Error analysis of a finite element method for the Willmore flow of graphs

Transcription

1 Interfaces and Free Boundaries 8 6, 46 Error analysis of a finite element metod for te Willmore flow of graps KLAUS DECKELNICK Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz, 396 Magdeburg, Germany AND GERHARD DZIUK Institut für Angewandte Matematik, Universität Freiburg Hermann-Herder-Str., 794 Freiburg, Germany [Received October 4 and in revised form 8 September 5] Te evolution of two-dimensional graps under Willmore flow is approximated by a continuousin-time finite element metod. Te igly nonlinear fourt order problem is split into two coupled second order problems using eigt and a weigted mean curvature as variables. We prove a-priori error estimates for te resulting time-continuous sceme and present results of test calculations. Matematics Subject Classification: 35K55, 65M5, 65M6. Keywords: Willmore flow; fourt order parabolic problem; finite element; mixed metod; error estimates.. Introduction Te purpose of tis paper is to analyze a finite element sceme for te approximation of twodimensional surfaces in R 3 wic evolve according to Willmore flow. Tis flow can be interpreted as te L -gradient flow for te Willmore functional Wf := H da, Γ = f M, Γ were f : M R 3 is a smoot immersion, H = κ + κ denotes te mean curvature of Γ and da is te area element. Furtermore, M denotes a fixed two-dimensional surface wit or witout boundary. Considering normal variations f ɛ x := f x + ɛφxνx, x M, were ν is a unit normal field to Γ and φ : M R is smoot and vanises near M, one obtains te formula W f, φ := d dɛ Wf ɛ ɛ= = φ Γ H + H 3 H K da. Here, K = κ κ is te Gauss curvature of Γ and Γ denotes te Laplace Beltrami operator. Te sign of H is cosen in suc a way tat H > for a spere wit outward pointing normal. Klaus.Deckelnick@matematik.uni-magdeburg.de gerd.dziuk@matematik.uni-freiburg.de Γ c European Matematical Society 6

2 K. DECKELNICK AND G. DZIUK For details on tis calculation and more information on te Willmore functional see [8]. Given a smoot, oriented surface Γ R 3, te Willmore flow problem now consists in finding a family Γ t t [,T ] of smoot, oriented surfaces wic satisfy V = Γ H + H 3 H K on Γ t,. Γ = Γ,. were V denotes te normal velocity. In tis paper we sall assume tat te surfaces Γ t are graps over some bounded domain R, i.e. Γ t = {x, ux, t x } oriented by te unit normal field ν = u, + u,.3 were u = u x, u x. In wat follows we sall assume tat as a smoot boundary. It will be convenient to use te abbreviation Q := + u.4 for te area element. Te evolution law. gives rise to a partial differential equation for te eigt function u. In order to write down tis equation we note tat te quantities V, H, K and Γ H appearing in. are expressed in terms of u as follows: V = u t Q, H = Γ H = Q QI u Q u u Q, K = det D u Q 4,.5 H,.6 were D u contains te second space derivatives. Te last relation can be rewritten as u u u u Γ H = I Q Q QH H I Q Q Q..7 Using te expression for H we conclude tat I Q and a calculation sows tat u u Q Inserting.8 and.9 into.7 we obtain u u Γ H = I Q Q = Q I Q = Q Q u Q u + H u Q.8 Q u Q Q u = K..9 u u Q QH QH + H K H H u Q + H K H Q u H 3.

3 WILLMORE FLOW OF GRAPHS 3 FIG.. A stationary solution of..: Willmore surface wit prescribed boundary and mean curvature equal to zero on te boundary. Te solution was computed wit te time dependent algoritm Comparing tis expression wit. and recalling tat V = u t /Q we obtain te following fourt order parabolic PDE for u: u t + Q I Q u u Q QH H Q Q u = in, T.. Note tat te above equation as after division by Q a nice divergence structure in wic te Gauss curvature K no longer appears. Tis structure was exploited by Droske and Rumpf in [8] for a level set approac to Willmore flow. We sall use te boundary conditions u = g, H = on, T,. for a given smoot function g : R. Te condition. is motivated by calculations in [4], were it is sown tat if Γ = f M is stationary for W wit respect to variations wic keep te boundary of Γ fixed, ten necessarily H = on Γ. Finally, we prescribe te initial condition u, = u in.. In wat follows we sall assume tat te initial-boundary value problem.. as a unique solution u, wic satisfies u L, T ; H 4, L, T ; H 5,.3 u t L, T ; H, L, T ; H 3,.4 u tt L, T ; L L, T ; H..5 Since te focus of our work is on te analysis of a finite element sceme approximating solutions of.., we do not address te question of weter a function u satisfying te regularity assumptions.3.5 exists. We expect a positive answer at least for small T, but at te moment little seems to be known about boundary value problems for Willmore flow. Te situation is different for te evolution of closed surfaces. In [7] it is sown tat a unique local solution of.,. exists provided tat Γ is a compact closed immersed and orientable C,α -surface in R 3. Te solution exists globally in time if Γ is sufficiently close to a spere in C,α. Using different metods, Kuwert & Scätzle [] obtain global existence of solutions provided tat Γ A is sufficiently small, were A denotes te trace-free part of te second fundamental form. Tey were subsequently able to remove te smallness assumption and to prove te existence of a global smoot

4 4 K. DECKELNICK AND G. DZIUK solution provided tat Wf 6π, were Γ = f S see [] and note tat our definition differs from te one used in [] by a factor of. Te numerical evidence of [3] indicates tat te above condition is optimal in te sense tat te flow develops a singularity if te initial surface as energy greater tan 6π.. Variational formulation and discretization In order to approximate solutions of.. by a finite element sceme, we need to derive a variational formulation of.. To begin, let us introduce Ep ij := δ ij, i, j =,, p R.. It is not difficult to verify tat + p p ip j + p Eq Ep c q p p, q R,. q Epq q + p 3 p, q R..3 Now,. suggests using w = QH rater tan H as te second variable in a splitting metod cf. [8]. If we divide. by Q, multiply by a test function ϕ H and integrate by parts we are led to u t ϕ Q + E u w ϕ + w Q 3 u ϕ = ϕ H,.4 wζ Q u ζ Q = ζ H,.5 were te second relation stems from te definition of w and.5. Before we proceed and base te discretization of our problem on.4,.5, we want to deduce te decrease in time of te Willmore energy Γ t H da = H Q from tese relations. Using ϕ = u t in.4 and observing tat u u t = Q t Q we deduce u t Q + E u w u t + w Q t Q =..6 Next, differentiating.5 wit respect to time gives w t ζ Q wζ Q t Q E u u t ζ = ζ H..7 If we insert ζ = w into.7 and combine te resulting identity wit.6 we obtain = u t Q w Q t ww t Q + Q = u t Q + d dt Observing tat w /Q = H Q we see tat te energy decreases in time. w Q..8

5 WILLMORE FLOW OF GRAPHS 5 Let us now turn to te discretization in space. Te variational formulation.4.5 suggests using a second order splitting sceme wit linear finite elements in order to approximate te pair u, w. Let T be a family of triangulations of wit maximum mes size := max S T diams. We suppose tat is te union of te elements of T so tat element edges lying on te boundary are allowed to be curved. Furtermore we suppose tat te triangulation is quasiuniform in te sense tat tere exists a constant κ > 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 defined by X := {v C v is a linear polynomial on eac S T }, wit suitable modifications for boundary elements. Tere exists an interpolation operator I : H X suc tat v I v + v I v c v H for all v H..9 Furtermore, let X := X H and suppose tat I v X for v H H. Here and trougout te paper we will denote te L -norm by. Our discrete problem now reads: find u t, w t, t T, suc tat u t I g X, w t X, u = u X and u t ϕ + E u w ϕ + Q Here, Q = + u. w ζ Q w Q 3 u ϕ = ϕ X,. u ζ Q = ζ X.. LEMMA. Te system.,. as a unique solution u, w on [, T ] for all T <. Proof. Local existence on an interval [, t follows from te teory of ordinary differential equations. Since u t, w t ave values in a finite-dimensional space, it is sufficient to bound some norm of u, w in order to obtain existence on [, T ]. If we repeat te argument wic led to.8 in te discrete setting, te result is t u t + w C, u, t < t.. Q Q Using ζ = u t X in. we obtain d/dt Q = w u t /Q, wic combined wit. yields t Q t c + w Q / t u t Q / c, T, u, t < t. Tis implies tat Q x, t c, T uniformly in x, t [, t, and. ten sows tat u t, w t remain bounded on [, t. We leave te proof of uniqueness to te reader. Our main result is te following error estimate:

6 6 K. DECKELNICK AND G. DZIUK THEOREM. Assume tat.. as a unique solution u on te interval [, T ], wic satisfies.3.5. Also suppose tat u = û, were û is defined as te projection of u see.7. Ten sup t T u u t + sup t T sup t T T T w w t c log,.3 u u t c,.4 u t u t dt c 4 log 4,.5 w w dt c..6 REMARK.3 We expect a similar result if te function g in. is allowed to depend on time or if we consider a Neumann boundary condition for u. Prescribing an inomogeneous Diriclet condition for H or replacing it by a Neumann condition is more difficult since our analysis uses te variable w = QH, so tat information on Q on, T is required in order to determine te boundary condition for w. A detailed proof of Teorem. will be given in Section 3. One of te crucial points of our error analysis is te use of suitable nonlinear Ritz projections of u and w wic we introduce next. To begin, let û be defined by: û I g X and û ζ Q = u ζ Q ζ X,.7 were Q = + û. Note tat t is just a parameter. Estimates for te error ρ u := u û in te time-independent case were first carried out in [], L -estimates are due to [5], [9]. For functions u wic depend on time it was proved in [4], [5] tat sup ρ u t + sup ρ u t c,.8 t T t T sup ρ u t L + sup ρ u t L c log,.9 t T t T sup t T ρ ut t c log,. ρ ut t c.. sup t T Wit te elp of û, we next define a projection ŵ X of w as follows: E û ŵ ϕ = E u w ϕ + u w Q 3 û Q 3 ϕ ϕ X.. Te proof of te following bounds on te error ρ w := w ŵ

7 will be given in Lemma A. of te Appendix: WILLMORE FLOW OF GRAPHS 7 sup ρ w t c,.3 t T sup t T ρ w t c log,.4 ρ wt t c,.5 sup t T sup ρ wt t c log..6 t T Note tat.3,.4,.9.,.3.6 togeter wit interpolation and inverse estimates imply tat û W,, û t W,, ŵ W,, ŵ t W, c.7 uniformly in. We end tis section wit a few remarks on numerical approaces to Willmore flow and related problems. In [3], a finite difference sceme is derived to approximate axisymmetric solutions of evolution laws V = f κ, κ. Te Willmore flow is studied in detail and numerical evidence is provided tat te flow may develop singularities in finite time. [8] derives a level set formulation using te level set function and a weigted mean curvature as variables. Te formal similarity to te grap approac motivated our coice of variables. For te parametric approac to Willmore flow, [6] derives a variational form wic employs position and mean curvature vector as variables and allows te use of linear finite elements to discretize in space. Tis approac is subsequently extended in [3] to surfaces wit boundaries and applied to problems in surface restoration. We also refer to [], were numerical simulations of anisotropic surface diffusion are carried out for a surface energy wic consists of a strongly anisotropic nonconvex part and te Willmore functional weigted by a small factor. Let us finally mention evolution by surface diffusion, V = Γ H on Γ t, wic coincides wit Willmore flow in te igest order term, but wic is simpler in tat te nonlinear terms in te principal curvatures are absent. In te grap case, it is possible to write down a splitting metod using eigt and mean curvature as variables. Finite element error bounds bot for time-continuous and fully discrete scemes ave been derived in [6], [], [7]. 3. Proof of Teorem. Before going into details let us sketc te main ideas in te proof of Teorem.. Combining te variational identities.4,.5 and.,. wit.7 and. we sall derive corresponding relations for e u := û u and e w := ŵ w. We ten try to mimick te derivation of te a-priori estimate.8 in order to gain control on T e ut dt + sup t,t e w t. 3.

8 8 K. DECKELNICK AND G. DZIUK Tis program is started in Lemmas 3.4 and 3.5. Unlike te case of.6.8 te integrals involving Q t and te matrix E will no longer cancel and require a subtle analysis. Anoter difficulty stems from te fact tat control of te quantities in 3. requires a uniform bound on te discrete area element Q. Te proof of suc a bound eavily relies on te coice of te nonlinear Ritz projections introduced above and a superconvergence property. To begin, let u, w be te discrete solution and denote by Q := + u te discrete area element. Furtermore, define C := sup Qx, t, C := sup wx, t. x, t T x, t T Te initial condition togeter wit a continuity argument ten yields, for small, sup Q x, t C, x sup w x, t C 3. x for small t. Define T := sup{t [, T ] 3. olds on [, t]}. 3.3 Our strategy is to first prove te error estimates on te interval [, T and subsequently use tese bounds to sow tat T = T. Tus, in wat follows we sall assume 3.. Let us decompose te errors u u and w w according to u u = ρ u + e u, were e u = û u, w w = ρ w + e w, were e w = ŵ w. Furtermore, it will be convenient to work wit te following discrete normals: We infer from Proposition in [5] tat ν := û, Q, ν := u, Q. ν ν û u + sup û Q ν ν, wic, combined wit 3. and.7, yields ν ν e u + cc ν ν. 3.4 In order to make te error analysis more transparent we split it up into a series of auxiliary results. LEMMA 3. Ten were R satisfies Suppose tat F : R n R is twice continuously differentiable and tat f H. F u F û f = f F uρ u + R, R C log f.

9 Proof. Clearly, were WILLMORE FLOW OF GRAPHS 9 F u F û f = F u ρ u f + F u s ρ u F u ρ u f = f F uρ u + R, R c ρ u L ρ u f C log f by.3,.8,.9,.7 and since F is twice continuously differentiable. LEMMA 3. For every ɛ > tere exists c ɛ suc tat e u t ɛ e w t + c ɛ e u t + c 4 log, t < T. Proof. In view of.5,. and.7 we ave û u w ϕ = Q Q Q w ϕ ϕ X. Q Using ϕ = û u = e u X and applying Lemma 3. wit F p = / + p and f = we u we derive û u Q Q e u = we u u Q 3 Q we u + Q w w e u we u + Q Q Q ρ u + c log we u + c e u e u + e w + ρ w e u c e u + c e u + e w + log e u, 3.5 in view of.3,.8 and.4. Observing tat e u, t = Q ν Q ν we may write û u e u = ν ν Q ν Q ν = Q Q ν ν Q + Q c e u by.7 and 3.4. Combining tis inequality wit 3.5 implies te result. LEMMA 3.3 e w t c e u t + e ut t + e w t + 4 log 4, t < T. Proof. First note tat in view of te definition of ŵ we ave, for all ϕ X, û t ϕ + E û ŵ ϕ + Q = û t u t ϕ Q + ŵ Q 3 û ϕ ϕ + ŵ w û Q 3 ϕ, u t Q Q

10 3 K. DECKELNICK AND G. DZIUK from wic we infer e ut ϕ + Q ŵ E u w ϕ + E û ρ ut ϕ = + u t Q Q Q Inserting ϕ = e w into 3.6 we derive E û ŵ E u w e w = ϕ + e ut e w Q ρ ut e w + u t Q Q Q Using.3,.,.7 and te fact tat Q C we obtain E û ŵ E u w e w = E u e w e w + + 4C 3 e w c e u e w Furtermore, we infer from Lemma 3. tat u t Q Q e w = Q Q c e u e w ŵ Q 3 û w Q 3 u ϕ ŵ w û Q 3 ŵ Q 3 e w + û w E û E u ŵ e w u t e w + 4C Q u u t e w Q 3 Q 3 ϕ. 3.6 u e w ŵ w û Q 3 3 e w c e u. u t e w c e u e w + c e w + c log e w, Q ρ u + c log u t e w e w. by.8. Te remaining terms are estimated in a straigtforward manner so tat we finally obtain + 4C 3 e w c e u + e ut e w + c e w + e u e w + ρ ut e w + c e u e w + c e w + c log e w + c ρ w e w. Te result now follows from Young s inequality,. and.4. Te following two lemmas constitute te first steps in proving Teorem.. As mentioned earlier, te strategy is to mimick te derivation of.8.

11 LEMMA 3.4 For t < T we ave e ut + ŵ E u w e ut + C E û d dt u u t Q 3 e uρ u + d dt WILLMORE FLOW OF GRAPHS 3 ŵ w û Q 3 Proof. Inserting ϕ = e ut into 3.6 and recalling 3. we obtain ŵ Q 3 û w Q 3 u e ut e u + c e u + c 4 log 4. ŵ e ut + ŵ E u w e ut + C E û Q 3 û w Q 3 ρ ut e ut + u t Q e ut + u t Q Q Q e ut + Q I + II + III + IV. We infer from. tat u e ut ŵ w û Q 3 e ut I + II ρ ut e ut + c e u e ut ɛ e ut + c ɛ 4 log 4 + e u, 3.7 wile Lemma 3. wit F p = / + p and f = u t e ut yields u u III = u t e ut Q 3 ρ u + R = u t Q 3 e ut ρ u d dt u + u t Q 3 e u ρ u + t u t u Q 3 e uρ u u t u Q 3 e uρ ut + R, were R C log u t e ut. Tus, III d u u t dt Q 3 e uρ u + c e ut ρ u + log + c e u ρ u + ρ ut d u u t dt Q 3 e uρ u + ɛ e ut + c e u + c ɛ 4 log by.8,.. Finally, IV = d ŵ dt w û Q 3 e u ŵ û w e u. Q 3 t ŵ ŵ t ww t û Q 3 e u We infer from.,.4 and.7 tat IV d ŵ dt w û Q 3 e u + c e u ρ w + ρ wt d ŵ dt w û Q 3 e u + c e u + c 4 log by.4,.6. Summing and coosing ɛ sufficiently small yields te result.

12 3 K. DECKELNICK AND G. DZIUK LEMMA 3.5 For t < T we ave d ew e w Q t dt Q Q Q t ŵ Q Q t Q e w E û û t E u u t e w ɛ e w + c ɛ e u + e w + c ɛ 4 log 4. Proof. Starting from.5 and recalling te definition of û we infer wξ Q = u ξ û ξ = ξ X, Q Q from wic we obtain after differentiation wit respect to time w t ξ Q wξ Q Q t E û û t ξ = ξ X. Similarly w t ξ Q w ξ Q Taking te difference of te above relations we obtain e wt ξ e w ξ Q Q Q t Q t Q Q t E u u t ξ = ξ X. ŵ Q t Q ξ E û û t E u u t ξ = ŵ t ξ ρ wt Q Q Q ξ Q t Q + ρ w Q ξ t ŵ Q If we use ξ = e w te result is d ew ew Q t Q t dt Q Q ŵ Q Q t Q e w E û û t E u u t e w = ŵ t e w ρ wt Q Q Q e Q t Q w + ρ w Q e t w ŵ Q I + + IV. Q t Q ξ. Q t Q e w In order to deal wit te first term we apply again Lemma 3. wit F p := / + p and f = ŵ t e w : I = Q ŵ t e w + Q Q ŵ t e w Q = Q u ŵ t e w ŵ t e w Q Q 3 ρ u + R,

13 WILLMORE FLOW OF GRAPHS 33 were R c log ŵ t e w. As a consequence, I c ŵ t L e u e w + e w ρ u + log e w + c ŵ t L e w ρ u ɛ e w + c ɛ 4 log + c e u + c e w 3. by.7. Next,.4 and.6 imply II + III c ρ wt + ρ w e w c e w + c 4 log Abbreviating G i p := p i / + p 3 and i =,, we obtain IV = u ut Q 3 û û t Q 3 ŵ e w = G i u G i û u txi ŵ e w G i u G i û ρ utxi ŵ e w + G i uρ utxi ŵ e w = G i uŵ e w u txi ρ u + R G i u G i û ρ utxi ŵ e w G i uρ ut ŵ e w G i uŵ xi ρ ut e w G i ue wxi ŵ ρ ut, x i were R c log ŵ e w u t. Using.7 we can estimate IV c ŵ W, e w H ρ u + c log e w + c ρ u L ρ ut e w + c ρ ut e w + c ŵ L ρ ut e w + c e w ŵ L ρ ut ɛ e w + c ɛ 4 log 4 + c e w. 3. Combining we finally obtain te result. We are now in a position to complete te proof of Teorem.. It follows from Lemmas 3.4 and 3.5 tat e ut + d C dt + ŵ Q 3 + e w Q û w Q 3 u e ut e w Q Q t Q t ŵ Q Q t Q e w E û ŵ E u w e ut E û û t E u u t e w d u u t dt Q 3 e uρ u + d ŵ dt w û Q 3 e u +c ɛ 4 log 4 + ɛ e w + c ɛ e u H + e w. 3.3

14 34 K. DECKELNICK AND G. DZIUK Te main problem now is to deal wit te terms appearing in te second and tird line of te above inequality. A sort calculation sows tat ŵ Q 3 û w Q 3 u e ut e w Q t Q Q t ŵ Q Q t Q e w Clearly, = ŵ Q t Q S + + S 5. ŵ Q 3 û u t w Q 3 u û t + ŵ Q t Q S = ŵ, S 4 = t Q ŵ t Observing tat ν ν = + û u we obtain Q Q S = ŵ û u t Q 3 + ŵ û t u Q 3 = ŵ û u + ŵ û t u t wile ŵ 3 Q Q t Q 3 Q 3 ŵ t Q 3 S 3 = ŵ w u û t Q 3 Summation of S,..., S 5 yields S + + S 5 = ŵ t Q, Q û u Q 3 Q 3 Q. 3 ŵ û u Q Q 4 t + 3 4ŵ ŵ w u û t Q 3 + 4ŵ 3 Q Q t Q 3 ν ν u û t e w Q 3 + ŵ u û û t Q 3 Q 3 ŵ û û t Q 3 = ŵ t Q u û t e w Q 3 Q t + ŵ w ŵ + w ŵ Q t Q Q û u Q 3 Q 3 Q Q t Q 3 ν ν + ŵ u û t. + ŵ u û t u û t Q 3 Q ŵ + ŵ u û û t Q 3 u û t Q 3 Q t Q 3 Q Q 3 Q 3 3 ŵ Q Q t Q 3 + ŵ w Q t Q + ŵ û û t Q 3 Q Q t Q 3 ν ν Q 3 4 Q + 3Q + Q 4 Q 3.

15 WILLMORE FLOW OF GRAPHS 35 A sort calculation sows tat D := Q Q û u Q 3 Q 3 can be written as D = Integration over togeter wit 3.4 gives ŵ Q 3 û w Q 3 u e ut d ŵ dt D c Q Q ν ν Q Q Q Q. 3.4 e w Q t Q e w Q Q t Q t ŵ Q Q t Q e w ŵ ŵ t D c e u c e w u û t Q 3 c 4 Q + 3Q + Q 4. Q 3 Since Q t Q û t u Q 3 = û t û Q u + u Q + Q Q Q û t Q Q Q Q Q Q we obtain It is easily sown tat Q t Q û t u Q 3 c ν ν c e u. 4 Q + 3Q + Q 4 cq Q c e u. Q 3 Recalling 3.4 we ave in conclusion Next, ŵ Q 3 d dt û w ŵ Q 3 { u e ut e w Q Q t Q t ŵ Q Q t Q e w Q Q ν ν } Q Q Q Q c e w + e u. 3.5 E û ŵ E u w e ut E û û t E u u t e w = E û E u û t u t ŵ E û E u û t e w. In order to deal wit te first term we introduce P, P R 3 3 by P,ij = δ ij ν i ν j, P,ij = δ ij ν i ν j, i, j =,, 3.

16 36 K. DECKELNICK AND G. DZIUK Since û u, t = Q ν Q ν we may calculate P Q Q P û t u t, t = P Q Q P Q t ν + Q ν t Q t ν Q ν t were Observe tat S = Q t Q Q = ν t Q t P ν Q P ν t Q t P ν Q P ν t + ν t Q Q Q Q = { ν ν Q } P ν + S, 3.6 t Q P ν + Q Q P t ν + ν t Q t Q P ν Q Q P ν t. and similarly Tis gives S = Q t Q Q + ν t Q t Q P ν = ν ν ν ν = ν ν + ν ν ν P ν = ν ν + ν ν ν. ν ν + ν ν ν + Q = Q t ν ν Q ν ν ν t ν t ν ν Q ν ν + ν ν ν Q ν t ν t ν ν Q ν + ν + Q t Q ν ν Q Q Q Q + Q ν ν Q Q Q Q Q ν t + ν t ν ν since ν ν t =. We infer from 3. and 3.4 tat Q Q Q Q, Q ν Q Q Q ν c e u, as well as wic implies In conclusion Q ν Q Q Q ν Q ν ν Q Q Q = Q ν ν Q Q Q Q Q c e u, S c e u. 3.7 E û E u û t u t ŵ = P Q Q P û t u t, t ŵ, t { Q = ν ν Q ν ν t Q Q ν } + S ŵ, t.

17 WILLMORE FLOW OF GRAPHS 37 If we integrate tis relation over and recall 3.7 and.7 we obtain E û E u û t u t ŵ = d Q ν ν Q ν ν dt Q Q ν ŵ, t Q ν ν Q ν ν Q Q ν ŵ t, t + S ŵ, t d Q ν ν Q ν ν dt Q Q ν ŵ, t c e u. 3.8 Recalling te definitions of Pˆ, P and observing 3. we may estimate E û E u û t e w = Pˆ Q P û t, t e w, t Q c e u e w ɛ e w + c ɛ e u. 3.9 If we insert 3.5, 3.8 and 3.9 into 3.3 we finally obtain e ut + d ew d u u t C dt Q dt Q 3 e uρ u + d ŵ dt w û Q 3 e u { d ŵ Q dt Q ν ν } Q Q Q Q d Q ν ν Q ν ν dt Q Q ν ŵ, t + ɛ e w + c ɛ e w + e u + 4 log 4. Integrating wit respect to time between and t t < T and taking into account Lemmas 3. and 3.3 yields t e ut ds + e w t c e u t ρ u t + e w t + e u t t t + ɛ e w ds + c ɛ e w + e u ds + c ɛ 4 log 4 t t ɛ e w t + e ut ds + c ɛ e u t + 4 log 4 + e u + e w ds. After coosing ɛ sufficiently small we obtain t e ut ds + e w t c e u t + c On te oter and, in view of e u = we ave e u t t t e u e ut ds e u + e w ds + c 4 log t t e ut ds + c e u ds.

18 38 K. DECKELNICK AND G. DZIUK If we use tis inequality in 3. we finally obtain t e ut ds + e w t + e u t c and Gronwall s inequality yields t e u + e w ds + c 4 log 4, t < T, e w t + e u t c 4 log 4, t T. 3. Lemma 3. implies tat e u t c log for t T, wic togeter wit.9 and an inverse estimate gives Q t L Qt L + c u û t L + c e u t L C + c log + c log 3 C, and similarly w t L 3 C, t T, provided tat. Now we are in a position to prove tat T = T. If not, te above argument would imply tat Q x, T 3 C and w x, T 3 C for x, and we could establis 3. on [, T + δ] for some δ >, contradicting te definition of T. Te estimates.3.6 now follow from 3., Lemma 3., Lemma 3.3 and te interpolation results for û, ŵ. 4. Numerical results For our numerical tests we ave to use a time discretization. We ave cosen a semi-implicit discretization wit respect to te time variable in te spatially discrete sceme.,.. For a generic function v we denote its evaluation at te m-t time level t m = mτ by v m = v, t m. Te time discretization is ten given by: τ u m+ Q m u m ϕ + E u m wm+ + w m+ ζ Q m ϕ w m Q m um+ 3 u m+ ζ Q m ϕ = f m ϕ ϕ X, 4. = ζ X, 4. t for m =,,..., mt wit τmt = T and Q m = + u m. We ave introduced an additional given rigt and side f, wic we will need for our numerical tests. Denote by {ϕ j } j=,...,n te usual nodal basis of X. Ten 4., 4. represents a linear system for te coefficients of u m+ and w m+ in te expansions ϕ i ϕ j Q m u m I g = N j= U m j ϕ j, w m = N j= W m j ϕ j. Denote by U m = U m,..., U N m, W m = W m,..., W N m te coefficient vectors and set Mi,j m =, Ei,j m = E u m ϕ i ϕ j, A m i,j =, ϕ i ϕ j Q m

19 WILLMORE FLOW OF GRAPHS 39 as well as Bi,j m = w m Q m ϕ 3 i ϕ j, Fi m = f m ϕ i for i, j =,..., N. Wit tese settings we can write te linear system 4., 4. in te form τ Mm U m+ + E m W m+ + B m U m+ = τ Mm U m + F m, 4.3 M m W m+ A m U m+ =. 4.4 Eliminating W m+ from te first equation by inverting te weigted mass matrix in te second equation leads to te linear system τ Mm + E m M m A m + B m U m+ = τ Mm U m + F m. 4.5 We solve tis nonsymmetric system by te biconjugate gradient metod. In te practical computations we use mass lumping, so tat M m becomes a diagonal matrix. We ave tested our algoritm wit te elp of te following problem: let = {x R x < },, T =,.5 and ux, t =.5 cosπt x 3 x 5. Te function f is calculated in suc a way tat u is a solution of te PDE u t Q + u u I Q Q QH H Q u = f in, T. In order not to destroy te second order convergence properties in some norms we coose te time step as τ =.. Tis as proved experimentally to be a good coice for our computations for te Willmore flow of graps. In Tables and we sow te absolute errors for u in te norms E,,u = max m=,...,mt um u m, E,, u = max m=,...,mt um u m, mt E,,ut = τ um t um um / τ, E,,u = max u m u m, m=,...,mt m= as well as for w in te norms E,,w = mt max m=,...,mt wm w m, E,, w = τ w m w m /, m= E,,w = max sup w m w m. m=,...,mt Te computational results confirm te teoretical results of Teorem.. Between two spatial grid levels wit maximal grid size and and errors E, E we computed te experimental order of convergence according to eoc, = log E log. E

20 4 K. DECKELNICK AND G. DZIUK TABLE u-errors for te test problem, τ =. E,,u eoc E,, u eoc E,,ut eoc E,,u eoc TABLE w-errors for te test problem, τ =. E,,w eoc E,, w eoc E,,w eoc gradient gradient Grad-6 Grad-8 Grad- Grad t x t x -3 FIG.. Maximal gradient of te solution u for initial data 4.6 versus time t.,.375 for global refinement levels 3, 4, 5, 6 left and rigt, and 7 left only of te grid. Te plot on te left corresponds to te coice δ = in 4.6, te plot on te rigt to δ =.5. We demonstrate te possible effect tat te gradient of te continuous solution may blow up at some time. Figure sows te norm gradientt = u, t L. From Lemma. we know tat te discrete solution exists for all times. Te continuous one may exist for finite time only. A blow up of te continuous gradient may be deduced practically from te beaviour of te discrete gradient. As initial function for te computations in Figure we ave cosen u x, x = δ sin π + x sin πx 4.6

21 WILLMORE FLOW OF GRAPHS 4 on te domain =,,. Figure sows te maximal gradient of te solution corresponding to δ = and δ =.5 respectively on various refinement levels. Te results indicate a gradient blow up in te first case wile te gradient remains bounded in te latter. We finally give a computational example for te Willmore flow of a grap. Te domain is =,, and te initial value for u is given by u x, x =.75 sin π + x sin πx +. sin 4πx sin 5πx. FIG. 3. Grap of u for te time steps t =.,.67,.448 and.57. FIG. 4. Level lines of u for te time steps from Figure 3.

22 4 K. DECKELNICK AND G. DZIUK FIG. 5. Level lines of w for te times t =.,.67,.448,.57. FIG. 6. Grap of w for te time steps t =.837,.34 and t =.479. In Figure 3 we sow te grap of te solution u at several time steps. Level lines of u are plotted in Figure 4. Te levels run from. to. wit an increment of.. We sow te grap of w for some time steps in Figure 6 and level lines in Figure 5. Te spatial grid size was =.8839 and in order to capture te rapid smooting of te solution te time step was cosen as τ = 6.35e 6. Appendix LEMMA A. Let ŵ X be given by.. Ten sup w ŵ t c t T sup w ŵ t c log, t T w t ŵ t t c, sup t T sup w t ŵ t t c log. t T A. A. A.3 A.4

23 Proof. Clearly,. implies E û w ŵ ϕ = E u w ϕ E û WILLMORE FLOW OF GRAPHS 43 u w Q 3 û Q 3 ϕ A.5 for all ϕ X. Inserting ϕ = I w ŵ we obtain E û w ŵ w ŵ = E û w ŵ w I w + E u w I w ŵ E û u w Q 3 û Q 3 I w ŵ, from wic we deduce A. in view of.3,.,.8, standard interpolation estimates and te uniform boundedness of Q. Te L -norm of w ŵ is estimated wit te elp of te usual duality argument. Solve dive u z = w ŵ in, z = on. Te ellipticity of E togeter wit our smootness assumptions on and u implies tat te above boundary value problem as a unique solution z H t is a parameter wit z H c w ŵ. Using A.5 wit ϕ = I z we derive w ŵ = dive u zw ŵ = E u z w ŵ = E u E û z w ŵ + E û z I z w ŵ + E u w I z E û I + + IV. We infer from.,.9, A. and te a-priori estimate for z tat u w Q 3 û Q 3 I z I c u û L z w ŵ c log w ŵ ɛ w ŵ + c ɛ 4 log. An interpolation estimate togeter wit A. implies II c D z w ŵ ɛ w ŵ + c ɛ 4. Next, Lemma 3. yields III = E ij û E ij uw xi z xj + E û E u w I z z = E ij uw x i z xj ρ u + R + E û E u w I z z, were R c log w L z c log z. Tis implies III c ρ u + log + ρ u z H ɛ w ŵ + c ɛ 4 log

24 44 K. DECKELNICK AND G. DZIUK and in a similar way it follows tat IV ɛ w ŵ + c ɛ 4 log. Combining te estimates for I,..., IV implies A.. Let us next turn to te estimates for w t ŵ t. Differentiating A.5 wit respect to time yields E û w t ŵ t ϕ = E pi û û t,xi w ŵ ϕ + E pi û û txi E pi uu txi w ϕ + E û E u w t ϕ u ww t Q 3 û Q 3 ϕ u w Q Q 3 ϕ t for all ϕ X. From tis relation for ϕ = I w t ŵ t it is not difficult to deduce A.3, using.,.8,., A. and an interpolation estimate. It remains to bound w t ŵ t. Denoting by z te solution of dive u z = w t ŵ t in, z = on, and proceeding in te same way as above we obtain w t ŵ t = E u E û z w t ŵ t + E û z I z w t ŵ t E pi û û txi w ŵ I z + E pi û û txi E pi uu txi w I z u + E û E u w t I z ww t Q 3 û Q 3 I z u w Q 3 û Q 3 I z t I + + I 7. Clearly, I + I c u û L z + c z H w t ŵ t ɛ w t ŵ t + c ɛ 4 log. Next, I 3 = E pi uu txi E pi û û txi w ŵ z + E pi û û txi w ŵ z I z E pi uu txi w ŵ z = I 3 + I 3 + I 33. Observing tat f L p cp f H for p > and using.8,., A. as well as an inverse estimate we deduce I 3 c u û + u t û t I w ŵ L p/p z L p + w W, z cp /p I w ŵ + c z H cp /p w t ŵ t.

25 Te coice p = log ten implies tat Clearly, WILLMORE FLOW OF GRAPHS 45 I 3 c log w t ŵ t ɛ w t ŵ t + c ɛ 4 log. I 3 c w ŵ z H c w t ŵ t ɛ w t ŵ t + c ɛ 4. Integration by parts togeter wit A. yields I 33 = E pi uu txi zw ŵ c log z H ɛ w t ŵ t + c ɛ 4 log. In order to deal wit I 4 we use te splitting I 4 = E pi û û txi E pi uu txi w I z z + E pi û E pi uû txi u txi w z + E pi û E pi uu txi w z + E pi uû txi u txi w z I I 44. Using.8,. and an interpolation estimate we obtain I 4 c u û + u t û t z H ɛ w t ŵt + c ɛ 4. Next,.9 and. imply I 4 c u û L u t û t z ɛ w t ŵ t + c ɛ 4 log. An application of Lemma 3. yields I 43 c log z H ɛ w t ŵ t + c ɛ 4 log. Finally, integration by parts togeter wit. implies tat I 44 c log z H ɛ w t ŵ t + c ɛ 4 log 4. Te remaining terms I 5, I 6 and I 7 can be dealt wit in a similar manner so tat we obtain A.4 after collecting te above estimates and coosing ɛ sufficiently small. Acknowledgements Tis work was supported by te Deutsce Forscungsgemeinscaft via DFG-Forscergruppe Nonlinear partial differential equations: Teoretical and numerical analysis and via DFG- Graduiertenkolleg: Nictlineare Differentialgleicungen: Modellierung, Teorie, Numerik, Visualisierung. Te grapical presentations were performed wit te packages GRAPE and Xgrap.

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