An Eulerian level set method for partial differential equations on evolving surfaces

Transcription

1 Computing an Visualization in Science manuscript No. (will be inserte by the eitor) G. Dziuk C. M. Elliott An Eulerian level set metho for partial ifferential equations on evolving surfaces Receive: ate / Accepte: ate Abstract In this article we efine an Eulerian level set metho for a partial ifferential equation on an evolving hypersurface Γ(t) containe in a omain R n+1. The key iea is base on formulating the partial ifferential equations on all level set surfaces of a prescribe time epenent function Φ whose zero level set is Γ(t). The resulting equation is then solve in one imension higher but can be solve on a fixe gri. In particular we formulate an Eulerian version of a scalar conservation law on Γ(t) an, in the case of a iffusive flux, erive an Eulerian transport an iffusion equation. Using Eulerian surface graients to efine weak forms of elliptic operators naturally generates weak formulations of elliptic an parabolic equations. The finite element metho is applie to the weak form of the conservation equation yieling an Eulerian Evolving Surface Finite Element Metho (EESFEM). The computation of the mass an element stiffness matrices are simple an straightforwar. Numerical experiments are escribe which inicate the power of the metho. We escribe how this framework may be employe in applications. Gerhar Dziuk Abteilung für Angewante Mathematik University of Freiburg Hermann-Herer-Straße 1 D 7914 Freiburg i. Br. Germany Tel.: Fax: ger.ziuk@mathematik.uni-freiburg.e Charles M. Elliott Department of Mathematics University of Sussex Falmer, Brighton BN1 9RF Unite Kingom Tel.: C.M.Elliott@sussex.ac.uk 1 Introuction Partial ifferential equations on evolving curves an surfaces occur in many applications. In [8] we introuce the evolving surface finite element metho (ESFEM) for the numerical solution of iffusion equations on prescribe moving hypersurfaces Γ(t) R n+1. The metho relies on approximating the partial ifferential equation on a triangulate surface (n=) or polygonal curve (n=1) which interpolates Γ(t), [1]. In [9] we treate an Eulerian surface finite element on implicit stationary surfaces. In this paper we propose an approach to the formulation an approximation of transport an iffusion of a material quantity on an evolving surface in R n+1 (n=1,) using an implicit representation of the surface. The evolving surface is just one level set of a prescribe function an the partial ifferential equation an its solution are extene to a neighbourhoo of the surface. This neighbourhoo is then iscretize by a finite element gri which is inepenent of the surface yieling an Eulerian ESFEM (EESFEM). We have in min a surface which not only evolves in the normal irection so as to efine the surface evolution but also has a tangential velocity associate with the motion of material points in the surface which avects material quantities such as heat or mass. For our purposes here we assume that the surface evolution is prescribe. 1.1 The avection iffusion equation Conservation of a scalar with a iffusive flux on an evolving hypersurface Γ(t) leas to the iffusion equation u + u Γ v Γ (A Γ u) = (1.1) on Γ(t). Here u enotes the covariant or avective surface material erivative, v is the prescribe velocity of the surface an Γ is the tangential

2 G. Dziuk, C. M. Elliott surface graient. We assume that Γ(t) is empty an so the equation oes not nee a bounary conition. 1. Applications Such a problem arises, for example, when moeling the transport of an insoluble surfactant on the interface between two fluis, [17, 13]. Here one views the velocity of the surface as being the flui velocity an hence the surfactant is transporte by avection via the tangential flui velocity (an hence the tangential surface velocity) as well by iffusion within the surface. The evolution of the surface itself in the normal irection is then given by the normal component of the flui velocity. Diffusion inuce grain bounary motion, [5, 11, 16,6], has the feature of coupling force mean curvature flow for the motion of a grain bounary with a iffusion equation for a concentration of mass in the grain bounary. In this case there is no material tangential velocity of the grain bounary so it is sufficient to consier the surface velocity as being in the normal irection. Another example is pattern formation on the surfaces of growing organisms moelle by reaction iffusion equations, [15]. Possible applications in image processing are suggeste by the article [14]. 1.3 Outline of paper The layout of the paper is as follows. We begin in section by efining notation an essential concepts from elementary ifferential geometry necessary to escribe the problem an numerical metho. The equations presente above are justifie in section 3. The weak form of the equations is erive in section 4 an the well poseness of the initial bounary value problem is establishe. In section 5 the finite element metho is efine an some preliminary approximations results are shown. Implementation issues are iscusse in section 6 an the results of numerical experiments are presente. Basic Notation an Surface Derivatives.1 Notation For each t [, T], T >, let Γ(t) be a compact smooth orientable hypersurface without bounary in R n+1 which has a representation efine by a smooth level set function Φ = Φ(x, t), x R n+1, t [, T] so that Γ(t) = {x : Φ(x, t) = } where is a boune omain in R n+1 with Lipschitz bounary. We assume that Φ satisfies the non-egeneracy conition Φ in (, T). (.1) We assume that Γ(t) is empty an set ν to be the unit outwar pointing normal to. We set T = (, T). In particular we suppose that for some k 3 an some < α < 1 Φ C 1 ([, T], C k,α ()). The orientation of Γ(t) is set by taking the normal ν to Γ to be in the irection of increasing Φ. Hence we efine a normal vector fiel by ν(x, t) = Φ(x, t) Φ(x, t) so that the normal ν Γ to Γ(t) is equal to ν Γ(t) an the normal velocity V of Γ is given by V (x, t) = Φ t(x, t) Φ(x, t). Observe that a possible choice for Φ is the signe istance function (x, t) to Γ(t) an in that case Φ = = 1 on T. The existence of T an an Φ for t [, T] such that the above hols is a consequence of the smoothness of Γ(t) in space an time. Remark 1 The above escription of Γ(t) relies solely on the normal velocity V. We return to this in section.. We efine the projection P Φ = I ν ν, (P Φ ) ij = δ ij ν i ν j. (.) (i, j = 1,...,n+1) so that P Φ ν =. For any function η on we efine its Eulerian surface graient by Φ η = η η ν ν = P Φ η where, for x an y in R n+1, x y enotes the usual scalar prouct an η enotes the usual graient on R n+1. Note that Φ η ν = an Γ η = Φ η Γ only epens on the values of η restricte to Γ an is the usual tangential (surface) graient on Γ. The Eulerian mean curvature is efine in the usual level set way by H Φ = ν = Φ Φ. Eulerian surface elliptic operators can then be efine in a natural way, for example the Eulerian Laplace-Beltrami operator is efine by Φ η = Φ Φ η. We recall the coarea formula

3 An Eulerian level set metho for partial ifferential equations on evolving surfaces 3 Lemma 1 (Coarea Formula) Let for each t [, T], Φ(, t) : R be Lipschitz continuous an assume that for for each r (inf Φ, sup Φ) the level set Γ r := {x : Φ(x, ) = r} is a smooth n-imensional hypersurface in R n+1. Suppose η : R is continuous an integrable. Then sup Φ ( ) r = η Φ. (.3) inf Φ Γ r η The Eulerian formula for integration by parts over level surfaces is given by the following Lemma. Lemma (Eulerian Integration by Parts) Assume that the following quantities exist. For a scalar function η an a vector fiel Q we have Φ η Φ = ηh Φ ν Φ (.4) + η(ν ν ν ν) Φ, Φ Q Φ = H Φ Q ν Φ (.5) + Q (ν ν ν ν) Φ, Φ Qη Φ + Q Φ η Φ (.6) = Q νηh Φ Φ + Q (ν ν ν ν)η Φ. Proof We employ the notation i = x i an ij = x i x j. Elementary calculations yiel i Φ = ν k ik Φ = (D Φν) i, Φ j ν k = jk Φ ν k (D Φν) j, Φ H Φ = Tr(D Φ) + ν D Φν, where D Φ is the Hessian matrix of secon erivatives, Tr( ) is the trace of a matrix an we employ the summation convention for repeate inices. Using the efinition of Φ we fin that the left han sie in (.4) is LHS := Φ η Φ = Φ ( η ν ην) an then we employ the stanar integration formula on. It follows that (LHS) i = η i Φ + η m (ν i ν m Φ ) + η Φ ((ν ) i ν i (ν ν )) = I + II + III. Straightforwar calculations yiel II = η(ν i Tr(D Φ) + (D Φν) i ν D Φνν i ). Combining I an II using the formula for H Φ gives the esire result. Remark The bounary terms in the integration by parts formulae isappear when ν = ν.. The material erivative an Leibniz formulae Let v : T R n+1 be a prescribe velocity fiel which has the ecomposition where v = V ν + v S V = v ν so that v S is orthogonal to ν an tangential to all level surfaces of Φ. By a ot we enote the material erivative of a scalar function η = η(x, t) efine on T : η = η + v η. (.7) t In particular we note that η restricte to a given level surface Γ r := {(x, t) : x, t (, T), Φ(x, t) = r} epens only on the values of η on that level surface in space-time. It is convenient to note that an Φ v = Φ (V ν) + Φ v S (.8) = V Φ ν + Φ v S = V H Φ + Φ v S Φ v = trace((p Φ v). For a scalar η we have v η = V ν η + v S η. (.9) Lemma 3 (Implicit surface Leibniz Formula) Let Φ be a level set function an η be an arbitrary function efine on T such that the following quantities exist. Then t = η Φ (.1) ( η + η Φ v) Φ η v ν Φ. Proof Noting that t Φ = ν Φ t straightforwar calculation yiels η Φ = (η t Φ + ην Φ t ). t Integration by parts on the secon term in the integran gives η Φ = Φ (η t + V η ν ηv νh Φ ) t ην ν V Φ.

4 4 G. Dziuk, C. M. Elliott The Eulerian ivergence theorem (.5) gives ηv νh Φ Φ = Φ (ηv) Φ ηv (ν ν ν ν) Φ an observing that V η ν + Φ (ηv) = v η + η Φ v yiels the esire result. 3 Conservation an iffusion 3.1 Eulerian conservation an iffusion Let Φ : T R be a prescribe non-egenerate level set function. Let Q : T R n+1 be a given flux. Then the Eulerian conservation law we consier is t R u Φ = (Q + Φ uv) ν R (3.1) R for each sub-omain R of where ν R is the outwar unit normal to R. In particular we consier a flux of the form Here A is a symmetric mobility tensor with the property that it maps the tangent space into itself, so that T = {ν R n+1 : ν ν = } Aν ν = ν T. Observe that (3.7) is a linear egenerate parabolic equation because P Φ has a zero eigenvalue in the normal irection ν. Another form of this PDE, less suitable for numerical purposes than (3.7) is given as u t +V u ν + Φ (uv S ) V H Φ u Φ (A Φ u) =. In Figure 1 we show an example for the influence of motion of the levels of Φ onto the solution u. We use = ( 1, 1) R an a level set function Φ (see (6.3)) which eforms straight lines in into curve lines. Constant initial ata u = 1 then become nonconstant uring the evolution. In Figure 1 we show a strip cut out of the computational omain. For a etaile escription we refer to Example 5. Q = Φ q Φ where q Φ : T R n+1 is a flux satisfying q Φ ν =. (3.) It follows by the implicit surface Leibniz formula (.1) that u Φ = ( u + u Φ v) Φ (3.3) t R R uv ν R Φ R an by Eulerian integration by parts an (3.) that q Φ ν R Φ = Φ q Φ Φ. (3.4) R R It follows that Φ ( u + u Φ v + Φ q Φ ) = R for every sub-omain R which implies the partial ifferential equation u + u Φ v + Φ q Φ = in. (3.5) Taking q Φ to be the iffusive flux q Φ = A Φ u (3.6) leas to the iffusion equation u + u Φ v Φ (A Φ u) =. (3.7) Fig. 1 Heating by motion: Solution of Example 5 for the times t =.,.39 an.1 on the strip δ. The variational form of (3.7) is obtaine in the following way. For each level surface of Φ we multiply equation (3.7) by a test function η an get ( u + u Φ v Φ (A Φ u))η Φ =. Observe that the Leibniz formula, (.1), gives uη Φ = ( u + u Φ v) η Φ t + u η Φ uηv ν Φ, an because of A Φ u ν = integration by parts (.6) gives A Φ u Φ η Φ = η Φ (A Φ u) Φ + A Φ u ν η Φ. In orer to procee we nee a bounary conition for u on. For the purpose of this paper we shall assume that v ν = on (3.8)

5 An Eulerian level set metho for partial ifferential equations on evolving surfaces 5 an impose the zero flux conition Φ A Φ u ν = on. (3.9) Finally we obtain uη Φ + A Φ u Φ η Φ = u η Φ. t (3.1) Remark 3 Taking η = 1 we fin the conservation equation u Φ =. t 4 Weak Form an Energy Estimate We introuce the notion of a weak solution of the linear egenerate Eulerian parabolic (3.7), for which we erive a variational form in (3.1). We efine the norme linear spaces L Φ ( T) = {η : η, η Φ < }, H 1 Φ ( T) = {η : η, η, Φ η L Φ ( T)}, where the norm is inuce by the inner prouct T η, ξ Φ = ηξ Φ, η, ξ H 1 Φ = η, ξ Φ + η, ξ Φ + Φ η, Φ ξ Φ, an we set η L Φ ( T ) = η, η Φ, η H 1 Φ ( T ) = η, η H 1 Φ. Definition 1 (Weak solution) A function u H 1 Φ ( T) is a weak solution of (3.7) an (3.9), if for almost every t (, T) uη Φ + A Φ u Φ η Φ = u η Φ t (4.1) for every η H 1 Φ ( T). Weak solutions satisfy the following basic energy estimate whose iscrete counterpart implies stability. Throughout we will assume the initial conition u(, ) = u ( ) on. (4.) Lemma 4 Let u satisfy (4.1). Then 1 u Φ + A Φ u Φ u Φ t + 1 u Φ v Φ =. (4.3) Proof We choose η = u in (4.1) an obtain u Φ + A Φ u Φ u Φ t = u u Φ = 1 ( u ) Φ Φ = 1 u Φ 1 u Φ v Φ t + 1 u v ν Φ, where the last term vanishes because of (3.8), an this was the claim. 5 Finite Element Approximation 5.1 Semi-iscrete Approximation Our Eulerian ESFEM is base on the the weak form (4.1) of the iffusion equation. We use fixe in time finite element functions so that the test functions now will satisfy η = v η. We assume that the omain is triangulate by an amissible triangulation T h = T T h T which consists of simplices T. The iscrete space then is S h = {U C () U T is a linear polynomial, T T h }. The iscrete space is generate by the noal basis functions χ i, i = 1,...,N, S h = span{χ 1,..., χ N }. It is possible to generalize the metho to higher orer finite elements. Definition (Semi-iscretization in space) Fin U(, t) S h such that Uη Φ + A Φ U Φ η Φ (5.1) t = Uv η Φ η S h. Using the Leibniz formula it is easily seen that an equivalent formulation is: Uη Φ + Uη Φ v Φ + A Φ U Φ η Φ = η S h. Setting U(, t) = N α j (t)χ j (, t) j=1

6 6 G. Dziuk, C. M. Elliott we fin that N α j χ j η Φ t j=1 N + A α j Φ χ j Φ η Φ j=1 N = α j χ j v η Φ j=1 forall η S h an taking η = χ k, k = 1,..., N, we obtain (M(t)α) + S(t)α = C(t)α (5.) t where M(t) is the evolving mass matrix M(t) kj = χ j χ k Φ, C(t) is a transport matrix C(t) kj = χ j v χ k Φ, an S(t) is the evolving stiffness matrix S(t) jk = A Φ χ j Φ χ k Φ. Since the mass matrix M(t) is uniformly positive efinite on [, T] an the other matrices are boune, we get existence an uniqueness of the semi-iscrete finite element solution. Remark 4 A significant feature of our approach is the fact that the matrices M(t), C(t) an S(t) epen only on the evaluation of the graient of the level set function Φ an the velocity fiel v. The metho oes not require a numerical evaluation of the curvature. 5. Stability The basic stability result for our spatially iscrete scheme from Definition is given in the following Lemma. Lemma 5 (Stability) Let U be a solution of the semi-iscrete scheme as in Definition with initial value U(, ) = U. Assume that an that T Φ v L () < Aξ ξ a ξ with some a > for every ξ R n+1 with ξ ν =. Then the following stability estimate hols: T sup U L ()+ Φ U Φ L Φ (,T) () c U L (). Φ (5.1) Proof The estimate follows from the Leibniz formula in Lemma 3 in the same way as this was one for the continuous equation in Lemma 4 an a stanar Gronwall argument. 6 Implementation an Numerical Results 6.1 Implicit Euler Scheme The time iscretization in our computations is one by an implicit metho. We iscretize the variational form (3.1) in time. The spatially iscrete problem is t = Uη Φ + Γ h (t) U η Φ η S h. A Φ U Φ η Φ (6.1) We introuce a time step size τ > an use upper inices for the time levels. Thus U m represents U(, mτ). With these notations we propose the following Algorithm. Algorithm 1 (Fully iscrete scheme) Let U S h be given. For m =,...,m T solve the linear system 1 τ U m+1 η Φ m+1 (6.) + A m+1 Φ m+1u m+1 Φ m+1η Φ m+1 = 1 U m η Φ m + U m v m η Φ m τ for all η S h. The transport term on the right han sie of (6.) is treate explicitly. Depening on the velocity v of the levels it may be necessary to employ techniques for iffusion convection equations. 6. Numerical Tests Example 1 To start with we solve the heat equation on a moving circle. We choose to be the annular region with outer raius 1 an inner raius.5. We set Φ(x, t) = x.75 + sin (4t)( x 1 )(1 x ) so that the bounary comprises level lines of Φ. The velocity v is chosen to be normal, v S =.

7 An Eulerian level set metho for partial ifferential equations on evolving surfaces 7 The function u(x, t) = exp ( t/ x )x 1 / x then solves u + u Φ v Φ u = f where we calculate f = V u( t r r ) from f = u t + V u r uv H 1 r u θθ where x = r(cos θ, sin θ). We are intereste in u on the moving curve Γ(t) = {x Φ(x, t) = 3 4 }. As initial ata we take u (x) = x 1 / x. We have chosen the coupling τ = h in orer to show the higher orer convergence for L an L errors. The time interval is T = 1.. In Figure we show part of the use gri with the curve Γ() cutting the triangulation. In Table 1 we show the absolute The most important errors are shown in Table 3. Here we show the asymptotic orer of the aequate norms on the curve Γ(t), L (L (Γ)) = sup u u h L (Γ), (,T) ( T L (H 1 (Γ)) = Φ (u u h ) L (Γ) ) 1 We observe that espite the quite irregular iscretization of the curves inuce by the triangulation of the orers are two for the L (L (Γ))- norm an one for the L (H 1 (Γ))-norm. As an aitional information we show the maximum norms L (L ()) = sup u u h L (), (,T) L (L (Γ)) = sup u u h L (Γ) (,T) on an Γ in Table 4. For an error E(h 1 ) an E(h ) for the gri sizes h 1 an h the experimental orer of convergence is efine as eoc(h 1, h ) = log E(h 1) E(h ) ( log h ) 1 1. h. Table 1 Heat equation on an evolving curve. Errors an experimental orers of convergence in Φ-norms for Example 1 on. Fig. Example 1: Triangulation of (black) an Curve Γ (re) for a quarter of the annulus. h L (L Φ()) eoc L (HΦ()) 1 eoc errors an the corresponing experimental orers of convergence for the Φ-norms on the full omain, abbreviate by L (L Φ ()) = sup u u h L Φ (), (,T) L (H 1 Φ ()) = ( T Φ (u u h ) L Φ () )1. For comparison we show in Table the usual L ()-norms for the error, L (L ()) = sup u u h L (), (,T) ( T L (H 1 ()) = (u u h ) L () ) 1. Table The same situation as in Table 1. Errors an experimental orers of convergence in the full L ()- norms. h L (L ()) eoc L (H 1 ()) eoc Example We solve the Φ-heat equation on concentric circles. We start with the stationary situation, i. e. Φ t =. We take = {x R 1 < x < 1},

8 8 G. Dziuk, C. M. Elliott Table 3 The same situation as in Table 1. Errors an experimental orers of convergence on Γ. h L (L (Γ)) eoc L (H 1 (Γ)) eoc Table 4 Maximum errors on an on Γ for the situation of Example 1. h L (L ()) eoc L (L (Γ)) eoc u (x) = sin (4ϕ) an Φ(x, t) = x.75 together with v S =. Since the mean value of u on concentric circles vanishes, the solution tens to zero as time tens to infinity. But this occurs at a rate which epens on the raius of the circle because of the ifferent iffusion coefficients on the ifferent circles. This effect is shown in Figure 3 for small times. In this computation we use the time step τ = 1 5 an spatial gri size h =.36. In Figure 4 we aitionally show the solution on a strip cut out of the omain by the level set function Φ. Example 3 We next compute the solution for a similar situation as above but now with purely tangential velocity v S = 1 ( Φ x, Φ x1 ). (6.1) Φ In Figure 5 we show the solution at some time steps. Example 4 Finally we compute the evolution with purely normal velocity. For this we have chosen Φ((r cosθ, r sin θ), t) (6.) = r.75 + sin (8θ)sin (4t)(r.5)(1 r). In Figure 6 we show some time steps. In orer to ientify the zero level set of Φ, on which we want to solve the PDE, we show a strip of with.5 aroun that zero level set. The colouring is as in the previous examples. The colours range from minimum to maximum of u on. Example 5 We choose = ( 1, 1) an constant initial ata u (x 1, x ) = 1. Fig. 3 Solution u of Example for the times t =., t =.11 an t =.3. The colour re stans for u = 1 an the colour blue for u = 1. Left column shows the solution on in this colour coe, right column: the levels. Fig. 4 Solution u of Example for the times t =., t =.1 an t =.5 on the strip.7 < x <.8. The colouring is the same as in Figure 3. The level set function is given by Φ(x 1, x, t) = x (1 x )sin (4πx 1)sin (t), (6.3) an we use τ =.3965 an h =.65. In Figure 7 we show the solution for times from one perio of the level set function Φ. The time epenent levels of Φ are shown in the same Figure. In Figure 1 we have shown the solution on the strip δ = {x Φ(x, t) < δ} for δ =.1 for some time steps to emonstrate the effect of heating by motion. Example 6 We finish with a three imensional example. We apply our numerical metho to the

9 An Eulerian level set metho for partial ifferential equations on evolving surfaces 9 Fig. 5 Solution u of Example 3 with tangential velocity (6.1) for the times t =., t =.46 an t =.37. The colour stans for the values of u between maximum (re) an minimum (blue). Fig. 6 Solution u of Example 4 with normal velocity inuce by (6.) for the time steps, 1 an 1. The colour stans for the values of u between maximum (re) an minimum (blue) on Ω. omain Ω = ( 1, 1)3 R3 an to the surfaces which are implicitly given by the function 1 π Φ(x, t) = x3 (1 x3 ) sin (πx1 ) sin (πx ) sin ( t). (6.4) As initial value we have chosen the constant function u = 1. In Figures 8 to 11 we show the results of the computation for some time steps. The colouring of the surfaces is given by the values of the solution u. Here, ark blue represents u =., re u =. an the colours are linearly interpolate. Green represents the value u = 1.. The computational ata were N = an τ =.315. Fig. 7 Solution (left) of Example 5 with levels of Φ (right) for the times t =.,.39,.46,.813,.36 an 3.13 an Blue represents the value u =.1 an re u = 1.1. Acknowlegements The work was supporte by the Deutsche Forschungsgemeinschaft via DFG-Forschergruppe Nonlinear partial ifferential equations: Theoretical an numerical analysis an by the UK EPSRC via the Mathematics Research Network: Computation an Numeri-

10 1 G. Dziuk, C. M. Elliott Fig. 8 Example 6: Values of the solution u for the times.47,.94 an. on the surface Φ =.75. Fig. 1 Example 6: Values of the solution u for the times.47,.94 an. on the surface Φ =.. Fig. 9 Example 6: Values of the solution u for the times.47,.94 an. on the surface Φ =.5. cal analysis for Multiscale an Multiphysics Moelling. Part of this work was one uring a stay of the first author at the ICM at the University of Warsaw supporte by the Alexaner von Humbolt Honorary Fellowship 5 grante by the Founation for Polish Science. The graphical presentations were performe with the package GRAPE. The three imensional computations have been one with the finite element package ALBERTA []. For technical help we thank C. Eilks, C.-J. Heine, M. Lenz an R. Klöfkorn. References 1. D. Aalsteinsson an J. A. Sethian Transport an iffusion of material quantities on propagating in- Fig. 11 Example 6: Values of the solution u for the times.47,.94 an. on the surface Φ =.75. terfaces via level set methos J. Comp. Phys. 185 (3) A. Schmit an K. G. Siebert Design of aaptive finite element software. The finite element toolbox ALBERTA. Springer Lecture Notes in Computational Science an Engineering 4 (4). 3. Th. Aubin Nonlinear analysis on manifols. Monge-Ampère equations Springer Berlin- Heielberg-New York (198) 4. M. Bertalmio, L. T. Cheng, S. Osher an G. Sapiro Variational problems an partial ifferential equations on implicit surfaces J. Comp. Phys. 174 (1) J. W. Cahn, P. Fife an O. Penrose A phase fiel moel for iffusion inuce grain bounary motion Acta Mater. 45 (1997) K. Deckelnick, C. M. Elliott an V. Styles Numerical iffusion inuce grain bounary motion Interfaces an Free Bounaries 3 (1)

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