Error estimates for a semi-implicit fully discrete finite element scheme for the mean curvature flow of graphs

Transcription

1 Interfaces and Free Boundaries 2, Error estimates for a semi-implicit fully discrete finite element sceme for te mean curvature flow of graps KLAUS DECKELNICK Scool of Matematical Sciences, University of Sussex, Falmer, Brigton BN 9QH, UK AND GERHARD DZIUK Institut für Angewandte Matematik, Universität Freiburg, Hermann Herder Str.0, 7904 Freiburg, Germany [Received 7 April 999 and in revised form 3 April 2000] Te efficient numerical simulation of te curvature-driven motion of interfaces is an important tool in several free- boundary problems. We treat te case of an interface wic is given as a grap. Te igly non-linear problem is discretized in space by piecewise linear finite elements. Altoug te problem is not in divergence form it can be written in a variational form wic allows te use of te modern adaptive tecniques of finite elements. Te time discretization is carried out in a semiimplicit way suc tat in every time step a linear system wit symmetric positive matrix as to be solved. Optimal error estimates are proved for te fully discrete problem under te assumption tat te time-step size is bounded by te spatial grid size.. Introduction Te aim of tis paper is to analyse a fully discrete finite element algoritm tat approximates te mean curvature flow of graps. A family Γ t t [0,T ] of n-dimensional surfaces in R n is said to flow by mean curvature if te normal velocity V of Γ t equals its mean curvature. We sall restrict our attention to two-dimensional surfaces wic can be written as graps over some bounded, smoot domain R 2, i.e. Γ t ={x, ux, t x }. Abbreviating by defining Qu = u 2,. te downward pointing normal νu to Γ t is given by u νu = Qu, Qu wile normal velocity and mean curvature wit respect to νu are calculated according to u t V = Qu, H = u Qu..3 Tus, te relation V = H translates into te quasi-linear partial differential equation u t Qu.2 u = 0 Qu in 0, T.4 c Oxford University Press 2000

2 342 K. DECKELNICK AND G. DZIUK to wic we add te following boundary and initial conditions: u, t = u 0 on 0, T u, 0 = u 0 in..5 Here, u 0 : R is a given smoot function. If te mean curvature of is non-positive wit respect to te exterior normal it was sown in [] and [9] tat.4,.5 as a global smoot solution. In general, owever, smoot solutions exist only locally in time and te gradient can blow up at te boundary; cf. [3]. Results for Neumann-type boundary conditions can be found in [, 2, 9]. Te differential equation.4 is not in divergence form and so we could expect tat a numerical metod like te finite element metod, wic is based on a variational formulation, migt not be applicable ere. But it is easy to see tat te energy equality u 2 t Qu d Qu = 0.6 dt olds for time-independent boundary values and tis will lead us to a variational form of our problem. It will be possible to discretize te equation in space wit piecewise linear finite elements and tis means tat te resulting sceme will be open to te use of modern adaptive metods. An error estimate for a semi-discrete sceme of tis form as been proved in [8] for te isotropic and in [7] for te anisotropic mean curvature flow of graps. In tis work we will derive a fully discrete sceme for.4 in wic te discretization is suc tat it linearizes te problem by a semi-implicit coice of te time discretization. Under te assumption tat τ δ 0, were τ is te time-step size and is te grid size in space, we will prove optimal asymptotic error estimates of te form sup u m u m H cτ. 0 m M Here we used te common sortand v m x = vx, mτ. Te main idea for te proof will be a form of te energy identity.6 togeter wit a superconvergence result for a non-linear Ritz Projection, wic we call te minimal surface projection. An important fact will be te adequate use of geometric quantities. We complete te paper wit some numerical tests wic confirm te results of te error estimates precisely. Tere are of course oter approaces to studying motion by mean curvature wic avoid te restriction tat Γ t as to be a grap. Let us briefly review some of te corresponding work, concentrating on fully discrete algoritms and teir numerical analysis. Numerical scemes for te level set approac were introduced by Oser & Setian [4], wile Walkington [5] proposes a finite element algoritm and studies its stability. Crandall & Lions [4] introduced a monotone, convergent finite difference sceme wic uses a regularization of te level set equation. An error estimate for tis sceme is proved in [5]. A furter possibility of approximating mean curvature motion is via te Allen Can equation, a singularly perturbed parabolic equation. Convergence results for fully discrete algoritms are proved in [3] and [2].

3 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS Discretization in space and time In order to derive our numerical algoritm we first rewrite.4 in te variational form u t ϕ Qu u ϕ = 0, ϕ H0, 0 t T 2. Qu togeter wit.5. Te fact, tat.4 is not in divergence form is reflected in 2. by te appearance of a denominator in te first integral wic complicates te analysis. Next, let us consider a family T of triangulations of allowing te boundary elements to ave a curved face wit maximum mes size := max S T diams for wic we assume te following regularity condition: tere exists a constant κ>0 wic is independent of suc tat eac S T is contained in a ball of radius κ and contains a ball of radius κ. Te discrete space is cosen to be X := {v C 0 v is a linear polynomial on eac S T } were te isoparametric modification is used in curved elements; see [6]. We also set X 0 := X H0. Furtermore, we denote by τ>0te time step and let M = T/τ. Recalling 2., it is natural to introduce te following sceme. ALGORITHM For u 0 = uˆ 0 find u m X, 0 m M suc tat u m I u 0 X 0 and u m u m ϕ u m ϕ τ Qu m Qu m = 0 for all ϕ X Here, I is te usual Lagrange interpolation operator and uˆ 0 denotes te minimal surface projection of u 0 wic will be defined below. Te above sceme is semi-implicit and requires te solution of a linear equation in eac time step. Furtermore, te following stability estimate olds. PROPOSITION Te solution u m, 0 m M of 2.2 satisfies for every m {,...,M} m τ V k 2 Qu k m k=0 k=0 m 2 k=0 Qu k Qu k 2 νu k νu k 2 Qu k Qu k Qu m = Qu were V k = uk u k /τ Qu k is te discrete normal velocity. Proof. See [8], Teorem 2. Note tat te above stability estimate is true witout any restriction on te time step τ. Te purpose of te present work is to analyse te convergence of te sceme 2.2. To formulate our result we define u m := u, t m, t m = mτ,0 m M. Ten we ave te following.

4 344 K. DECKELNICK AND G. DZIUK THEOREM Let u be a smoot solution of.4,.5 and u m exists δ 0 > 0 suc tat sup u m u m cτ 2 log 2 0 m M sup u m u m cτ 0 m M provided τ δ 0. Here, denotes te L 2 norm. te solution of 2.2. Ten tere Te precise regularity assumptions on u appear in Proposition 3 below. Altoug we formulated our error bounds in terms of te usual L 2 -norms, we sall see tat it is muc more appropriate to work wit te geometric quantities νu and Qu. Tis is not entirely surprising as we deal wit a geometric problem but it is interesting to see ow te use of tese quantities can simplify te analysis or even make it possible. Tus, rater tan trying to estimate u m u m we sall focus on te quantity E m := νu m νûm 2 Qu m. 2.4 Here, û m denotes te minimal surface projection of um to be defined below. Te main part of te proof of Teorem ten consists in deriving te estimate E m cτ 2 4 log 4, 0 m M. Note tat tere is superconvergence in as we obtain 4 log 4 rater tan 2 and tis effect will be crucial for our argument. 3. Error estimates Let us start wit some useful relations involving te geometric quantities Q and ν, wic will be used frequently trougout te paper. PROPOSITION 2 Let u and v be in H,. Ten we ave a.e. in : Qu Qv QuQv νu νv 3. v u sup v Qu νu νv, 3.2 v Qv Qu2 v u = Qv Qu Qu 2Qu v u 2, 3.3 2Qu u v 2 = Qu Qv 2 νu νv 2 QuQv. 3.4 Proof. Te first relation follows from te fact tat Qu, are te last components of Qv νu, νv respectively. Te second inequality is a consequence of v v u = Qu Qv u Qu Qu Qu v. Qv Finally, 3.3 and 3.4 follow from elementary calculations.

5 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 345 Before we start wit te proof of Teorem we introduce an important tool for te analysis. For a given function v H 2 H, we define te minimal surface projection ˆv X by te relations I v ˆv X 0 and v ϕ Qv = ˆv ϕ Q ˆv for all ϕ X 0. Existence and uniqueness of ˆv, togeter wit an error analysis, were establised in [0]. For functions wic also depend on time te following bounds are available. PROPOSITION 3 Let u L 0, T ; H 4 L 2 0, T ; H 5 wit u t L 0, T ; H 2 L 2 0, T ; H 3, u t L 0, T, u tt L 0, T ; L 2 L 2 0, T ; H. Ten te error between u and û can be estimated as follows: sup u û sup u û c 2, 3.5 0,T 0,T sup u û L sup u û L c 2 log, 3.6 0,T 0,T sup u t û,t c 2 log 2, 3.7 0,T sup u t û,t c ,T Proof. Te proof is carried out in [6] for te case of zero boundary values of u, but it can be extended to inomogeneous boundary values in a straigtforward way. As for 3.8 we note tat T in [6] te weaker estimate u t û,t 2 c 2 is proved wic was a consequence of 0 see [6], p. 202 u t û,t 2 c u t û,t u t L u t H 2 c 2 u t L u t H 2. However, our assumption u t L 0, T wic was not made in [6] implies te stronger bound 3.8. We sall keep te above assumptions concerning te regularity of u trougout tis work. Using te minimal surface projection and evaluating 2. at t = t m we obtain u t, t m ϕ Qu m û m ϕ Qû m = 0 for all ϕ X 0 and terefore u m u m ϕ û m ϕ τ Qu m Qû m = τ R m ϕ Qu m û m û m ϕ Qû m 3.9 were R m cτ 2, 0 m M. 3.0

6 346 K. DECKELNICK AND G. DZIUK Let us decompose te error e m = u m u m in te following way: e m = u m û m ûm um =: ɛm e m. Te difference between te continuous 3.9 and te discrete 2.2 equation evaluated at a discrete test function ten gives u m u m τ Qu m = τ R m ϕ Qu m um u m Qu m û m û m ϕ û m ϕ Qû m Qû m um Qu m ϕ for all ϕ X 0. Te weigts /Q witin te integrals will be important for our reasoning and so we will use te following form of te above equation: e m e m ϕ τ Qu m = u m u m τ We coose û m Qû m um Qu m ϕ = e m Qu m Qu m as a test function and get e m e m 2 τ 2 Qu m τ = e m e m ɛ m ɛ m τ 2 Qu m τ 2 ϕ τ ϕ 3. e m /τ = em e m /τ ɛ m ɛ m /τ û m Qû m um Qu m τ û m u m u m e m e m Qu m Qu m Before we start our estimates we prove a useful lemma. R m ϕ Qu m û m û m ϕ Qû m. e m e m û m em e m Qû m 3.2 τ 2 LEMMA Suppose tat 2 E m γ 2 for some γ. Ten we ave R m e m e m Qu m. sup Qu m c, 3.3 e m 2 ce m, 3.4 e m L cγ. 3.5 Proof. We infer from 3.6 tat û m is bounded uniformly in m and. Tus, an inverse estimate

7 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 347 togeter wit 3.2 implies sup Qu m um L c um ûm L c c u m ûm c c c c sup Qu m E m 2 sup Qu m c, νu m νûm 2 Qu m 2 2 wic implies 3.3. To see 3.4, note tat by 3.2 and 3.3 e m 2 = u m ûm 2 c νu m νûm 2 Qu m 2 ce m. Finally, 3.5 again follows from an inverse estimate. In wat follows we sall assume tat te condition 2 E m γ is satisfied so tat te results of Lemma are available. We will justify tis assumption at te end of our proof witin an induction argument. Let us now return to 3.2. Te crucial term is te second one on te left-and side wic we sall bound from below by a suitable discrete time derivative minus some error terms. Since te corresponding calculations are quite tecnical we sketc te argument in te continuous case for te convenience of te reader. For two functions v and w wic are smoot enoug one can sow te following inequality: v Qv w v t w t = d QvQw v w Qw dt Qv w v t Qv w Qw v v w v Qv Qv 2 Qv d νv νw 2 Qw v t νv νw 2 Qw. 2 dt Tis relation translates into te time-discrete setting as follows. LEMMA 2 For sufficiently small γ we ave û m τ Qû m um Qu m e m e m 2τ Em E m e m e m 2 4τ Qû m ce m E m cτ 2. Proof. Let us denote te integral on te left-and side by L. Clearly, L = τ û m Qû m um Qu m û m û m um u m.

8 348 K. DECKELNICK AND G. DZIUK From 3.3 and te trivial identity 2 νu u v νv 2 = QuQv 3.7 we infer τ L = = Qû m Qû m Qum Qû m Qû m 2 2Qû m û m û m 2 2Qû m Qû m Qu m Qu m Qum Qu m 2 2Qu m um u m 2 2Qu m û m u m 2 νûm νu m 2 Qu m Qû m Qû m 2 2Qû m û m û m 2 2Qû m û m u m Qû m 2 νûm νum 2 Qu m Qum Qu m 2 2Qu m um u m 2 2Qu m um û m Qu m S m 3.8 were S m = Qû m Qû m ûm u m Qû m = Qû m Qû m ûm u m ûm um Qû m Qû m Qu m û m u m ûm u m Qû m um û m Qu m Qû m Qu m u m û m Qû m Qu m ûm û m um u m Qû m.

9 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 349 Using again 3.7 tis term can be rewritten as follows: S m = 2 νûm νu m 2 Qu m Qûm Qu m 2 νum νû m 2 Qu m Qûm Qu m Qû m Qû m ûm û m um u m Qû m = 2 νûm νu m 2 Qu m Qûm Qu m 2 νum νû m 2 Qu m Qûm Qu m Qûm Qû m Qum Qu m Qu m ûm û m um u m Qû m. Inserting te above expression into 3.8 we obtain 2τ L = νû m νu m 2 Qu m νû m νum 2 Qu m Qû m Qû m 2 Qu m Qu m Qû m 2 Qu m û m û m 2 u m u m Qû m 2 Qu m Qû m νû m νu m 2 Qu m Qu m νû m νum 2 Qu m Qûm Qu m Qû m Qû m 2 Qum Qu m Qu m û m û m 2 um u m Qû m

10 350 K. DECKELNICK AND G. DZIUK = νû m νu m 2 Qu m νû m νum 2 Qu m Qû m Qu m Qû m Qum 2 e m e m Qu m 2 Qû m Qû m Qû m 2 Qû m Qu m u m u m 2 Qu m Qû m Qû m νû m νu m 2 Qu m Qu m νû m νum 2 Qu m Qûm Qu m νû m νu m 2 Qu m νû m νum 2 Qu m Qû m Qu m Qû m Qum 2 e m e m Qu m 2 Qû m νu m νu m 2 Qu m Qum Qû m Qu m Qu m 2 Qû m Qû m 2 Qu m Qû m Qû m νû m νu m 2 Qu m Qu m νû m νum 2 Qu m Qûm Qu m, = were we used 3.4 in order to rewrite u m u m 2. Recalling te definition 2.4 of E m we finally arrive at 2τ L = E m E m e m e m 2 Qû m Qû m Qu m Qû m Qum 2 Qu m 4 S i, 3.9 i= were S,...,S 4 are te last four terms in te equation above. We are going to examine tese integrals separately. To begin, note tat 3. and 3.3 imply Qum Qû m = Qû m Qûm Qum Qum νûm νum c νûm νum,

11 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 35 so tat We define S c νu m νu m 2 νu m νûm Qum q m := Qum Qû m Qu m Qûm. Te estimate 3.3 togeter wit te boundedness of û m yields Qu m c q m. Using tis in 3.20 we get for small enoug δ>0 S c νu m νu m 2 νu m νûm qm q m δ 2 Qu m c νu m νu m δ 2 νu m νûm. 3.2 Here we used again 3.3 as well as te fact tat νu. Next we observe tat 3.4 implies in view of Qu, Qv for u,v H,. In particular, we derive νu νv u v a.e. in 3.22 νu m νu m um u m em e m ûm û m e m e m cτ sup û,t L 0,T e m e m cτ 3.23 wic follows from 3.8 togeter wit an inverse estimate. We insert 3.23 into 3.2 and use 3.22, 3.5 to find q m S δ 2 Qu m c e m e m δ 2 e m τ 2 νu m νûm q m δ 2 Qu m c e m δ e m L e m 2 Qû m τ 2 νu m νûm Qû m Qu m Qû m δ Qum 2 Qu m c δ γ e m e m 2 Qû m c δ τ 3 τ E m, were te last estimate follows from Hölder s inequality and te definitions of q m and Em. Next, letting a = Qu m Qu m and b = Qûm Qû m we ave a b = qm qm as above and a 2 b 2 = a b 2 2ba b 2 a b 2 b 2 2 q m 2 cτ 2.

12 352 K. DECKELNICK AND G. DZIUK Since Qu is te last component of νu we obtain similarly as above S 2 c cγ Finally, note tat S 3 S 4 = q m 2 τ 2 νu m νûm Qû m Qu m Qû m Qum 2 Qu m cτ 3 τ E m. Qû m νû m νu m 2 Qu m Qu m νû m νu m νû m νûm 2 Qu m = νû m νu m 2 Qum Qu m Qûm Qû m 2νû m νu m νû m νûm νû m νûm 2 Qu m. Qûm Qu m Tus, using similar arguments as in te estimate for S we conclude Qûm Qu m S 3 S 4 cτ E m cτ νu m νû m τ νu m νûm Qum cτ E m cτ νu m νû m τ νu m νûm qm δ Qû m Qu m Qû m Qum 2 Qu m c δ τ 3 τ E m τ E m. Combining 3.9 wit te above estimates and coosing δ and γ small enoug completes te proof of te lemma. Let us now return to 3.2 and estimate te four terms on te rigt-and side of tis relation. i Since ɛ m ɛ m cτ sup ɛ t cτ 2 log 2 by 3.7 we obtain 0,T e m e m ɛ m ɛ m τ 2 Qu m δ τ 2 δ τ 2 e m e m 2 Qu m c δτ 2 ɛm ɛ m 2 e m e m 2 Qu m c δ 4 log 4.

13 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 353 ii Since e m e m X 0, integration by parts yields û m û m em e m τ Qû m = τ û m û m τ Qû m um u m Qu m e m e m u m u m τ Qu m e m e m c ɛ m ɛ m τ Qû m u m u m Qu m Qû m c τ um u m H,2 e m e m c τ ɛm ɛ m τ ɛ m e m c e m e m ɛ m ɛ m c sup δ τ 0,T e m u m u m e m e m Qu m ɛ t c e m e m e m e m e m e m 2 Qû m δ e m e m 2 τ 2 Qu m c δ τ 2 4 log 4 were we used 3.5, 3.8 and once again 3.3. Combining te above estimates we arrive at τ δ τ û m em e m Qû m e m e m 2 Qû m δ e m e m 2 τ 2 û m Qu m c δ τ 2 4 log 4. iii To begin, let us rewrite te difference Qu m Qu m as follows: Qu m Qu m = bm em were b m = 0 Bs u m s um ds, B i = p i p 2.

14 354 K. DECKELNICK AND G. DZIUK If we let in addition b m := B u m we may write τ 2 u m u m e m e m Qu m Qu m = τ 2 u m u m e m e m bm em τ 2 u m u m e m e m bm ɛ m τ 2 u m u m e m e m bm bm ɛ m I II III. Since b m uniformly in and m we obtain for 0 <δ wit te elp of 3.4 e m em δ e m e m 2 τ 2 Qu m c δ em 2 I c e m τ 2δ e m e m 2 τ 2 Qu m 2δ e m e m 2 τ 2 Qu m c 4 log 4 c δ Em. Next, as b m bm c e m we may estimate III c τ ɛm L e m e m em δ τ 2 e m e m 2 Qu m c δ 4 log 4 c δ Em. 2δ τ 2 c τ 2 ɛm ɛ m 2 c δ Em e m e m 2 Qu m c δ ɛm 2 L ɛm 2 e m 2 It remains to examine II. Integration by parts gives II = τ 2 b m u m u m e m e m ɛm τ 2 b m e m e m um u m ɛ m τ 2 b m u m u m e m e m ɛm II II 2 II 3. Using te regularity of u and similar arguments as above we get II II 3 c τ e m e m ɛm 2δ e m e m 2 τ 2 Qu m c δ 4.

15 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 355 Observing tat u m u m = τu t, t m R m were R m satisfies 3.0 and letting d m := u t, t m b m, te remaining term II 2 can be rewritten as II 2 = d m e m e m τ ɛm τ 2 b m e m e m Rm ɛ m = d m e m ɛ m d m e m τ τ ɛm τ d m e m ɛ m ɛ m τ τ 2 d m d m e m ɛ m b m e m e m Rm ɛ m. We sall keep te discrete time derivative and estimate te last tree integrals. To begin, note tat d m d m = u t, t m B u m u t, t m B u m = tm t m u tt, s dsb u m u t, t m tm Recalling tat u t, u t L 0, T we infer d m d m tm c u tt, s ds cτ, so tat 3.5 and 3.6 imply τ c τ t m d m d m e m ɛ m ɛ m L tm t m t m u tt., s ds τ ɛm c sup u tt ɛ m L ɛ m e m 0,T c 2 log e m e m em δ τ Furtermore, using 3.0 and 3.6 we ave d m e m ɛ m ɛ m τ τ 2 c e m τ em B u, s u t, s ds. e m e m e m 2 Qû m ce m c δ 4 log c τ ɛm ɛ m e m em δ τ b m e m e m ɛm ɛ m c τ 2 e m e m e m 2 Qû m c δ 4 log 4 ce m. e m Rm ɛ m c em e m e m Rm ɛ m e m ɛm L

16 356 K. DECKELNICK AND G. DZIUK Collating our estimates we finally get τ 2 u m u m e m e m Qu m Qu m d m e m ɛ m d m e m τ τ ɛm 6δ e m e m 2 τ 2 Qu m 2δ e m e m 2 τ Qû m c δ 4 log 4 c δ Em E m. iv From 3.0 and 3.3 we obtain R m e m e m τ 2 Qu m c em e m c em e m ɛ m ɛ m δ e m e m 2 τ 2 Qu m c τ 2 δ cτ2 log 2. If we apply Lemma 2 and our estimates i iv to 3.2 we obtain, by coosing δ sufficiently small, e m e m 2 2τ 2 Qu m e m e m 2 8τ Qû m 2τ Em E m 3.25 d m e m ɛ m d m e m τ τ ɛm ce m E m cτ 2 4 log 4. Now we are in position to complete te proof of Teorem. In a first step we claim tat 2 E m γ 2 for all 0 m M 3.26 provided τ δ 0, 0 and γ is cosen according to Lemmas and 2. Since E 0 = 0, te assertion clearly olds for m = 0. Assuming tat 3.26 is true for k m we may multiply 3.25 wit m replaced by jby τ and sum from j = 0,...,k: 2 Ek d k e k ɛ k cτ 4 τek cτ k E j E j cτ 2 4 log 4 j=0 k E j cτ 2 4 log 4. Here, we used te inequality d k e k ɛ k 4 Ek c 4 log 2 j=0 wic can be derived in essentially te same way as If τ is small enoug we obtain E k cτ k E j cτ 2 4 log 4, 0 k m 3.27 j=0

17 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 357 and a discrete Gronwall argument implies In particular, E k cτ 2 4 log 4, 0 k m E m cτ 2 4 log 4 cδ log 4 γ 2 2 provided τ δ 0, 0 and δ 0 is cosen sufficiently small. Tus, 3.26 is proved and we can start again from 3.25 in order to sow tat E k cτ 2 4 log 4 for all 0 k M as well as M m=0 e m e m 2 τ Qu m cτ 2 4 log 4. Since Qu m is uniformly bounded, tis implies Finally, e k e 0 k e m e m c 2 m=0 cτ 2 log 2. M τ m=0 e m e m 2 2 u m u m em ɛm c E m cτ by 3.5 and 3.4. Tis completes te proof of Teorem. 4. Implementation and numerical tests Te sceme 2.2 is suc tat in every time step a linear system of equations as to be solved. Assume tat X = span{ϕ,...,ϕ N } wit te nodal basis functions ϕ j j =,...,N. Ten u m as te form u m x = N j= u m j ϕ jx wit coefficients u m = u m,...,um N. For given um te linear system ten is τ Mm S m u m = τ Mm u m wit mass matrix M m and stiffness matrix S m Mij m ϕ i ϕ j = Qu m, Sm ij = ϕ i ϕ j Qu m, i, j =,...,N. Te matrices are symmetric and M m /τ S m is positive definite and we can use a suitable conjugate gradient algoritm to solve te linear system. We are going to verify te asymptotic error estimates of te Teorem. For tis we use an exact solution of te equation for te mean curvature flow of graps wit a given rigt-and side, i.e. wit

18 358 K. DECKELNICK AND G. DZIUK TABLE Absolute errors in L 0, T ; L 2 and experimental orders of convergence EOC for te test problem τ = 2 τ = 0.25 τ = 0.5 τ = TABLE 2 Absolute errors in L 0, T ; H and experimental orders of convergence EOC for te test problem τ = 2 τ = 0.25 τ = 0.5 τ = prescribed mean curvature. Tis introduces additional error terms in te analysis not presented in tis paper. We solve te geometric equation u u t Qu = fqu 4. Qu on te unit disk ={x R 2 x < } wit zero boundary conditions u = 0on. Te solution is given by ux, t = sin x 2 t sin t. Te solution does not decay for large time t and te norm L 0, T ; H is quite large. We ave cosen T = 4 for te test computations. Te rigt-and side is calculated from equation 4. and ten used in te program to compute te discrete solution. For two sucessive grids wit grid sizes and 2 we computed te absolute errors Err j,j =, 2 between discrete solution and exact solution for certain norms. Te experimental order of convergence ten was defined by EOC, 2 = log Err Err 2 log. 2 In Table Err = sup 0 m M u m u m wit Mτ = T and in Table 2 Err = sup 0 m M u m u m. We see tat te computations confirm te results of te Teorem precisely. In order to demonstrate te dependency on te coupling parameter δ 0 in te condition τ δ 0 we provide computations for δ 0 =, 0.25, 0.5 and.0.

19 ERROR ESTIMATES FOR THE CURVATURE FLOW OF GRAPHS 359 Acknowledgements Tis work was supported by te German Science Foundation DFG via Graduiertenkolleg Nictlineare Differentialgleicungen: Modellierung, Teorie, Numerik, Visualisierung. REFERENCES. ALTSCHULER, S. J. & WU, L.-F. Convergence to translating solutions for a class of quasi-linear parabolic boundary problems. Mat. Ann. 295, ALTSCHULER, S.J.&WU, L.-F. Translating surfaces of te non-parametric mean curvature flow wit prescribed contact angle. Calc. Var. 2, CHEN, X., ELLIOTT, C. M., GARDINER, A., & ZHAO, J. J. Convergence of numerical solutions to te Allen Can equation. Appl. Anal. 69, CRANDALL, M. G. & LIONS, P. L. Convergent difference scemes for non-linear parabolic equations and mean curvature motion. Numer. Mat. 75, DECKELNICK, K. Error analysis for a difference sceme approximating mean curvature flow. Interfaces and Free Boundaries 2, DECKELNICK, K.& DZIUK, G. Convergence of a finite element metod for non-parametric mean curvature flow. Numer. Mat. 72, DECKELNICK, K.&DZIUK, G. Discrete anisotropic curvature flow of graps. Mat. Modelling Numer. Anal. 33, DZIUK, G. Numerical scemes for te mean curvature flow of graps. In: ARGOUL, P.,FRÉMOND, M. &NGUYEN, Q. S. eds, IUTAM Symposium on Variations of Domains and Free-Boundary Problems in Solid Mecanics. pp Kluwer Academic Publisers, Dordrect HUISKEN, G. Non-parametric mean curvature evolution wit boundary conditions. J. Diff. Eqns 77, JOHNSON, C.& THOMÉE, V. Error estimates for a finite element approximation of a minimal surface. Mat. Comput. 29, LIEBERMAN, G. Te first initial-boundary value problem for quasi-linear second-order parabolic equations. Ann. Scuola Norm. Sup. Pisa 3, NOCHETTO, R. H.& VERDI, C. Convergence past singularities for a fully discrete approximation of curvature driven interfaces. SIAM J. Numer. Anal. 34, OLIKER, V. I. & URALTSEVA, N. N. Evolution of non-parametric surfaces wit speed depending on curvature II. Te mean curvature case. Commun. Pure Appl. Mat. 46, OSHER, S.& SETHIAN, J. A. Fronts propagating wit curvature dependent speed: algoritms based on Hamilton Jacobi formulations. J. Comput. Pys. 79, WALKINGTON, N. J. Algoritms for computing motion by mean curvature. SIAM J. Numer. Anal. 33, ZLAMAL, M. Curved elements in te finite element metod. Part I. SIAM J. Numer. Anal. 0, Part II. SIAM J. Numer. Anal.,