A FINITE ELEMENT METHOD FOR ELLIPTIC EQUATIONS ON SURFACES

Transcription

1 A FINITE ELEMENT METHOD FOR ELLIPTIC EQUATIONS ON SURFACES MAXIM A. OLSHANSKII, ARNOLD REUSKEN, AND JÖRG GRANDE Abstract. In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the H 1 and in the L 2 -norm. Results of numerical experiments are included that confirm this optimality. Key words. Surface, interface, finite element, level set method, two-phase flow, Marangoni AMS subject classifications. 58J32, 65N15, 65N30, 76D45, 76T99 1. Introduction. Moving hypersurfaces and interfaces appear in many physical processes, for example in multiphase flows and flows with free surfaces. Certain mathematical models involve elliptic partial differential equations posed on such surfaces. This happens, for example, in multiphase fluids if one takes so-called surface active agents (surfactants into account. These surfactants induce tangential surface tension forces and thus cause Marangoni phenomena [9, 10]. Numerical simulations play an important role for a better understanding and prediction of processes involving this or other surface phenomena. In mathematical models surface equations are often coupled with other equations that are formulated in a (fixed domain which contains the surface. In such a setting a common approach is to use a splitting scheme that allows to solve at each time step a sequence of simpler (decoupled equations. Doing so one has to solve numerically at each time step an elliptic type of equation on a surface. The surface may vary from one time step to another and usually only some discrete approximation of the surface is available. A well-known finite element method for solving elliptic equations on surfaces, initiated by the paper [5], consists of approximating the surface by a piecewise polygonal surface and using a finite element space on a triangulation of this discrete surface, cf. [3, 9]. If the surface is changing in time, then this approach leads to time-dependent triangulations and time-dependent finite element spaces. Implementing this requires substantial data handling and programming effort. Another approach has recently been introduced in [2]. The method in that paper applies to cases in which the surface is given implicitly by some level set function and the key idea is to solve the partial differential equation on a narrow band around the surface. Unfitted finite element spaces on this narrow band are used for discretization. In this paper we introduce a new technique for the numerical solution of an elliptic equation posed on a hypersurface. The main idea is to use time-independent finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface. Our method is particularly Department of Mechanics and Mathematics, Moscow State M.V. Lomonosov University, Moscow , Russia; Maxim.Olshanskii@mtu-net.ru. This author was partially supported by the Russian Foundation for Basic Research through projects and Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D Aachen, Germany; reusken@igpm.rwth-aachen.de This work was supported by the German Research Foundation through SFB

2 suitable for problems in which the surface is given implicitly by a level set or VOF function and in which there is a coupling with a flow problem in a fixed outer domain. If in such problems one uses finite element techniques for the discetization of the flow equations in the outer domain, this setting immediately results in an easy to implement discretization method for the surface equation. The new approach does not require additional surface elements. If the surface varies in time, one has to recompute the surface stiffness matrix using the same data structures each time. Moreover, quadrature routines that are needed for these computations are often available already, since they are needed in other surface related calculations, for example surface tension forces. Opposite to the method in [2] we do not use an extension of the surface partial differential equation but instead use a restriction of the outer finite element spaces. We prove that the method has optimal order of convergence in H 1 and L 2 norms. The analysis requires shape regularity of the outer triangulation, but does not require any type of shape regularity for discrete surface elements. The number of unknowns in the resulting algebraic systems is almost the same as in the approach based on the surface finite element spaces. All these properties make the new method very attractive both from the theoretical and the practical (implementation point of view. Although our primal objective is to solve efficiently equations on moving and implicitly defined surfaces, the method is also well suited for problems with steady and/or explicitly given surfaces. The remainder of the paper is organized as follows. In section 2 we present the finite element method for the model example of the Laplace-Beltrami equation. Section 3 contains the main theoretical results of the paper concerning the approximation properties of the finite element spaces and discretization error bounds for the new method. Finally, in section 4 results of numerical experiments are given, which support the theoretical analysis of the paper. 2. Laplace-Beltrami equation and finite element discretization. In applications, the finite element method that is presented in this section is particularly suited for discretization of elliptic equations on a moving manifold = (t. In this paper, however, we restrict ourselves to the case of a fixed sufficiently smooth manifold (= (t n without boundary. As a model problem for an elliptic equation we consider the pure diffusion (i.e., Laplace-Beltrami equation. We assume that Ω is an open subset in R 3 and a connected C 2 compact hypersurface contained in Ω. For a sufficiently smooth function g : Ω R the tangential derivative (along is defined by g = g g n n. (2.1 By we denote the Laplace-Beltrami operator on. We consider the Laplace- Beltrami problem in weak form: For given f L 2 ( with fds = 0, determine u H 1 ( with u ds = 0 such that u v ds = fv ds for all v H 1 (. (2.2 The solution u is unique and satisfies u H 2 ( with u H2 ( c f L2 ( and a constant c independent of f, cf. [5]. For the discretization of this problem one needs an approximation h of. We assume that this approximate manifold is constructed as follows. Let {T h } h>0 be a family of tetrahedral triangulations of a fixed domain Ω R 3 that contains. These 2

3 triangulations are assumed to be regular, consistent and stable [1]. Take T h {T h } h>0 and denote the set of tetrahedra that form T h by {S}. We assume that h is a C 0,1 surface without a boundary and h can be partitioned in planar segments, triangles or quadrilaterals, consistent with the outer triangulation T h. This can be formally defined as follows. For any tetrahedron S T T h such that meas 2 (S T h > 0 define T = S T h. We assume that each T is planar, i.e., either a triangle or a quadrilateral. Thus h can be decomposed as h = T Fh T, (2.3 where F h is the set of all triangles or quadrilaterals T such that T = S T h for some tetrahedron S T T h. Note that if T coincides with a face of an element in T h than the corresponding S T is not unique. In this case, we chose one arbitrary but fixed tetrahedron S T which has T as a face. Remark 1. We briefly explain an approach for the construction of an approximation h of that is used in our applications in two-phase flow problems, cf. [6, 8, 7]. The interface is represented as the zero level of a (unknown level set function φ. The level set equation for φ is discretized with continuous piecewise quadratic finite elements on the tetrahedral triangulation T h. The use of piecewise quadratics (instead of piecewise linears allows an accurate discretization of the surface tension force (which depends on the curvature of. The (given piecewise quadratic finite element approximation of φ on T h is denoted by φ h. We now introduce one further regular refinement of T h, resulting in T h = T. Let I(φ h h be the continuous piecewise 2 linear function on T h which interpolates φ h at all vertices of all tetrahedra in T h. The approximation of the interface is defined by h := {x Ω I(φ h (x = 0 }. (2.4 and consists of piecewise planar segments. The mesh size parameter h is the maximal diameter of these segments. This maximal diameter is approximately the maximal diameter of the tetrahedra in T h that contain the discrete interface, i.e., h = h is approximately the maximal diameter of the tetrahedra in T h that are close to the interface. In Figure 2.1 we illustrate this construction for the two-dimensional case. h T h T h Fig Construction of approximate interface for 2D case. Each of the planar segments of h is either a triangle or a quadrilateral. This construction of h satisfies the assumptions made above. It can be shown that under reasonable assumption, as explained in remark 7 below, the approximation h is close to in the following sense (cf. (3.14, (3.15 : dist( h, c 0 h 2, and ess sup x h n(x n h (x c 0 h, where n is the extension of n in a neighborhood of and n h a unit normal on h. In Fig. 2.2 we show a part of h that is constructed as explained above for a two-phase flow application with a rising droplet. 3

4 Fig Example of a part of h in a two-phase flow application. The main new idea of this paper is that for discretization of the problem (2.2 we use a finite element space induced by the continuous linear finite elements on T h. This is done as follows. We define a subdomain that contains h : We introduce the finite element space ω h := T Fh S T. (2.5 V h := { v h C(ω h v ST P 1 for all T F h }. (2.6 where P 1 is the space of polynomials of degree one. The space V h induces the following space on h : V h := {ψ h H 1 ( h v h V h : ψ h = v h h }. (2.7 This space is used for a Galerkin discretization of (2.2: determine u h Vh with h u h ds h = 0 such that h u h h ψ h ds h = h f h ψ h ds h h for all ψ h Vh, (2.8 with f h an extension of f such that h f h ds h = 0, cf. section 3.3. Due the Lax- Milgram lemma this problem has a unique solution u h. In section 3 we present a discretization error analysis of this method that shows that under reasonable assumptions we have optimal error bounds. In section 4 we show results of numerical experiments that confirm the theoretical analysis. As far as we know this method for discretization of a partial differential equation on a surface is new. In the remarks below we give some comments related to this approach. Remark 2. The family {T h } h>0 is shape-regular but the family {F h } h>0 in general is not shape-regular. In our numerical experiments, cf. section 4, F h contains a significant number of strongly deteriorated triangles that have very small angles. Moreover, neighboring triangles can have very different areas, cf. Fig As we will prove in section 3, optimal discretization bounds hold if {T h } h>0 is shape-regular; for {F h } h>0 shape-regularity is not required. 4

5 Remark 3. Let (ξ i 1 i m be the collection of all vertices of all tetrahedra in ω h and φ i the nodal linear finite element basis function corresponding to ξ i. Then Vh is spanned by the functions φ i h, 1 i m. These functions, however, are not necessarily independent. In computations we use this generating system φ i h, 1 i m, for solving the discrete problem (2.8. Properties that are of interest for the numerical solution of the resulting linear system, such as conditioning of the mass and stiffness matrix are analyzed in the forthcoming paper [11]. Remark 4. In the implementation of this method one has to compute integrals of the form h φ j h φ i ds, f h φ i ds for T F h. T T The domain T is either a triangle or a quadrilateral. The first integral can be computed exactly. For the second one standard quadrature rules can be applied. Remark 5. Each quadrilateral in F h can be subdivided into two triangles. Let F h be the induced set consisting of only triangles and such that T Fh T = h. Define Wh := { ψ h C( h ψ h T P 1 for all T F h }. (2.9 The space Wh is the space of continuous functions that are piecewise linear on the triangles of h. Clearly Vh W h holds. There are, however, situations in which Vh W h. A 2D illustration of this is given in Fig Fig Example In this example ω h consists of 10 triangles (shaded. The nodal basis functions correponding to these basis functions are denoted by {φ i } 1 i 10. The line segments of the interface h (denoted by - - intersect midpoints of edges of the triangles. The space Wh consists of piecewise linears on h and is spanned by the 1D nodal basis functions at the intersection points labeled by boldface 1,...,10. Clearly dim(wh = 10. In this example we have dim(vh = 9. For the piecewise linear function v = 10 i=1 α iφ i with α i = 1 for i = 1, 2, 3 and α i = 1 for i = 4,...,10 we have v h = 0. The example in remark 5 shows that the finite element space Vh can be smaller then Wh, and therefore approximation properties of V h do not follow directly from those of Wh. Moreover, the triangulations { F h } h>0 of h are not shape regular, cf. remark 2 and Fig Thus it is not clear how (optimal approximation error bounds for the standard linear finite element space Wh in (2.9 can be derived. 5

6 3. Discretization error analysis. In this section we derive discretization error bounds, both in the H 1 - and the L 2 -norm on h. We first collect some preliminaries in section 3.1, then derive approximation error bounds in section 3.2 and finally present discretization error bounds in section Preliminaries. We will need a Poincare type inequality that is given in the following lemma. Lemma 3.1. Consider a bounded domain Ω R n and a subdomain S Ω. Assume that Ω is such that the Neumann-Poincare inequality is valid: f L2 (Ω C P f L2 (Ω for all f H 1 (Ω with Then for any f H 1 (Ω the following estimate holds: Ω f dx = 0. (3.1 f 2 L 2 (Ω Ω ( 2 f 2 L S 2 (S + 3C2 P f 2 L 2 (Ω. (3.2 Proof. The proof uses a technique developed by Sobolev ([13], Ch.I for building equivalent norms on W l q(ω (Sobolev spaces. We consider the simple case with q = 2, l = 1, i.e. H 1 (Ω. Introduce the projectors Π k : H 1 (Ω R, k = 1, 2: Π 1 f := Ω 1 Ω f dx, Π 2 f := S 1 S f dx. Since (I Π 1 f 2 L 2 (Ω = f 2 L 2 (Ω Ω Π 1f 2, the Neumann-Poincare inequality (3.1 can be rewritten in the equivalent form: f 2 L 2 (Ω Ω Π 1f 2 + C 2 P f 2 L 2 (Ω for all f H 1 (Ω. (3.3 For any f H 1 (Ω with Π 1 f = 0 the Cauchy and Neumann-Poincare inequality implies Π 2 f = S 1 S f dx S 1 2 f L2 (S S 1 2 f L 2 (Ω C p S 1 2 f L 2 (Ω. (3.4 Define M := C P S 1 2. Note that for f H 1 (Ω we have Π 1 (I Π 1 f = 0 and thus from (3.4 we obtain: (Π 2 Π 1 f = Π 2 (I Π 1 f M (I Π 1 f L2 (Ω = M f L2 (Ω. Hence, for any f H 1 (Ω we have Π 1 f 2 + M 2 f 2 L 2 (Ω 2 Π 2f (Π 2 Π 1 f 2 + M 2 f 2 L 2 (Ω 2 Π 2 f 2 + 3M 2 f 2 L 2 (Ω. (3.5 6

7 Estimates (3.3 and (3.5 imply: ( f 2 L 2 (Ω max{ Ω, C2 PM 2 } Π 1 f 2 + M 2 f 2 L 2 (Ω ( = Ω Π 1 f 2 + M 2 f 2 L 2 (Ω ( Ω 2 Π 2 f 2 + 3M 2 f 2 L 2 (Ω ( Ω 2 S 1 f 2 L 2 (S + 3M2 f 2 L 2 ( (Ω = Ω S 1 2 f 2 L 2 (S + 3C2 P f 2 L 2 (Ω, which proves the inequality in (3.2. Remark 6. In the analysis below we shall apply lemma 3.1 for the case of convex domain Ω. For convex domains the following upper bound is well-known [12] for the Poincare constant: C P diam(ω. (3.6 π We define a neighborhood of : U = {x R 3 dist(x, < c }, with c sufficiently small and assume that h U. Let d : U R be the signed distance function, d(x := dist(x, for all x U. Thus is the zero level set of d. We assume d < 0 on the interior of and d > 0 on the exterior. Note that n = d on. We define n(x := d(x for all x U. Thus n = n on and n(x = 1 for all x U. Here and in the remainder denotes the Euclidean norm. The Hessian of d is denoted by H: H(x = D 2 d(x R 3 3 for all x U. (3.7 The eigenvalues of H(x are denoted by κ 1 (x, κ 2 (x and 0. For x the eigenvalues κ i (x, i = 1, 2, are the principal curvatures. We will need the orthogonal projection P(x = I n(xn(x T for x U. Note that the tangential derivative can be written as g(x = P g(x for x. We introduce a locally orthogonal coordinate system by using the projection p : U : p(x = x d(xn(x for all x U. We assume that the decomposition x = p(x+d(xn(x is unique for all x U. Note that n(x = n(p(x for all x U. We use an extension operator defined as follows. For a function v on we define v e (x := v(x d(xn(x = v(p(x for all x U, 7

8 i.e., v is extended along normals on. We define a discrete analogon of the orthogonal projection P: P h (x := I n h (xn h (x T for x h, x not on an edge. Here n h (x denotes the (outward pointing normal at x h (x not on an edge. The tangential derivative along h can be written as h g(x = P h (x g(x for x h (not on an edge. In the analysis we use techniques from [3, 5]. For example, the formula u e (x = (I d(xh(x u(p(x a.e. on U (3.8 (cf. section 2.3 in [3], which implies, h v e (x = P h (x ( I d(xh(x v(p(x a.e. on h. (3.9 Furthermore, for u sufficiently smooth and µ = 2, the inequality D µ u e (x c( D µ u(p(x + u(p(x a.e. on U (3.10 µ =2 holds, cf. lemma 3 in [5]. We define an h-neighborhood of : U h = {x R 3 dist(x, < c 1 h } and assume that h is sufficiently small, such that ω h U h U and 5c 1 h < ( max κ 1. i L ( (3.11 i=1,2 From (2.5 in [3] we have the following formula for the principal curvatures κ i : κ i (x = Hence, from (3.11 and (3.12 it follows that κ i (p(x, for x U. ( d(xκ i (p(x d L (U h max i=1,2 κ i L (U h 1 4 (3.13 holds. In the remainder we assume that ess sup x h d(x c 0 h 2, (3.14 ess sup x h n(x n h (x c 0 h, (3.15 holds. Remark 7. Related to these assumptions we note the following. Consider an approach as outlined in remark 1 in which the approximation h of is constructed using a level set method and a piecewise quadratic finite approximation φ h of the level set function φ. We assume that the level set function φ equals the signed distance function d, i.e., φ = d and that for the finite element approximation an error bound φ h φ L (ω h + h φ h φ H 1, (ω h ch 3 k φ H 3 k, (ω h, k = 0, 1, 2, (3.16 8

9 holds (which is reasonable for the case of piecewise quadratics and if φ is sufficiently smooth. Let I be the nodal interpolation operator on the vertices of the triangulation ω h. Using standard properties of this operator and the error bound in (3.16 one obtains Iφ h φ L (ω h I(φ h φ L (ω h + Iφ φ L (ω h φ h φ L (ω h + ch 2 φ H 2, (ω h c h 2 φ H 2, (ω h, and thus for x h we have d(x = I(φ h (x φ(x c h 2, hence (3.14 is satisfied. We also have Iφ h φ H 1, (ω h I(φ h φ H 1, (ω h + Iφ φ H 1, (ω h c φ h φ H 1, (ω h + ch φ H 2, (ω h ch φ H 2, (ω h. Using this and φ = 1 we then have I(φ h (x = 1 + O(h for x h. For x h (not on an edge we obtain n h (x n(x = I(φ h(x I(φ h (x φ(x 1 I(φ h (x 1 I(φ h (x + I(φ h (x φ(x c h, and thus (3.15 is satisfied (for h sufficiently small. Lemma 3.2. There are constants c 1 > 0 and c 2 independent of h such that for all u H 2 ( the following inequalities hold: c 1 u e L 2 (U h h u L 2 ( c 2 u e L 2 (U h, (3.17 c 1 u e L2 (U h h u L2 ( c 2 u e L2 (U h, (3.18 D µ u e L 2 (U h c 2 h u H 2 (, µ = 2. (3.19 Proof. Note that u H 2 ( is continuous and thus u e is well-defined. Define From (2.20, (2.23 in [3] we have µ(x := ( 1 d(xκ 1 (x ( 1 d(xκ 2 (x, x U h. µ(xdx = drds(p(x, x U, where dx is the measure in U h, ds the surface measure on and r the local coordinate at x in the direction n(p(x = n(x. Using (3.13 we get 9 25 µ(x for all x U h. (3.20 Using the local coordinate representation x = (p(x, r, for x U, we have c1h u e (x 2 µ(xdx = [u e (p(x, r] 2 ds(p(xdr U h c 1h c1h = [u(p(x, 0] 2 ds(p(x = 2c 1 h u 2 L 2 (. c 1h 9

10 Combining this with (3.20 yields the result in (3.17. From (3.8 we have that u e H 1 (U h. Note that U h [ u e (x] 2 µ(xdx = c1h c 1h [ (I d(xh(x u(p(x ] 2 ds(p(xdr. Using this in combination with d(xh(x 1 4 for all x U h (cf. (3.13 and the bounds in (3.20 we obtain the result in (3.18. Finally, using similar arguments and the bound in (3.10 one can derive the bound in ( Approximation error bounds. Let I h : C(ω h V h be the nodal interpolation operator. We use the approximation property of the linear finite element space V h : For v H 2 (ω h v I h v H k (ω h C h 2 k v H 2 (ω h, k = 0, 1. (3.21 A consequence of this approximation result is given in the following lemma. Lemma 3.3. For u H 2 ( and k = 0, 1 we have u e I h u e Hk (ω h C h 5 2 k u H2 (. (3.22 Proof. From (3.21 and (3.18 we obtain u e I h u e H k (ω h C h 2 k u e H 2 (ω h C h 2 k u e H 2 (U h C h 5 2 k u H 2 (, which proves the result. The following two lemmas play a crucial role in the analysis. In both lemmas we use a pull back strategy based on lemma 3.1. For this we introduce a special local coordinate system as follows. For a subdomain ω R n let ρ(ω be the diameter of the largest ball that is contained in ω. Take an arbitrary planar segment T of h, i.e., T F h. Let S T T h be the tetrahedron such that h S T = T. There exists a planar extension T e of T such that T e U, T e is convex, p(s T p(t e and diam(t e ρ(t e h, (3.23 cf. remark 8. This extension T e is used to define a coordinate system in the neighborhood N T := {x U p(x p(t e }. Note that S T N T. Every x N T has a unique decomposition of the form x = s + d(xn(x, with s T e, d(x := ± s x. (3.24 On which side of the plane T e the point x lies determines the sign of d(x. Note that d is a signed distance, along the normal n(x, to the planar segment T e. The representation in this coordinate system is denoted by Φ, i.e., Φ(x = (s(x, d(x. This coordinate system is illustrated, for the 2D case, in Fig For x T e we thus have Φ(x = (s(x, 0. Due to the shape-regularity of T h there exists, in the Φ-coordinate system, a cylinder B T that has the following properties: B T = T e b [d 0, d 1 ] S T, T e b T e, T e b h2, d 1 d 0 h. (3.25 This coordinate system and the cylinder B T S T are used in the analysis below. 10

11 x = (s, d(x d PQ : p(s T Q AB : T e P S T T s B A Fig D Illustration of coordinate system Remark 8. The following shows that an extension T e of T with the properties described above exists. Take a fixed x 0 T. Let W be the tangent plane at p(x 0. The normal vector of W is n(x 0. There is a subdomain w of this plane such that p(w = p(s T. Due to shape regularity of T h this subdomain is such that diam(w ρ(w h holds. Let w x0 be a planar subdomain that is parallel to w, contains x 0 and such that p(w = p(w x0. Using assumption (3.14 it follows that diam(w x0 ρ(w x0 h holds. The point x 0 belongs to the planar subdomains w x0 and T, which have normals n(x 0 and n h (x 0, respectively. Due to assumption (3.15 the angle between these normals is bounded by ch and thus there exists a planar extension T e of T such that T e U and p( T e = p(w x0, now we set T e to be a minimal convex envelope for T e. This T e has the property (3.23. Lemma 3.4. Let v h be a linear function on N T and u H 2 (. There exists a constant c independent of v h, u and T such that the following inequality holds: h (u e v h L 2 (T e ch 1 2 (u e v h L 2 (S T + h u H 2 (p(t e. (3.26 Here h denotes the projection of the gradient on T e. Proof. Using lemma 3.1, (3.6 and (3.10 we obtain h (u e v h 2 L 2 (T e c h (u e v h 2 L 2 (T e b + ch2 2 h u e 2 L 2 (T e c h (u e v h 2 L 2 (T e b + ch2 u 2 H 2 (p(t e. (3.27 We consider the first term in (3.27. We write v h =: c T and use the notation x = (s(x, d(x =: (s, y in the Φ-coordinate system. From (3.8 we have u(p(x = u e (s, y + d(xh(x u(p(x. 11

12 Using this and (3.9 we obtain h (u e v h 2 L 2 (T e b = h u e P h c T 2 L 2 (T e b 2 P h ( u p P h c T 2 L 2 (T e b + 2 P hdh( u p 2 L 2 (T e b c ( u p c T 2 L 2 (Tb e + ch2 u 2 H 1 (p(s T = c u(p(s, 0 c T 2 ds + ch 2 u 2 H 1 (p(s T T e b d1 ch 1 d 0 d1 ch 1 d 0 T e b T e b u(p(s, 0 c T 2 dsdy + ch 2 u 2 H 1 (p(s T u e (p(s, y c T 2 dsdy + ch 2 u 2 H 1 (p(s T ch 1 (u e v h 2 L 2 (B T + ch2 u 2 H 1 (p(s T ch 1 (u e v h 2 L 2 (S T + ch2 u 2 H 1 (p(s T. Combination of this result with the one in (3.27 completes the proof. Lemma 3.5. There are constants c i independent of h such that for all u H 2 ( and all v h V h the following inequality holds: u e v h L 2 ( h c 1 h 1 2 u e v h L 2 (ω h +c 2 h 1 2 u e v h H 1 (ω h +c 3 h 2 u H 2 (. (3.28 Proof. We consider an arbitrary element T h. Let T e be its extension as defined above. Take v h V h. The extension of v h to a linear function on T e is denoted by v h, too. Using lemma 3.1 and (3.6 we get: u e v h 2 L 2 (T ue v h 2 L 2 (T e T = (u e (s, 0 v h (s, 0 2 ds e c (u e (s, 0 v h (s, 0 2 ds T e b ( ch 2 T e h ( u e (s, 0 v h (s, 0 2 ds. We consider the first term on the right handside of (3.29. For a linear function g and 0 δ 0 < δ 1 we have g(δ i 2 6 δ1 δ 1 δ 0 δ 0 g(t 2 dt for i = 0, 1 and g(0 = δ g(δ 0 1 δ δ 1 δ 0 g(δ 1 0 δ 1 δ 0. Hence, g(0 2δ1 δ 1 δ 0 max i=0,1 g(δ i and thus g( ( δ1 δ 1 δ δ 1 δ 0 δ1 δ 0 g(t 2 dt (3.30 holds. Without loss of generality we can assume that d 0, d 1 from (3.25 satisfy 0 d d 0 < d 1. Furthermore, we have i d 1 d 0 c for i = 1, 2, with c independent of h. Using this and the result in (3.30 applied to the linear function y c + v h (s, y we obtain d1 (u e (s, 0 v h (s, 0 2 ds ch 1 (u e (s, 0 v h (s, y 2 dyds Tb e Tb e d 0 d1 = ch 1 (u e (s, y v h (s, y 2 dy ds = ch 1 u e v h 2 (3.31 L 2 (B T d 0 T e b ch 1 u e v h 2 L 2 (S T. 12

13 For the second term on right handside of (3.29 we can apply lemma 3.4 and thus we get u e v h 2 L 2 (T ch 1 u e v h 2 L 2 (S T + ch (ue v h 2 L 2 (S T + ch4 u 2 H 2 (p(t e. Summation over all triangles in F h gives (3.28. Lemma 3.6. There are constants c 1, c 2 independent of h such that for all u H 2 ( and all v h V h the following inequality holds: u e v h H 1 ( h c 1 h 1 2 u e v h H 1 (ω h + c 2 h u H 2 (. (3.32 Proof. Take u H 2 ( and v h V h. By definition of the H 1 -norm on h we get u e v h 2 H 1 ( h = ue v h 2 L 2 ( h + h (u e v h 2 L 2 ( h. For the first term on the right handside we can apply lemma 3.5 and use h 1 2 u e v h L2 (ω h + c 2 h 1 2 u e v h H1 (ω h ch 1 2 u e v h H1 (ω h. We now consider the second term h ( u e v h 2 L 2 ( h = T F h h ( u e v h 2 L 2 (T. Take a T F h and extend v h linearly outside T. This extension is denoted by v h, too. Using lemma 3.4 we get h ( u e v h 2 L 2 (T h ( u e v h 2 L 2 (T e Summation over T F h yields ch 1 (u e v h 2 L 2 (S T + h2 u 2 H 2 (p(t e. h (u e v h 2 L 2 ( h c h 1 u e v h 2 H 1 (ω h + ch2 u 2 H 2 ( and thus the proof is completed. As a direct consequence of the previous two lemmas we obtain the following main theorem. Theorem 3.7. For each u H 2 ( the following holds inf u e v h L 2 ( h u e (I h u e h L 2 ( h C h 2 u H 2 (, (3.33 v h Vh inf v h Vh u e v h H1 ( h u e (I h u e h H1 ( h C h u H2 (, (3.34 with a constant C independent of u and h. Proof. Combine the results in the lemmas 3.5 and 3.6 with the result in lemma

14 3.3. Finite element error bounds.. In this section we prove optimal discretization error bounds both in the H 1 ( h and the L 2 ( h norm. The arguments are very close to those in [5]. A difference is that in [5] the convergence results are derived in the H 1 ( and the L 2 ( norms by lifting the discrete solutions from h on, whereas we consider the error between the finite element solution u h V h and the extension u e of the continuous solution to the discrete interface. This difference is of minor importance since error bounds in H 1 ( h imply similar bounds in H 1 (, cf. remark 9. In the analysis we need a few results from [3]. For x h define P h (x = I n h (xn(x T /(n h (x T n(x. In (2.19 in [3] the following representation of the surface gradient of u H 1 ( in terms of h u e is given: u(p(x = ( I d(xh(x 1 Ph (x h u e (x a.e. on h. (3.35 For x h define µ h (x = (1 d(xκ 1 (x(1 d(xκ 1 (xn(x T n h (x. The integral transformation formula µ h (xds h (x = ds(p(x, x h, (3.36 holds, where ds h (x and ds(p(x are the surface measures on h and, respectively (cf. (2.20 in [3]. From n(x n h (x 2 = 2 ( 1 n(x T n h (x, assumption (3.15 and d(x ch 2, κ i (x c we obtain with a constant c independent of h. ess sup x h 1 µ h (x ch 2, (3.37 Theorem 3.8. Let u H 2 ( be the solution of (2.2 and u h Vh the solution of (2.8 with f h = f e c f, where c f is such that h f h ds = 0. The following discretization error bound holds h (u e u h L 2 ( h c h f L 2 ( (3.38 with a constant c independent of f and h. Proof. Using (3.37 we obtain 1 1 µ h (x ch2 on h. Define c f := f e ds h, δ f := (1 µ h f e c f. h Note that f h = f e c f and due to f ds = 0 we get c f = f e ds h = f( 1 1ds ch 2 f h µ L 2 (. h Furthermore, δ f L 2 ( h ess sup 1 µ h (x f e L 2 ( h + h 1 2 cf ch 2 f L 2 (. (3.39 x h 14

15 Using relation (3.35 and (3.36 we obtain u v ds = A h h u e h v e ds h for all v H 1 (, (3.40 h with A h (x = µ h (x P h (x(i d(xh(x 2 Ph (x. Any ψ h H 1 ( h can be lifted on by defining ψ l h (p(x := ψ h(x. Then ψ l h H1 ( holds. From the definition of the discrete solution u h in (2.8 we get, for arbitrary ψ h V h : h u h h ψ h ds h = f h ψ h ds h = (f c f µ h (x 1 ψh l ds h h = fψh l ds + δ f ψ h ds h h = u ψh l ds + δ f ψ h ds h h = A h h u e h ψ h ds h + δ f ψ h ds h. h h Using this we obtain, for arbitrary ψ h V h, h (u e u h h ψ h ds h = (I A h h u e h ψ h ds h δ f ψ h ds h h h h = P h (I A h h u e h ψ h ds h δ f ψ h ds h. h h Therefore we get (3.41 h (u e u h 2 L 2 ( h = h (u e u h h (u e ψ h ds h h + P h (I A h h u e h (ψ h u h ds h h δ f (ψ h u h ds h. h (3.42 From P h A h ch 2 a.e. on h and P h Ph = P h we obtain, for x h, P h (x(i A h (x = P h (x( P h (x A h (x ch 2. (3.43 Furthermore, using (3.9 we get h u e L 2 ( h ess sup x h P h (x(i dh(x u L 2 ( c f L 2 (. (3.44 Introduce the notation E h := h (u e u h L 2 ( h. Note that by taking ψ h = (I h u e h and using the approximation result (3.34 we have h (u h ψ h L 2 ( h E h + h (u e ψ h L 2 ( h E h + ch f L 2 (. 15

16 For the third term on the right handside in (3.42 we have the bound δ f (ψ h u h ds h δ f L 2 ( h ψ h u h L 2 ( h h ch 2 f L2 (( ψh u e L2 ( h + u e u h L2 ( h ch 2 f L 2 (( ch 2 f L 2 ( + u e u h L 2 ( h. Note that u e u h L 2 ( h u e L 2 ( h + u h L 2 ( h c ( u L 2 ( + h u h L 2 ( h c ( u L 2 ( + h u h L 2 ( h c f L 2 (. Combination of these results leads to Eh 2 E h ch f L 2 ( + ch 2 f L (( 2 Eh + ch f L 2 ( + f L 2 ( 1 2 E2 h + ch2 f 2 L 2 (. This yields the bound in (3.38. Remark 9. We indicate how the error bound (3.38 in H 1 ( h yields a similar bound in H 1 (. For this we need the extension of functions defined on h along the normals n on : for v C( h we define, for x h, v e,h (x + αn(x := v(x for all α R with x + αn(x U. The following holds (cf. [3], Lemma 3.3 in [8]: v e,h L 2 ( c h v L 2 ( h for all v H 1 ( h C( h. Using this for the error v = u e u h and noting that (u e e,h = u on the bound (3.38 yields (u u e,h h L 2 ( ch f L 2 (, i.e., an optimal error bound in H 1 (. We now apply a duality argument to obtain an L 2 ( h -error bound. Theorem 3.9. Let u and u h be as in theorem 3.8. The following error bound holds u e u h L 2 ( h c h 2 f L 2 ( (3.45 with a constant c independent of f and h. Proof. Denote e h := (u e u h h and let e l h be the lift of e h on and c e := el h ds. Consider the problem: Find w H 1 ( with w ds = 0 such that w v dσ = (e l h c e v ds for all v H 1 (. (3.46 The solution w satisfies w H 2 ( and w H 2 ( c e l h L 2 (/R with e l h L 2 (/R := e l h c e L 2 (. Furthermore, h w e L 2 ( h c e l h L 2 (/R and w e L 2 ( h 16

17 c w L 2 ( c w L 2 ( c e l h L 2 (/R. Due to (3.46 and (3.41 we have, for any ψ h Vh, e l h 2 L 2 (/R = w (e l h c eds = w e l h ds = A h h e h h w e ds h h = h e h h (w e ψ h ds h + P h (A h I h e h h w e ds h h h + P h (I A h h u e h ψ h ds h δ f ψ h ds h. h h Introduce E h := e l h L 2 (/R. Thanks to the approximation property (3.34 one can choose ψ h such that h (w e ψ h L 2 ( h ch w H 2 ( che h. Using Cauchy- Schwarz and triangle inequalities and the bounds in (3.39, (3.43 we get Eh 2 h e h L 2 ( h che h + ch 2 h e h L 2 ( h h w e L 2 ( h + ch 2 h u e L 2 ( h ( h w e L 2 ( h + che h + ch 2 ( f L 2 ( w e L 2 ( h + che h ch 2 f L2 (E h + ch 2 f L2 (( Eh + che h. Hence, E h ch 2 f L 2 ( holds. We have and thus c e = u u e h ds = u e h ds = (µ h 1u e h ds h ch 2 f L 2 (, h e h L 2 ( h c µ 1 2 h e h 2 L 2 ( h = c el h L 2 ( c(e h + c e ch 2 f L 2 (, which completes the proof. 4. Numerical experiments. In this section we present results of numerical experiments. As a first test problem we consider the Laplace-Beltrami equation on the unit sphere: u = f on, with = {x R 3 x 2 = 1} and Ω = ( 2, 2 3. The source term f is taken such that the solution is given by u(x = a ( 3x 2 x 3 1 x 2 x 3 2, x = (x1, x 2, x 3 Ω, with a = 12. Using the representation of u in spherical coordinates one can verify that u is an eigenfunction of : u(r, φ, θ = a sin(3φsin 3 θ, u = 12u =: f(r, φ, θ. (4.1 The right handside f satisfies the compatibility condition f ds = 0, likewise does u. Note that u and f are constant along normals at. A family {T l } l 0 of tetrahedral triangulations of Ω is constructed as follows. We triangulate Ω by starting with a uniform subdivision into 48 tetrahedra with mesh size h 0 = 3. Then we apply an adaptive red-green refinement-algorithm (implemented 17

18 in the software package DROPS [4] in which in each refinement step the tetrahedra that contain are refined such that on level l = 1, 2,... we have h T 3 2 l for all T T l with T. The family {T l } l 0 is consistent and shape-regular. The interface is the zero-level of ϕ(x := x 2 1. Let ϕ h := I(ϕ where I is the standard nodal interpolation operator on T l. The discrete interface is given by hl := {x Ω I(φ h (x = 0 }, cf. (2.4. Let {φ i } 1 i m be the nodal basis functions corresponding to the vertices of the tetrahedra in ω h, as explained in remark 2. The entries h h φ i h φ j ds h of the stiffness matrix are computed within machine accuracy. For the right handside of the Galerkin discretization (2.8 we need an extension f h of f. In order to be consistent with the theoretical analysis we take the constant extension of f along the normals at, i.e. we take f h (r, φ, θ = f(1, φ, θ + c h, with f(r, φ, θ as in (4.1 and c h such that h f h ds h = 0. For the computation of the integrals T f hψ h ds h we use a quadrature-rule that is exact up to order five. The computed solution u h is normalized such that h u h ds h = 0. The discrete problem is solved using a standard CG method with symmetric Gauss-Seidel preconditioner to a relative tolerance of The number of iterations needed on level l = 1, 2,..., 7, is 14, 25, 50, 101, 209, 417, 837, respectively. The discretization errors in the L 2 ( h -norm are given in table 4.1. The extension u e of u is given by u e (r, φ, θ := u(1, φ, θ, cf. (4.1. level l u e u h L 2 ( h factor Table 4.1 Discretization errors and error reduction. These results clearly show the h 2 behaviour as predicted by our theoretical analysis. To illustrate the fact that in this approach the triangulation of the approximate manifold h is strongly shape-irregular we show a part of this triangulation in Figure 4.1. The discrete solution is visualized in Fig To demonstrate the flexibility of the method with respect to the form of we repeat the previous experiment but now with a torus instead of the unit sphere. Ω = ( 2, 2 3 with = {x Ω r 2 = x ( x x2 2 R2 }. We take R = 1 and r = 0.6. In the coordinate system (ρ, φ, θ with x = R cosφ sin φ + ρ cosφcos θ sin φcosθ 0 sin θ the ρ-direction is normal to, x ρ for x. Thus the following solution u and 18

19 Fig Detail of the induced triangulation of h. Fig Level lines of the discrete solution u h corresponding right-hand side f are constant in normal direction: u(x = sin(3φcos(3θ + φ, f(x = r 2 (9 sin(3φcos(3θ + φ (R + r cos(θ 2 ( 10 sin(3φcos(3θ + φ 6 cos(3φsin(3θ + φ (r(r + r cos(θ 1 (3 sin(θsin(3φsin(3θ + φ (4.2 Both u and f satisfy the zero mean compatibility condition. level l u e u h L 2 ( h factor Table 4.2 Torus: Discretization errors and error reduction. The discretization errors in the L 2 ( h -norm are given in table 4.2. The extension u e of u is given by u e (ρ, φ, θ := u(r, φ, θ, cf. (4.2. Again we observe the h 2 behaviour 19

20 Fig Torus: Level lines of the discrete solution u h as predicted by the theoretical analysis. The discrete solution is visualized in Fig Acknowledgements. The authors thank the referees for their comments and suggestions. These led to a significantly improved revised version. REFERENCES [1] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3d edition, Cambridge University Press, [2] K. Deckelnick, G. Dziuk, C. M. Elliott, and C.-J. Heine, An h-narrow band finite element method for elliptic equations on implicit surfaces, IMA J. Numer. Anal., doi: /imanum/drn049 (2009. [3] A. Demlow and G. Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal., 45 (2007, pp [4] DROPS package. [5] G. Dziuk, Finite elements for the beltrami operator on arbitrary surfaces, in Partial differential equations and calculus of variations, S. Hildebrandt and R. Leis, eds., vol of Lecture Notes in Mathematics, Springer, 1988, pp [6] S. Groß, V. Reichelt, and A. Reusken, A finite element based level set method for two-phase incompressible flows, Comp. Vis. Sci., 9 (2006, pp [7] S. Groß and A. Reusken, An extended pressure finite element space for two-phase incompressible flows, J. Comp. Phys., 224 (2007, pp [8], Finite element discretization error analysis of a surface tension force in two-phase incompressible flows, SIAM J. Numer. Anal., 45 (2007, pp [9] A. James and J. Lowengrub, A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant, J. Comp. Phys., 201 (2004, pp [10] M. Muradoglu and G. Tryggvason, A front-tracking method for computation of interfacial flows with soluble surfactant, J. Comput. Phys., 227 (2008, pp [11] M. A. Olshanskii and A. Reusken, A finite element method for surface pdes: Matrix properties, Preprint 287, IGPM, RWTH Aachen, [12] L. E. Payne and H. F. Weinberger, An optimal poincare inequality in convex domains, Arch. Rat. Mech. Anal., 5 (1960, pp [13] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics. Third Edition, AMS,