Surface finite elements for parabolic equations

Transcription

1 1 Surface finite elements for parabolic equations G. Dziuk Abteilung für Angewandte Mathematik University of Freiburg Hermann-Herder-Straße 1 D 7914 Freiburg i. Br. Germany C. M. Elliott Department of Mathematics University of Sussex Falmer, Brighton BN1 9RF United Kingdom Abstract In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces in R n+1. The key idea is based on the approximation of by a polyhedral surface h consisting of a union of simplices (triangles for n = 2, intervals for n = 1) with vertices on. A finite element space of functions is then defined by taking the continuous functions on h which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth oder linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. 1 Introduction Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics and materials science and more recently in the mathematics of images. In this paper we propose a mathematical approach to the formulation and finite element approximation of parabolic equations on a surface in R n+1 (n=1,2). We give examples of linear and nonlinear equations. In particular we

2 2 show how surface level set and phase field models can be used to compute the motion of curves on surfaces. 1.1 The diffusion equation Conservation on a hypersurface of a scalar u with a diffusive flux D w, where D is the diffusivity tensor and w is a scalar, leads to the diffusion equation u t (D w) = (1.1) on. Here is the tangential or surface gradient. If is empty then the equation does not need a boundary condition. Otherwise we can impose Dirichlet or no flux boundary conditions on. Choosing various constitutive relations to define the relationship between the flux and u leads to a variety of second and fourth order linear and nonlinear parabolic equations. For example the constitutive relations w = u and w = u lead to linear second and fourth order order diffusion equations. 1.2 The finite element method In this paper we propose a finite element approximation based on the variational form u t ϕ + D w ϕ = (1.2) where ϕ is an arbitrary test function defined on the surface. This provides the basis of our surface finite element method (SFEM) which is applicable to arbitrary n dimensional hypersurfaces in R n+1 (curves in R 2 ) with or without boundary. Indeed this is the extension of the method from [1] for the Laplace-Beltrami equation, which was extended to linear second order diffusion equations on moving surfaces in [11]. The principal idea is to use a polyhedral approximation of based on a triangulated surface. It follows that a quite natural local piecewise linear parameterisation of the surface is employed rather than a global one. The finite element space is then the space of continuous piecewise linear functions on the triangulated surface. The implementation is thus rather similar to that for solving the diffusion equation on flat stationary domains. For example, for w = u, the backward Euler time discretization leads to the SFEM scheme 1 ( Mα m+1 Mα m) + Sα m+1 = τ where M and S are the surface mass and stiffness matrices and α m is the vector of nodal values for the approximation of u at time t m. Here, τ denotes the time step size. Observe that this approach to evolutionary surface partial differential equations was used in [14] to evolve a surface by mean curvature flow. See also [5]. 1.3 Level set or implicit surface approach An alternative approach to our method based on the use of (1.2) is to embed the surface in a family of level set surfaces [3, 4, 2, 1, 29, 12, 13]. This Eulerian approach can be discretized on a Cartesian grid in R n+1 and has the usual advantages and disadvantages of level set methods. Equations on surfaces also arise in phase field models [18, 7, 24].

3 3 1.4 Applications Models involving partial differential equations on surfaces arise in many areas including material science, bio-physics, fluid mechanics and image processing. For example, phase formation of surface alloying by spinodal decomposition resulting in two dimensional structures has been modelled by the Cahn-Hilliard equation on surfaces, [8]. See also [25, 26] for studies of the Allen-Cahn and Cahn-Hilliard equations in the context of phase ordering on surfaces. Other examples in the physical sciences include diffusion induced grain boundary motion [18, 7, 22] and the Ginzburg-Landau model for superconductivity [9]. In image processing we mention geodesic flow of curves on surfaces and active contours for segmentation on surfaces, [21, 23]. 1.5 Outline of paper The layout of the paper is as follows. We begin in section 2 by defining notation and essential concepts from elementary differential geometry necessary to describe the problem and numerical method. Several linear and nonlinear partial differential equations of second and fourth order are described in section 3 together with a number of computational results. In section 4 the finite element method is defined and some preliminary approximation results are shown. Error bounds for the semi-discretization in space are proved in section 5. Implementation issues are discussed in section 6. 2 Basic notation and surface derivatives Let be a compact smooth connected and oriented hypersurface in R n+1 (n = 1, 2). In order to formulate the model it is convenient to use a level set description of. We assume the existence of a smooth level set function d = d(x), x R n+1, so that = {x N d(x) = } where N is an open subset of R n+1 in which d and chosen so that d C 2 (N ). The orientation of is set by taking the normal ν to to be in the direction of increasing d. Hence we define a normal vector field by Throughout this paper we denote by ν(x) = d(x) d(x). P(x) = I ν(x) ν(x) (2.1) the projection onto the tangent space of at x. Observe that a possible choice for d is a signed distance function and in that case d = 1 on N. For later use we mention that N can be chosen such that for every x N and there exists a unique a(x) such that x = a(x) + d(x)ν(a(x)), (2.2)

4 4 where here d denotes the signed distance function to. For any function η defined on an open subset N of R n+1 containing we define its tangential gradient on by η = η η ν ν = P η where, for x and y in R n+1, x y denotes the usual scalar product and η denotes the usual gradient on R n+1. The tangential gradient η only depends on the values of η restricted to and η ν =. The components of the tangential gradient will be denoted by η = ( D 1 η,..., D n+1 η ). The Laplace-Beltrami operator on is defined as the tangential divergence of the tangential gradient: n+1 η = η = D i D i η. Let have a boundary whose intrinsic unit outer normal, tangential to, is denoted by µ. Then the formula for integration by parts on is η = ηhν + ηµ, (2.3) where H denotes the mean curvature of with respect to ν, which is given by i=1 H = ν. (2.4) The orientation is such that for a sphere = {x R n+1 x x = R} and the choice d(x) = R x x the normal is pointing into the ball B R (x ) = {x R n+1 x x < R} and the mean curvature of is given by H = n. Note that H is the sum of R the principle curvatures rather than the arithmetic mean and hence differs from the common definition by a factor n. The mean curvature vector Hν is invariant with respect to the choice of the sign of d. Green s formula on the surface is ξ η = ξ η µ ξ η. (2.5) If is closed then is empty and the boundary terms in (2.3) do not appear. For these facts about tangential derivatives we refer to [19], pp Note that, in general, higher order tangential derivatives do not commute. We shall use Sobolev spaces on surfaces. For a given surface we define H 1 () = {η L 2 () η L 2 () n+1 } and H 1 () in the obvious way, if. For smooth enough we analogously define the Sobolev spaces H k () for k N. See [28] for additional information.

5 5 3 Parabolic equations on surfaces 3.1 Conservation and diffusion on a surface Let u be the density of a scalar quantity on (for example mass per unit area n = 2 or mass per unit length n = 1). The basic conservation law we wish to consider can be formulated for an arbitrary portion M of. The law is that, for every M, d dt M u = M q µ (3.1) where, M is the boundary of M (a curve if n = 2 and the end points of a curve if n = 1) and µ is the conormal on M. Thus µ is the unit normal to M pointing out of M and tangential to. The surface flux is denoted by q. Observe that components of q normal to M do not contribute to the flux, so we may assume that q is a tangent vector. With the use of integration by parts, (2.3), we obtain q µ = q + q νh = q, so that M M M M (u t + q) =, which implies the pointwise conservation law We take q to be the diffusive flux M u t + q = on. (3.2) q = D w (3.3) where D is a symmetric mobility tensor with the property that it maps the tangent space into itself at every point of, i. e. for every tangent vector ν. This leads to the equation Dν ν = (3.4) u t (D w) = on. (3.5) If =, i.e. the surface has no boundary, then there is no need for boundary conditions. For example, this would be the case if is the bounding surface of a domain. If is non-empty then we impose the homogeneous Dirichlet boundary condition u = on. (3.6) Again if is non-empty then we could impose the no flux condtion D w µ = on. (3.7)

6 6 The variational form (1.2) then is an easy consequence. We multiply equation (1.1) by an adequate test function ϕ and integrate over. We then obtain using integration by parts (2.3): u t ϕ + D w ϕ =. (3.8) Here and in all subsequent equations we have to impose an initial condition u(, ) = u. In the following we will not mention this condition when it is obviously required. Remark 3.1. Note that for arbitrary D this weak equation implies u t (PD w) = on. (3.9) Also, in general, constant coefficient mobility tensors D will not satisfy assumption (3.4) and P will not be constant coefficient. Remark 3.2 (Conservation). Taking ϕ = 1 yields the conservation equation d u =. dt Example 3.3 (Linear diffusion). Setting w = u and D = I we find the heat equation on surfaces u t = u. (3.1) Clearly this can be generalized to the inhomogenous equation u t u = f. (3.11) In Figure 1 we display the solution at three successive times of (3.11) on the torus = {x R 3 ( x x 2 2 1) 2 + x 2 3 = 1 16 } (3.12) with the right hand side being a regularized version of the characteristic function f(x, t) = 1 χ G (x), x, with G = {x x (, 1, ) <.25} and with initial value u =. Figure 1: Heating up a torus. Solution at times.6934,.3467 and For the choice D = A = (a ij (x, t)) i,j=1,...,n+1 with a symmetric matrix A which satisfies (3.4) and is positive definite on the space orthogonal to ν we obtain the linear parabolic PDE u t = n+1 i,j=1 D i ( aij D j u ). (3.13)

7 7 Example 3.4 (Nonlinear diffusion). Setting we find the nonlinear diffusion equation where w = f(u) and D = m(u)i u t = (a(u) u) (3.14) a(u) = m(u)f (u). Clearly one recovers linear diffusion and the porous medium equation by suitable choices. Example 3.5 (Parabolic surface p Laplace equation). Setting w = u and, for 1 < p, D = u p 2 I yields the following parabolic surface p Laplace equation which is L 2 ()-gradient flow for the energy E p (u) = 1 p u t = ( u p 2 u ) (3.15) u p. (3.16) Example 3.6 (Total variation flow). Setting w = u and taking D = u 1 I leads formally to the surface total variation flow u t = Example 3.7 (Fourth order linear diffusion). The choice leads to the fourth order linear diffusion equation u u. (3.17) w = u (3.18) u t = (D u). (3.19) Figure 2: We solve the Cahn-Hilliard equation on this surface. Views from the x 3 -axis, tilted, and view from the x 1 -axis. Example 3.8 (Surface Cahn-Hilliard equation). Setting w = ɛ u + 1 ɛ Ψ (u), (3.2)

8 8 where Ψ : R R typically is a double well potential, Ψ(u) = 1 4 (1 u2 ) 2, (3.21) leads to the fourth order Cahn-Hilliard equation ( u t = D (ɛ u 1 ) ɛ Ψ (u)). (3.22) Using second order splitting as introduced in [16] we formulate the problem as a system of second order equations in space. We solved the Cahn-Hilliard equation for D = I on a strongly deformed surface (see Figure 2). The topology of the surface is cylindrical: = F ( ), where F (x 1, x 2, x 3 ) = f(x 1, x 2, x 3 )(x 1, x 2, ) + (,, x 3 ) with f(x) = sin (14ϕ) sin (15x 3 ) and ϕ is the polar angle in the x 1, x 2 -plane. As reference surface we have chosen the cylinder = {x R 3 x x 2 2 < 1, < x 3 < 1.2}. As initial value we have taken u (x) = sin (4x 1 ) sin (3x 2 ) sin (5x 3 ). The grid contained N = 2496 nodes. We have used ε =.1. Figure 3: Solution of the Cahn-Hilliard equation on the surface from Figure 2. View from the x 1 -axis with x 3 -axis pointing to the right. The colours represent the magnitude of the solution between 1 and 1. Times from left to right: t =.,.1247, Figure 4: The same situation as in Figure 3 seen from the x 3 -axis. Figure 5: The same situation as in Figure 3: tilted view. 3.2 Equations in non-conservation form Here we formulate several time dependent equations on surfaces which are not in conservation form. Example 3.9 (Surface Allen-Cahn equation). Consideration of the L 2 ()-gradient flow for the gradient energy functional E(v) = ( ɛ 2 v 2 + Ψ(v) ), (3.23) ɛ (ɛ > ) leads to ɛu t = ɛ u 1 ɛ Ψ (u). (3.24)

9 9 Here the potential (3.21) gives the classical Allen-Cahn equation on a surface. In Figure 3.9 we show results of the numerical solution of the surface Allen-Cahn equation on the torus (3.12) with initial data u (x) = sin (3πx 3 ) cos (3ϕ) where ϕ denotes the polar angle in the x 1, x 2 -plane. We observe the rapid evolution to a checkerboard pattern on the torus (first row). After some time this pattern is dissolved and the solution evolves to an interesting stationary solution (second row). It is interesting to observe that the red and blue regions are connected, so that there is just one diffuse interface which approximates a curve of zero geodesic curvature. Figure 6: Solution of the surface Allen-Cahn equation on a torus. In Figure 7 we show the typical decomposition effect of the Allen-Cahn equation on a deformed cylinder. Here we used for the initial value u a random distribution of matter on the surface. Figure 7: Solution of the Allen-Cahn equation on a deformed cylinder (surface, initial distribution, distribution after some time). Example 3.1 (Level set geodesic mean curvature flow). We formulate the level set equation u u t = u u. (3.25) For example, we have in mind the evolution, on the fixed surface in R 3, of a closed curve C(t) which is evolving in the intrinsic normal direction ν g with a velocity V g given by minus it s geodesic curvature κ g. The curve C(t) is given by the zero level set on of u(, t) and ν g = u u, κ u g = u, V g = u t u. Figure 8 shows how circles on a dumbbell shaped surface move under geodesic mean curvature flow. The surface is given as = F (S 2 ) where F (x) = (x 1, η(x)x 2, η(x)x 3 ) with η(x) = 1 x (1 x 2 1 ) 2 / x x 2 3 for x S 2. The initial function is u (x) = x In Figure 8 we display all the level lines {x u(x, t) = c} for c between 1.5 and 1.25 with intervals of.2. We observe that circles shrink and either move to the center of the dumbbell s neck or shrink to round points at the extreme ends of the dumbbell. Each of these possibilities is displayed in Figure 9 for a single circle. Finally in Figure 1 we show the evolution of many level sets on the sphere. We see topology change of the level sets. In this computation we use the initial value u (x) = sin (5π(x 1.25)) sin (3π(x )). In all these computions of geodesic curve shortening flow we regularized the equations by replacing u by ε 2 + u 2, and we have taken the parameter ε proportional to the grid size h.

10 1 Figure 8: Geodesic curve shortening flow on a dumbbell. Figure 9: Geodesic curve shortening flow for two circles on a dumbbell. Figure 1: Level set mean curvature flow on the sphere. Example 3.11 (Level Set Surface Active Contours). We formulate the level set equation ( u t = u f ) u (3.26) u where f = I σ 2. (3.27) Here I σ is a smoothed image which is essentially a characteristic function with sharp edges. The evolution of the zero level set curve C(t) is designed to detect the edge. Example 3.12 (Anisotropic geodesic level set mean curvature flow). We formulate the level set equation for anisotropic mean curvature flow on the given surface as µ( u)u t = u Dγ( u) (3.28) where γ : R n+1 \ {} (, ), γ() =, is an anisotropy function, smooth and positively homogeneous of degree one. Here Dγ denotes the gradient of γ. µ is a positive and -homogeneous function. Example 3.13 (Level set geodesic surface diffusion). We formulate the level set equation u t = ( u (I ν g ν g ) w). (3.29) w = u u. (3.3) For example, we have in mind the evolution, on the fixed surface in R 3, of a closed curve C(t) which is evolving in the intrinsic normal direction ν g with a velocity V given by the geodesic Laplacian of the geodesic curvature κ g. The curve C(t) is given by the zero level set on of u(, t). Example 3.14 (Level set geodesic Willmore flow). We formulate the level set equation ( ) 1 u t = u u (I ν g ν g ) w + 1 ( ) w 2 2 u u. (3.31) u 3 u

11 11 u w = u u (3.32) Here the zero level set of u is the curve C(t) constrained to lie on which evolves according to L 2 gradient flow for the energy E C = 1 κ 2 2 g. (3.33). 4 Finite element approximation 4.1 Finite elements on surfaces The smooth surface ( = ) is approximated by a surface h N, ( h = ), which is of class C,1. In particular for n = 2, h is a triangulated (and hence polyhedral) surface consisting of triangles e in T h with maximum diameter being denoted by h and inner radius bounded below by σ ch with some c >. The vertices {X j } N j=1 of the triangles are taken to sit on so that h is an interpolation. Note that by (2.2) for every triangle e h there is a unique curved triangle T = a(e). In order to avoid a global double covering (see [11]) we assume that, C for each point a there is at most one point x h with a = a(x). (4.1) This implies that there is a bijective correspondence between the triangles on h and the induced curvilinear triangles on. For any continuous function η defined on the discrete surface h we may define an extension or lift onto by η l (a) = η(x(a)) a (4.2) where by (2.2) and our assumptions, x(a) is defined as the unique solution of x = a + d(x)ν(a). (4.3) Furthermore we understand by η l (x) the constant extension from in the normal direction ν(a). An application of the chain rule for differentiation gives where P h is the discrete projection We have a finite element space h η = P h (I dh) η l (a(x)) (4.4) P h,ij = δ ij ν h,i ν h,j. S h = { φ C ( h ) φ e is linear affine for each e T h } = span{χj j = 1,..., N}. By χ j we denote the common nodal basis function from S h which is 1 at the node X j and zero at all other nodes. It is convenient to introduce S l h = { η l C () η l (a) = η(x(a)), η S h and x(a) given by (4.3) } In the error analysis of the finite element scheme we shall need the following technical Lemmas [1].

12 12 Lemma 4.1. Assume and h are as above and let d denote the oriented distance function of. Then d L ( h ) ch 2. (4.5) Let δ h be the quotient between the smooth and discrete surface measures da and da h so that δ h da h = da, and let R h = 1 δ h P(I dh)p h (I dh), H ij = d xi x j. Then we have the following estimates sup 1 δ h ch 2 (4.6) h and sup (I R h )P ch 2. (4.7) h Lemma 4.2. Let η : h R with lift η l : R. Then for the planar and curved triangles e h and T the following estimates hold. There is a constant c > independent of h such that 1 c η L 2 (e) η l L 2 (T ) c η L 2 (e) (4.8) 1 c h η L 2 (e) η l L 2 (T ) c h η L 2 (e) (4.9) 2 h η L 2 (e) c 2 η l L 2 (T ) + ch η l L 2 (T ) (4.1) It is convenient to introduce a piecewise linear interpolant which is constructed in an obvious way. Observing that the pointwise linear interpolation Ĩhη S h is well defined for η belonging to H 2 (), I h η is defined by lifting Ĩhη onto according to (4.2), so that I h η = (Ĩhη) l. The following interpolation results hold, [1]. Lemma 4.3. For given η H 2 () there exists a unique I h η S l h such that η I h η L 2 () + h (η I h η) L 2 () ch 2 ( 2 η L 2 () + h η L 2 ()). (4.11) 4.2 Semi-discrete approximation Our SFEM, based on the finite element spaces introduced in this section, is applicable to all the partial differential equations described in Section 3. Here we give some representative examples. They show that solving partial differential equations on surfaces in R n+1 is analogous to solving equations on domains in R n. The fourth order problems considered here are treated by the second order splitting approach of [16] Conservation and diffusion We begin by describing the semi-discretization of (3.8) in space. Let (U(, t), W (, t)) S h S h be such that U t φ + h D l h W h φ = h φ S h. (4.12) Setting U(, t) = N α j (t)χ j ( ), W (, t) = j=1 N β j (t)χ j ( ) j=1

13 13 we find that, φ S h, h j=1 N α j,t χ j φ + h D l and taking φ = χ k, k = 1,..., N we obtain where N β j (t) h χ j h φ = j=1 M α + Sβ = (4.13) M jk = χ j χ k, h S jk = D l h χ j h χ k. h Here D l is such that its lift is the diffusivity D so that ( D l) l = D. Example 4.4 (Linear diffusion). In this case W=U so that the problem becomes: Find U(, t) S h such that U t φ + h D l h U h φ = h φ S h (4.14) yielding the time dependent ordinary differential equations M α + Sα =. (4.15) Example 4.5 (Fourth order linear diffusion). In this case we employ the following finite element approximation of (3.18) W φ h h U h φ = h φ S h (4.16) which yields where Substituting into (4.13) yields Mβ S α = Sjk = h χ j h χ k. h M α + SM 1 S α =. (4.17) Example 4.6 (Surface Cahn Hilliard equation). Find (U(, t), W (, t)) S h S h such that U t φ + D l h W h φ = φ S h. (4.18) h h ɛ h U h φ + ĨhΨ (U) φ = W φ φ S h. (4.19) h ɛ h

14 14 Note the use of ĨhΨ (U). The discrete equations become and eliminating β we obtain M α + Sβ = Mβ ɛs α MΨ (α) = ɛ Here we use the notation that M α + ɛsm 1 S α + SΨ (α) ɛ =. (4.2) (Ψ (α)) j = Ψ (α j ) Equations in non-conservation form Example 4.7 (Surface Allen-Cahn equation). The approximate problem is to find U(, t) S h such that ɛ U t φ + (ɛ h U h φ + ĨhΨ (U) φ) = φ S h (4.21) h h ɛ which may be rewritten in matrix form as ɛm α + ɛs α + MΨ (α) ɛ =. (4.22) Example 4.8 (Level set geodesic mean curvature flow). The semi-discrete finite element approximation is: Find U(, t) S h such that U t h U φ + h φ = φ S h. (4.23) h U ɛ h U ɛ h h Here we have introduced the following regularised l 2 norm p ɛ = p 2 + ɛ 2 in order to avoid dividing by zero when h U =. It follows that the discrete equations may be written as M U α + S U α = (4.24) where the weighted mass and stiffness matrices are χ j χ k h χ j h χ k (M U ) jk =, (S U ) jk =. h U ɛ h h U ɛ h Example 4.9 (Level set geodesic surface diffusion). Find (U(, t), W (, t)) S h S h such that U t φ + h U Pg h h W h φ = φ S h (4.25) h h h U W φ + h U h φ = φ S h. (4.26) h

15 15 Here we have used the notation ν h g = h U h U, P h g = I ν h g ν h g. The discrete equations become M α + S h g β = Mβ S U α = and eliminating β we obtain M α + S h g M 1 S U α = (4.27) where (Sg h ) jk = h U Pg h h χ j h χ k. h 5 Error bounds In this section we will prove convergence results for the parabolic plate equation on a surface. We do this in order to give a nontrivial example for our approximation methods. Several of the equations discussed in the previous sections can be treated along the lines of the convergence proof for the parabolic plate equation on a surface. In order to simplify the presentation we treat the case D = I only. 5.1 Linear Diffusion Lemma 5.1 (Stability). Let U, W be a solution of the semi-discrete scheme as in Example 3.7 with initial value U(, ) = U and U l, W l its lift according to (4.2). Then the following stability estimates hold: T sup U l 2 L 2 () + W l 2 L 2 () c U l 2 L 2 (), (5.1) (,T ) T Ut l 2 L 2 () + sup W l 2 L 2 () c W l (, ) 2 L 2 (), (5.2) (,T ) T sup U l 2 L 2 () + W l 2 L 2 () c( U l 2 L 2 () + W l (, ) 2 L 2 ()). (5.3) (,T ) Proof. If we test the diffusion equation (4.12) with φ = U t and take the time derivative of the equation (4.16) and then test that equation with φ = W and take the sum of the two resulting equations we get Ut d W 2 = h 2 dt h which leads to the estimate (5.2).

16 16 Choosing φ = U in (4.12), φ = W in (4.16) and taking the sum leads to 1 d U 2 + W 2 = 2 dt h h and this gives the estimate (5.1). The estimate (5.3) follows by taking φ = U in (4.16) with the use of the previous estimates. Similarly the choice φ = W in (4.12) gives the estimate for the gradient of W. We now can prove a convergence result for piecewise linear finite elements on two dimensional surfaces. Note that as discrete initial value we take a realistic function which can be computed. Theorem 5.2 (Convergence). Let u, w = u be a sufficiently smooth solution of u t + 2 u =, u(, ) = u on on the sufficiently smooth closed surface and let U, W be the discrete solution according to (4.12) and (4.16). As initial value U we take the discrete Ritz projection of the continuous initial value u defined by h U h φ = h h u l h φ h φ S h (5.4) and h U =. With the lifts U l of U and W l of W we then have the error estimates and ( T where and sup u(, t) U l (, t) L 2 () + h sup (u(, t) U l (, t)) L 2 () C 1 h 2 (5.5) t (,T ) t (,T ) ) 1 ( w(, t) W l (, t) 2 2 T ) 1 L 2 () dt + h (w(, t) W l (, t)) 2 2 L 2 () dt ( T ) 1 C 1 = c( sup u(, t) H 4 () + u t (, t) 2 2 H 2 () dt + u H 4 ()) t (,T ) ( T ) 1 ( C 2 = c( u(, t) 2 2 T ) 1 H 4 () + u t (, t) 2 2 H 2 () dt + u H 4 ()). C 2 h 2 (5.6)

17 17 Proof. In order to make the proof clearer we only treat the case D = I here. The proof is easily extended to the general case. The error bounds follow the classic Galerkin error analysis for parabolic equations [27] and rely on a suitable form of the error equation. In order to compare discrete and continuous solution both should be defined on the same surface which we take to be the continuous surface. The continuous equations read u t ϕ + w ϕ =, wψ u ψ = (5.7) for all ϕ, ψ H 1 (), and the discrete equations are given by U t φ + h h W h φ =, h W ψ h h U h ψ = h (5.8) for all φ, ψ S h. We lift the discrete equations to the continuous surface as it was described in (4.2). We define U l, W l and φ l, ψ l by U(x, t) = U l (a(x), t), W (x, t) = W l (a(x), t), φ(x) = φ l (a(x)), ψ(x) = ψ l (a(x)) for points x h. For better understanding of the following, we introduce the notation u h (x, t) = U l (x, t), w h (x, t) = W l (x, t), x. We recall the abbreviation (see Lemma 4.1) R h (x) = 1 δ h (x) P(x)(I d(x)h(x))p h(x)(i d(x)h(x)), x h together with its lifted version Rh l (a(x)) = R h(x), x h. Thus (5.8) leads to 1 u h,t ϕ h + R l 1 δ h h l w h ϕ h =, w h ψ h R δ h l h l u h ψ h = (5.9) for all ϕ h, ψ h Sh l. We take the differences of the equations (5.7) at ϕ h, ψ h and (5.9). The error relation between continuous and lifted discrete solution then is given by t (u 1 ( ) u δh l h,t )ϕ h + w Rh l w h ϕ h =, (w 1 ( ) w δh l h )ψ h u Rh l u h ψ h = (5.1) for all ϕ h, ψ h Sh l, and with the use of the estimates from Lemma 4.1 we obtain (u u h ) t ϕ h + (w w h ) ϕ h ch 2 ( w h ϕ h + u h,t ϕ h ), (w w h )ψ h (u u h ) ψ h ch 2 ( u h ψ h + w h ψ h )(5.11) for all ϕ h, ψ h S l h.

18 18 We define the Ritz-Galerkin projection ṽ h of a function v as the unique solution ṽ h Sh l with ṽh = of ṽ h ϕ h = v ϕ h ϕ h Sh. l (5.12) It is easily shown, with the help of Lemma 4.3, that for a sufficiently smooth function v one has the estimates v ṽ h L 2 () + h (v ṽ h ) L 2 () ch 2 v H 2 () (5.13) and similar estimates for the time derivative of v if v t H 2 (). We now use ϕ h = ũ h u h and ψ h = w h w h as test functions in (5.11), take the sum of the two resulting equations, and arrive at 1 d 2 dt ũ h u h 2 L 2 () + w h w h 2 L 2 () ũ h,t u t L 2 () ũ h u h L 2 () + w h w L 2 () w h w h L 2 () +ch ( ) 2 w h L 2 () (ũ h u h ) L 2 () + u h,t L 2 () ũ h u h L 2 () +ch ( ) 2 u h L 2 () ( w h w h ) L 2 () + w h L 2 () w h w h L 2 () ε ũ h u h 2 L 2 () w h w h 2 L 2 () +c ε h 4 ( u h 2 L 2 () + w h 2 L 2 () + u h,t 2 L 2 () + w h 2 L 2 () + u t 2 H 2 () + w 2 H 2 () +ε (ũ h u h ) 2 L 2 () + ε ( w h w h ) 2 L 2 (). (5.14) Here ε is a positive number which will be chosen later. The last two terms on the right hand side of this inequality are treated as follows. In (5.1) we choose ϕ h = w h w h and ψ h = ũ h,t u h,t as test functions and subtract the resulting equations. Thus ( w h Rh l w h ) ( w h w h ) + ( ũ h Rh l u h ) (ũ h,t u h,t ) ( = (u t 1 u δh l h,t )( w h w h ) (w 1 ) w δh l h )(ũ h,t u h,t ). (5.15) We estimate the terms in this equation separately. For the first term on the left hand side we have with Lemma 4.1 ( w h Rh l w h ) ( w h w h ) ( w h w h ) 2 ch 2 w h ( w h w h ) 1 ( w h w h ) 2 ch 4 w h 2. (5.16) 2 )

19 19 The second term on the left hand side of (5.15) is treated as follows: ( ũ h Rh l u h ) (ũ h,t u h,t ) 1 d R 2 dt h l (ũ h u h ) (ũ h u h ) + d (I R dt h) l ũ h (ũ h u h ) ε (ũ h u h ) 2 c ε h 4 ũ h,t 2. (5.17) Here we have used the symmetry of the matrix PRh l P. And the right hand side of (5.15) is rewritten as ( (u t 1 u δh l h,t )( w h w h ) (w 1 ) w δh l h )(ũ h,t u h,t ) = d u h )( dt (u 1 w δh l h w) u h )( (u 1 w δh l h,t w t ) + (u t ũ h,t )(w w h ) + t w h ũ h,t w h )(1 (u 1 ) δh l d u h )( dt (u 1 w δh l h w) ε (ũ h u h ) 2 ε ( w h w h ) 2 c ε h 4 ( u 2 H 2 () + u t 2 H 2 () + w 2 H 2 () + w t 2 H 2 ()). (5.18) We now collect the estimates (5.16), (5.17) and (5.18), integrate with respect to time. For h h this gives the estimate t (ũ h u h ) 2 L 2 () + ( w h w h ) 2 L 2 () dt t t ε ũ h u h 2 L 2 () + ε ũ h u h 2 H 1 () dt + ε w h w h 2 L 2 () dt ( t ) +c ε h 4 u 2 H 2 () + w 2 H 2 () + u 2 H 2 () + w 2 H 2 () dt + A,h. (5.19) By A,h we denote the contribution of the initial values A.h = 1 R 2 h l (ũ h (, ) u h (, )) (ũ h (, ) u h (, )) + (I Rh) l ũ h (, ) (ũ h (, ) u h (, )) + u h (, ))( (u 1 w δh l h (, ) w(, )) Our choice of u h = U l (see (5.4)) leads to (ũ h (, ) u h ) L 2 () ch 3 u H 2 (), (5.2) ũ h (, ) u h L 2 () ch 2 u H 2 (). (5.21)

20 2 This estimate is left to the reader. It is an easy consequence of the definition of the discrete initial value, the geometric estimates from Lemma 4.1 and the assumptions on the mean values of ũ h (, ) and u h. Altogether we arrive at the following estimate for the gradient terms: (ũ h u h ) 2 L 2 () + t ( w h w h ) 2 L 2 () dt (5.22) t t ε ũ h u h 2 L 2 () + ε ũ h u h 2 H 1 () dt + ε w h w h 2 L 2 () dt ( t ) +c ε h 4 u 2 H 2 () + w 2 H 2 () + u 2 H 2 () + w 2 H 2 () dt + u 2 H 2 (). We now integrate estimate (5.14) with respect to time and get ũ h u h 2 L 2 () + t ε t w h w h 2 L 2 () dt (5.23) t ũ h u h 2 H 1 () dt + ε ( w h w h ) 2 L 2 () dt ( t ) +c ε h 4 u h 2 L 2 () + w h 2 H 1 () + u h,t 2 L 2 () + u t 2 H 2 () + w 2 H 2 () dt +ch 4 u 2 H 2 () We now add the estimates (5.22) and (5.23) and choose ε small enough. ũ h u h 2 H 1 () + t t ũ h u h 2 H 1 () dt w h w h 2 H 1 () dt (5.24) t +ch 4 u t 2 H 2 () + u 2 H 2 () + w 2 H 2 () dt t + ch4 u h 2 L 2 () + w h 2 H 1 () + u h,t 2 L 2 () dt +ch 4 ( u 2 H 2 () + w 2 H 2 () + u 2 H 2 () + w(, ) 2 H 2 () After an additional Gronwall argument we deduce the final estimate T sup ũ h u h 2 H 1 () + w h w h 2 H 1 () dt (,T ) T ch 4 ( u h,t 2 L 2 () + u h 2 L 2 () + w h 2 H 1 () dt T + u 2 H 4 () + u t 2 H 2 () + u 2 H 4 () dt). The stability estimates from Lemma 5.1 together with the error estimates for the Ritz- Galerkin projections finally prove the Theorem. )

21 21 We finally mention how the initial value of W is treated. For the estimate of W l (, ) L 2 () we use an inverse inequality. From (4.16) and (5.4) we infer W (, ) 2 L 2 ( h ) = h U h W (, ) = h u l h W (, ) h h = Rh l u W l (, ) = (Rh l I) u W l (, ) + u W l (, ) ch 2 u L 2 () W l (, ) L 2 () + u L 2 () W l (, ) L 2 () ch 2 u L 2 () h W (, ) L 2 ( h ) + c u H 2 () W (, ) L 2 () c u H 2 () W (, ) L 2 ( h ), and so, W (, ) L 2 ( h ) c u H 2 (). 6 Implementation For our computations we used the same time discretizations as in the Cartesian case. For the linear problems these were the standard first order implicit time discretizations. For the nonlinear problems time discretizations were used to linearize the problem. A survey of such time discretizations can be found in [6, 15, 17]. The resulting linear systems were solved with CG algorithms. We would like to point out that the implementations of the fully discrete schemes are nearly the same as for plane problems. The only difference for the finite element code is that the nodes lie in R n+1. A finite element program sets up the stiffness matrix, the mass matrix and the right hand side of the linear system by looping over all triangles. In this loop it visits each element once and computes the element stiffness matrix, the element mass matrix and the element right hand side and sums the result to the correct places in the matrices or the right hand side. In order to demonstrate the simplicity of the algorithm we give a program for the computation of the element stiffness matrix Sij e = χ e i χ e j = e χ e i e χ e j, i, j =, 1, 2 e e for two dimensional surfaces in three space dimensions. Here χ e, χ e 1, χ e 2 are the element basis functions, i. e. the restictions of the global basis functions to the element e. Note that in the definition of the element stiffness matrix the tangential gradients become Cartesian gradients because e is planar. In the following denotes the vector product in R 3. Algorithm 6.1 (Element stiffness matrix). For a given triangle e with vertices a, a 1, a 2 R 3 calculate the area area = 1 2 (a 1 a ) (a 2 a ),

22 22 calculate the vectors b k = a k+1 a k (k =, 1, 2 mod 3), c k = b k b k+1 b k 1 b k b k 1 b k+1 (k =, 1, 2 mod 3), and then set up the element stiffness matrix as S e ij = 7 Concluding remarks 1 16 area c i c j (i, j =, 1, 2 mod 3). The approach described here is directly applicable to other boundary conditions when is non-empty such as the non-homogeneous Dirichlet condition or Neumann boundary condition u = g on. u µ = g on. The method could be developed to apply to a coupling with field equations away from the surface. Observe that the approximating surfaces are polyhedral. It is a challenge to extend this approach to higher order approximations of the surface and higher order finite element methods. Although the exposition has been concerned with triangulated surfaces in R 3, immediately applicable to curves, the methodology is also applicable to hypersurfaces in higher space dimensions. Furthermore equations can be solved by this approach on surfaces with self-intersections and on non-oriented surfaces. Acknowledgements: This work was began whilst the authors participated in the 23 programme Computational Challenges in Partial Differential Equations at the Isaac Newton Institute, Cambridge, UK. The work was supported by the Deutsche Forschungsgemeinschaft via DFG-Forschergruppe Nonlinear partial differential equations: Theoretical and numerical analysis and by the UK EPSRC via the Mathematics Research Network: Computation and Numerical analysis for Multiscale and Multiphysics Modelling. Part of this work was done during a stay of the first author at the ICM at the University of Warsaw supported by the Alexander von Humboldt Honorary Fellowship 25 granted by the Foundation for Polish Science. The graphical presentations were performed with the package GRAPE. References [1] D. Adalsteinsson and J. A. Sethian Transport and diffusion of material quantities on propagating interfaces via level set methods J. Computational Physics 185 (23) [2] Th. Aubin Nonlinear analysis on manifolds. Monge-Ampère equations Springer Berlin-Heidelberg-New York (1982)

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