2_H2_(Γ)+∂_h•uL_h2_L2_(Γ)+uL_h2_H1_(Γ)dt

+ch4τ n

k=1

|∇Γukh|2k+cτ2

_tn

0

u2_H2_(Γ)+∂•u2_H1_(Γ)dt.

The theorem now follows by a Gronwall argument, the error decomposition together with the bounds for the Ritz projection, and assumption (2.24) on the initial condition together with our stability estimates from Lemma 4.1.

Appendix. Approximation results. The results in this section are taken from [6].

A.1. Geometric perturbation errors. We begin with the bounding of the geometric perturbation errors in the bilinear forms.

Lemma A.1. _{For any (Wh(}·_{, t), φh(}·_{, t))}∈_Sh(t)×_{Sh(t)with corresponding lifts}

(w_h(·, t), ϕ_h(·, t))∈S_hl(t)×Sl_h(t) the following bounds hold:

|m(w_h, ϕ_h)−m_h(W_h, φ_h)| ≤ch2w_hL2_(Γ(t))ϕ_hL2_(Γ(t)),

(A.1)

|a(w_h, ϕ_h)−a_h(W_h, φ_h)| ≤ch2∇_Γw_hL2_(Γ(t))∇_Γϕ_hL2_(Γ(t)),

(A.2)

|g(v;w_h, ϕ_h)−g_h(V_h;W_h, φ_h)| ≤ch2w_hH1_(Γ(t))ϕ_hH1_(Γ(t)).

(A.3)

Note that (A.3) holds withv replaced byv_h.

A.2. Approximation errors. The following result gives estimates for the ap-proximation of the continuous material derivative by the discrete material derivative.

Lemma A.2. The following bounds hold for the material derivatives on Γ(t):

∂•z−∂_h•z_L2_(Γ)≤ch2z_H1_(Γ), z∈H1(Γ),

(A.4)

∇Γ(∂•z−∂h•z)L2_(Γ)≤chz_H2_(Γ), z∈H2(Γ).

(A.5)

It is convenient in the error analysis to use the Ritz projectionR_h:H1(Γ)→S_hl

defined as follows: Forz∈H1(Γ),_Γz= 0,

(A.6) a(R_hz, ϕ_h) =a(z, ϕ_h) ∀ϕ_h∈S_hl

and_Γ

hRhz= 0.

Lemma A.3. The error in the Ritz projection satisfies the bounds

z− R_hz_L2_(Γ)+h∇_Γ(z− R_hz)_L2_(Γ)≤ch2z_H2_(Γ),

(A.7)

∂•z−∂•R_hz_L2_(Γ)+h∇_Γ(∂•z−∂•R_hz)_L2_(Γ)

(A.8)

≤ch2z_H2_(Γ)+∂•z_H2_(Γ)

if h≤h₀.