A STABILIZED FINITE ELEMENT METHOD FOR ADVECTION-DIFFUSION EQUATIONS ON SURFACES

Transcription

1 A SABILIZD FINI LMN MHOD FOR ADVCION-DIFFUSION QUAIONS ON SURFACS MAXIM A. OLSHANSKII, ARNOLD RUSKN, AND XIANMIN XU Abstract. A recently developed ulerian finite element method is applied to solve advectiondiffusion equations posed on hypersurfaces. When transport processes on a surface dominate over diffusion, finite element methods tend to be unstable unless the mesh is sufficiently fine. he paper introduces a stabilized finite element formulation based on the SUPG technique. An error analysis of the method is given. Results of numerical experiments are presented that illustrate the performance of the stabilized method. Key words. surface PD, finite element method, transport equations, advection-diffusion equation, SUPG stabilization AMS subject classifications. 58J32, 65N12, 65N30, 76D45, Introduction. Mathematical models involving partial differential equations posed on hypersurfaces occur in many applications. Often surface equations are coupled with other equations that are formulated in a (fixed) domain which contains the surface. his happens, for example, in common models of multiphase fluids dynamics if one takes so-called surface active agents into account [1]. he surface transport of such surfactants is typically driven by convection and surface diffusion and the relative strength of these two is measured by the dimensionless surface Peclet number P e s = UL D s. Here U and L denote typical velocity and lenght scales, respectively, and D s is the surface diffusion coefficient. ypical surfactants have surface diffusion coefficients in the range D s cm 2 /s [2], leading to (very) large surface Peclet numbers in many applications. Hence, such applications result in advection-diffusion equations on the surface with dominating advection terms. he surface may evolve in time and be available only implicitly (for example, as a zero level of a level set function). It is well known that finite element discretization methods for advection-diffusion problems need an additional stabilization mechanism, unless the mesh size is sufficiently small to resolve boundary and internal layers in the solution of the differential equation. For the planar case, this topic has been extensively studied in the literature and a variety of stabilization methods has been developed, see, e.g., [3]. We are, however, not aware of any studies of stable finite element methods for advection-diffusion equations posed on surfaces. In the past decade the study of numerical methods for PDs on surfaces has been a rapidly growing research area. he development of finite element methods for solving elliptic equations on surfaces can be traced back to the paper [4], which considers a piecewise polygonal surface and uses a finite element space on a triangulation of this discrete surface. his approach has been further analyzed and extended in several directions, see, e.g., [5] and the references therein. Another approach has been intro- Department of Mathematics, University of Houston, Houston, exas and Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia, (Maxim.Olshanskii@mtu-net.ru). Institut für Geometrie und Praktische Mathematik, RWH-Aachen University, D Aachen, Germany (reusken@igpm.rwth-aachen.de,xu@igpm.rwth-aachen.de). LSC, Institute of Computational Mathematics and Scientific/ngineering Computing, NCMIS, AMSS, Chinese Academy of Sciences, Beijing , China 1

2 2 M. A. OLSHANSKII, A. RUSKN, AND X. XU duced in [6] and builds on the ideas of [7]. he 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. Another surface finite element method based on an outer (bulk) mesh has been introduced in [8] and further studied in [9, 10]. he main idea of this method is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface by considering traces of the bulk finite element space on the surface, instead of extending the PD off the surface, as in [7, 6]. he method is particularly 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 differential equation in a fixed outer domain. If in such problems one uses finite element techniques for the discetization of equations in the outer domain, this setting immediately results in an easy to implement discretization method for the surface equation. he approach does not require additional surface elements. In this paper we reconsider the volume mesh finite element method from [8] and study a new aspect, that has not been studied in the literature so far, namely the stabilization of advection-dominated problems. We restrict ourselves to the case of a stationary surface. o stabilize the discrete problem for the case of large mesh Peclet numbers, we introduce a surface variant of the SUPG method. For a class of stationary advection-diffusion equation an error analysis is presented. Although the convergence of the method is studied using a SUPG norm similar to the planar case [3], the analysis is not standard and contains new ingredients: Some new approximation properties for the traces of finite elements are needed and geometric errors require special control. he main theoretical result is given in heorem It yields an error estimate in the SUPG norm which is almost robust in the sense that the dependence on the Peclet number is mild. his dependence is due to some insufficiently controlled geometric errors, as will be explained in section 3.7. he remainder of the paper is organized as follows. In section 2, we recall equations for transport-diffusion processes on surfaces and present the stabilized finite element method. Section 3 contains the theoretical results of the paper concerning the approximation properties of the finite element space and discretization error bounds for the finite element method. Finally, in section 4 results of numerical experiments are given for both stationary and time-dependent advection-dominated surface transport-diffusion equations, which show that the stabilization performs well and that numerical results are consistent with what is expected from the SUPG method in the planar case. 2. Advection-diffusion equations on surfaces. Let Ω be an open domain in R 3 and be a connected C 2 compact hyper-surface contained in Ω. For a sufficiently smooth function g : Ω R the tangential derivative at is defined by g = g ( g n )n, (2.1) where n denotes the unit normal to. Denote by the Laplace-Beltrami operator on. Let w : Ω R 3 be a given divergence-free (div w = 0) velocity field in Ω. If the surface evolves with a normal velocity of w n, then the conservation of a scalar quantity u with a diffusive flux on (t) leads to the surface PD: u + (div w)u ε u = 0 on (t), (2.2)

3 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 3 where u denotes the advective material derivative, ε is the diffusion coefficient. In [11] the problem (2.2) was shown to be well-posed in a suitable weak sense. In this paper, we study a finite element method for an advection-dominated problem on a steady surface. herefore, we assume w n = 0, i.e. the advection velocity is everywhere tangential to the surface. his and div w = 0 implies div w = 0, and the surface advection-diffusion equation takes the form: u t + w u ε u = 0 on. (2.3) Although the methodology and numerical examples of the paper are applied to the equations (2.3), the error analysis will be presented for the stationary problem ε u + w u + c(x)u = f on, (2.4) with f L 2 () and c(x) 0. o simplify the presentation we assume c( ) to be constant, i.e. c(x) = c 0. he analysis, however, also applies to non-constant c, cf. section 3.7. Note that (2.3) and (2.4) can be written in intrinsic surface quantities, since w u = w u, with the tangential velocity w = w (w n )n. We assume w H 1, () L () and scale the equation so that w L () = 1 holds. Furthermore, since we are interested in the advection-dominated case we take ε (0, 1]. Introduce the bilinear form and the functional: a(u, v) := ε u v ds + (w u)v ds + c uv ds, f(v) := fv ds. he weak formulation of (2.4) is as follows: Find u V such that with V = a(u, v) = f(v) v V, (2.5) { {v H 1 () v ds = 0} if c = 0, H 1 () if c > 0. Due to the Lax-Milgram lemma, there exists a unique solution of (2.5). For the case c = 0 the following Friedrich s inequality [12] holds: v 2 L 2 () C F v 2 L 2 () v V. (2.6) 2.1. he stabilized volume mesh FM. In this section, we recall the volume mesh FM introduced in [8] and describe its SUPG type stabilization. Let { h } h>0 be a family of tetrahedral triangulations of the domain Ω. hese triangulations are assumed to be regular, consistent and stable. o simplify the presentation we assume that this family of triangulations is quasi-uniform. he latter assumption, however, is not essential for our analysis. We assume that for each h a polygonal approximation of, denoted by h, is given: h is a C 0,1 surface without boundary and h can be partitioned in planar triangular segments. It is important to note that h is not a triangulation of in the usual sense (an O(h 2 ) approximation of, consisting of regular triangles). Instead, we (only) assume that h is consistent with the outer triangulation h in the following sense. For any tetrahedron S h

4 4 M. A. OLSHANSKII, A. RUSKN, AND X. XU such that meas 2 (S h ) > 0, define = S h. We assume that every h is a planar segment and thus it is either a triangle or a quadrilateral. ach quadrilateral segment can be divided into two triangles, so we may assume that every is a triangle. An illustration of such a triangulation is given in Figure 2.1. he results shown in this figure are obtained by representing a sphere implicitly by its signed distance function, constructing the piecewise linear nodal interpolation of this distance function on a uniform tetrahedral triangulation h of Ω and then considering the zero level of this interpolant. Fig Approximate interface h for a sphere, resulting from a coarse tetrahedral triangulation (left) and after one refinement (right). Let F h be the set of all triangular segments, then h can be decomposed as h =. (2.7) F h Note that the triangulation F h is not necessarily regular, i.e. elements from may have very small internal angles and the size of neighboring triangles can vary strongly, cf. Figure 2.1. In applications with level set functions (that represent implicitly), the approximation h can be obtained as the zero level of a piecewise linear finite element approximation of the level set function on the tetrahedral triangulation h. he surface finite element space is the space of traces on h of all piecewise linear continuous functions with respect to the outer triangulation h. his can be formally defined as follows. We define a subdomain that contains h : ω h = S, (2.8) F h an a corresponding volume mesh finite element space V h := {v h C(ω h ) v h S P 1 F h }, (2.9) where P 1 is the space of polynomials of degree one. V h induces the following space on h : V h := {ψ h H 1 ( h ) v h V h such that ψ h = v h h }. (2.10) When c = 0, we require that any function v h from Vh satisfies h v h ds = 0. Given the surface finite element space Vh, the finite element discretization of (2.5) is as

5 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 5 follows: Find u h Vh such that ε h u h v ds + (w e h u)v ds + c uv ds = f e v ds (2.11) h h h h for all v h Vh. Here we and f e are the extensions of w and f, respectively, along normals to (the precise definition is given in the next section). Similar to the plain Galerkin finite element for advection-diffusion equations, the method (2.11) is unstable unless the mesh is sufficiently fine such that the mesh Peclet number is less than one. We introduce the following stabilized finite element method based on the standard SUPG approach, cf. [3]: Find u h Vh such that a h (u h, v h ) = f h (v h ) v h V h, (2.12) with a h (u, v) :=ε h u h v ds + c uvds h h + 1 [ ] (w e h u)v ds (w e h v)u ds 2 h h + ( ε h u + w e h u + c u)w e h v ds, F h δ f h (v) := f e vds + δ h F h (2.13) f e (w e h v) ds. (2.14) he stabilization parameter δ depends on S. he diameter of the tetrahedron S is denoted by h S. Let Pe := h S w e L ( ) be the cell Peclet number. We 2ε take δ 0 h S w δ e if Pe > 1, L ( ) = and δ δ 1 h 2 = min{ δ, c 1 }, (2.15) S if Pe 1, ε with some given positive constants δ 0, δ 1 0. Since u h V h is linear on every we have h u h = 0 on, and thus a h (u h, v h ) simplifies to a h (u h, v h ) = ε h u h h v h ds + 1 [ ] (w e h u h )v h (w e h v h )u h ds h 2 h + c u h (v h + δ(x)w e h v h ) ds + δ(x)(w e h u h )(w e h v h ) ds, (2.16) h h where δ(x) = δ for x. 3. rror analysis. he analysis in this section is organized as follows. First we collect some definitions and useful results in section 3.1. In section 3.2, we derive a coercivity result. In section 3.3, we present interpolation error bounds. In sections 3.4 and 3.5, continuity and consistency results are derived. Combining these analysis we

6 6 M. A. OLSHANSKII, A. RUSKN, AND X. XU obtain the finite element error bound given in section 3.6. In the error analysis we use the following mesh-dependent norm: u := ( ) 1 ε h u 2 ds + δ(x) w e h u 2 ds + c u 2 2 ds. (3.1) h h h Here and in the remainder denotes the uclidean norm for vectors and the corresponding spectral norm for matrices Preliminaries. For the hypersurface, we define its h-neighborhood: U h := {x R 3 dist(x, ) < c 0 h}, (3.2) and assume that c 0 is sufficiently large such that ω h U h and h sufficiently small such that 5c 0 h < ( max i=1,2 κ i L ()) 1 (3.3) holds, with κ i being the principal curvatures of. Here and in what follows h denotes the maximum diameter for tetrahedra of outer triangulation: h = max S ω h diam(s). Let d : U h R be the signed distance function, d(x) = dist(x, ) for all x U h. hus is the zero level set of d. We assume d < 0 in the interior of and d > 0 in the exterior and define n(x) := d(x) for all x U h. Hence, n = n on and n(x) = 1 for all x U h. he Hessian of d is denoted by H(x) := 2 d(x) R 3 3, x U h. (3.4) he eigenvalues of H(x) are denoted by κ 1 (x), κ 2 (x), and 0. For x the eigenvalues κ i, i = 1, 2, are the principal curvatures. For each x U h, define the projection p : U h by p(x) = x d(x)n(x). (3.5) Due to the assumption (3.3), the decomposition x = p(x) + d(x)n(x) is unique. We will need the orthogonal projector P(x) := I n(x)n(x), for x U h. Note that n(x) = n(p(x)) and P(x) = P(p(x)) for x U h holds. he tangential derivative can be written as g(x) = P g(x) for x. One can verify that for this projection and for the Hessian H the relation HP = PH = H holds. Similarly, define P h (x) := I n h (x)n h (x), for x h, x is not on an edge, (3.6) where n h is the unit (outward pointing) normal at x h (not on an edge). he tangential derivative along h is given by h g(x) = P h (x) g(x) (not on an edge). Assumption 3.1. In this paper, we assume that for all F h : ess sup x d(x) c 1 h 2 S, (3.7) ess sup x n(x) n h (x) c 2 h S, (3.8)

7 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 7 where h S denotes the diameter of the tetrahedron S that contains, i.e., = S h and constants c 1, c 2 are independent of h,. he assumptions (3.7) and (3.8) describe how accurate the piecewise planar approximation h of is. If h is constructed as the zero level of a piecewise linear interpolation of a level set function that characterizes (as in Fig. 2.1) then these assumptions are fulfilled, cf. Sect. 7.3 in [1]. In the remainder, A B means A cb for some positive constant c independent of h and of the problem parameters ε and c. A B means that both A B and B A. For x h, define µ h (x) = (1 d(x)κ 1 (x))(1 d(x)κ 2 (x))n (x)n h (x). he surface measures ds and ds h on and h, respectively, are related by he assumptions (3.7) and (3.8) imply that µ h (x)ds h (x) = ds(p(x)), x h. (3.9) ess sup x h (1 µ h ) h 2, (3.10) cf. (3.37) in [8]. he solution of (2.4) is defined on, while its finite element approximation u h Vh is defined on h. We need a suitable extension of a function from to its neighborhood. For a function v on we define v e (x) := v(p(x)) for all x U h. (3.11) he following formula for this lifting function are known (cf. section 2.3 in [13]): u e (x) = (I d(x)h) u(p(x)) a.e. on U h, (3.12) h u e (x) = P h (x)(i d(x)h) u(p(x)) a.e. on h, (3.13) with H = H(x). By direct computation one derives the relation 2 u e (x) = (P d(x)h) 2 u(p(x))(p d(x)h) (n u(p(x))h (H u(p(x)))n n(h u(p(x))) d H : u(p(x)). (3.14) For sufficiently smooth u and µ 2, using this relation one obtains the estimate D µ u e (x) D µ u(p(x)) + u(p(x)) a.e. on U h, (3.15) µ =2 (cf. Lemma 3 in [4]). his further leads to (cf. Lemma 3.2 in [8]): D µ u e L 2 (U h ) h u H 2 (), µ 2. (3.16) he next lemma is needed for the analysis in the following section. Lemma 3.1. he following holds: div h w e L ( h ) h w L ().

8 8 M. A. OLSHANSKII, A. RUSKN, AND X. XU Proof. We use the following representation for the tangential divergence: div w(x) = tr( w(x)) = tr(p w(x)). (3.17) ake x h, not lying on an edge. Using (3.12) we obtain div h w e (x) = tr(p h w e (x)) = tr (P h (I d(x)h) w(p(x))) = tr (P w(p(x))) + tr ((P h P) w(p(x))) d(x) tr (P h H w(p(x))). he first term vanishes due to tr (P w(p(x))) = div w(p(x)) = 0. he second and the third term can be bounded using (3.7), (3.8): his proves the lemma. P h P h, d(x)p h H h Coercivity analysis. In the next lemma we present a coercivity result. We use the norm introduced in (3.1). Lemma 3.2. he following holds: a h (v h, v h ) 1 2 v h 2 for all v h V h. (3.18) Proof. For any v h Vh, we have a h (v h, v h ) = v h 2 + c δ(x)v h (w e h v h ) ds. (3.19) h he choice of δ, cf. (2.15), implies c δ(x) 1. Hence the last term in (3.19) can be estimated as follows: c δ(x)v h (w e h v h ) ds h 1 c vh 2 ds + 1 c δ(x) 2 (w e h v h ) 2 ds 1 2 h 2 h 2 v h 2. his yields (3.18). As a consequence of this result, we obtain the well-posedness of the discrete problem (2.12) Interpolation error bounds. Let I h : C( ω h ) V h be the nodal interpolation operator. For any u H 2 () the surface finite element function (I h u e ) h Vh is an interpolant of u e in Vh. For any u H 2 () the following estimates hold [8]: u e (I h u e ) h L2 ( h ) h 2 u H2 (), (3.20) h u e h (I h u e ) h L 2 ( h ) h u H 2 (). (3.21) Using these results we easily obtain an interpolation error estimate in the - norm:

9 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 9 Lemma 3.3. For any u H 2 () the following holds: u e (I h u e ) h h(ε 1/2 + h 1/2 + c 1 2 h) u H2 (). (3.22) Proof. Define φ := u e (I h u e ) h H 1 ( h ). Using the definition (2.15) of δ(x), we get h δ(x) w e h φ 2 ds = δ w e h φ 2 ds F h (3.23) h h φ 2 L 2 ( h ) h3 u 2 H 2 (). he remaining two terms in u e (I h u e ) are estimated in a straightforward way using (3.20) and (3.21). his and (3.23) imply the inequality (3.22). he next lemma estimates the interpolation error on the edges of the surface triangulation. In the remainder, h denotes the set of all edges in the interface triangulation F h. Lemma 3.4. For all u H 2 () the following holds: ( h (u e I h u e ) 2 h ds) 1/2 h 3/2 u H 2 (). (3.24) Proof. Define ϕ := u e I h u e H 1 (ω h ). ake h and let F h be a corresponding planar segment of which is an edge. Let W be a side of the tetrahedron S such that W. From Lemma 3 in [14] we have ϕ 2 L 2 () h 1 ϕ 2 L 2 (W ) + h ϕ 2 H 1 (W ). From the standard trace inequality w 2 L 2 ( S ) h 1 w 2 L 2 (S ) + h w 2 H 1 (S ) for all w H 1 (S ), applied to ϕ and xi ϕ, i = 1, 2, 3, we obtain h 1 ϕ 2 L 2 (W ) h 2 ϕ 2 L 2 (S ) + ϕ 2 H 1 (S ), h ϕ 2 H 1 (W ) ϕ 2 H 1 (S ) + h2 u e 2 H 2 (S ). From standard error bounds for the nodal interpolation operator I h we get ϕ 2 L 2 () h 2 ϕ 2 L 2 (S ) + ϕ 2 H 1 (S ) + h2 u e 2 H 2 (S ) h2 u e 2 H 2 (S ). Summing over h and using u e H 2 (ω h ) h 1 2 u H 2 (), cf. (3.16), results in ϕ 2 L 2 () h2 u e 2 H 2 (ω h ) h3 u 2 H 2 (), h which completes the proof.

10 10 M. A. OLSHANSKII, A. RUSKN, AND X. XU n h n h + m + h h 1 h 2 m h Fig Continuity estimates. In this section we derive a continuity estimate for the bilinear form a h (, ). If one applies partial integration to the integrals that occur in a h (, ) then jumps across the edges h occur. We start with a lemma that yields bounds for such jump terms. Related to these jump terms we introduce the following notation. For each F h, denote by m h the outer normal to an edge in the plane which contains element. Let [m h ] = m + h + m h be the jump of the outer normals to the edge in two neighboring elements, c.f. Figure 3.1. Lemma 3.5. he following holds: P(x)[m h ](x) h 2 a.e. x. (3.25) Proof. Let be the common side of two elements 1 and 2 in F h, and n + h, n h, m+ h and m h are the unit normals as illustrated in Figure 3.1. Denote by s h the unit (constant) vector along the common side, which can be represented as s h = n + h m+ h = m h n h. he jump across is given by [m h ] = s h (n + h n h ). For each x and p(x), let n = n(p(x)) be the unit normal to at p(x) and P = P(x) = I nn the corresponding orthogonal projection. Using (3.8), we get n h n+ h n+ h n + n h n h S 1 + h S2 h. Since n h = n+ h = n = 1, the above estimate implies We also have n + h n h = ch2 n + e 1, e 1 n, e 1 h. s h = n + h m+ h = (n + (n+ h n)) m+ h = n m+ h + e 2, e 2 h. We use the decomposition P[m h ] = P [ (n m + h + e 2) ( ch 2 n + e 1 )]. Since e 1 n we have (n m + h ) e 1 n and thus P ( (n m + h ) e 1) = 0. herefore, we get P[m h ] h 2 + e 1 e 2 h 2, (3.26)

11 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 11 i.e., the result (3.25) holds. In the analysis below, we need an inequality of the form v h L2 ( h ) v h for all v h Vh. his result can be obtained as follows. First we consider the case c = 0. hen the functions v h Vh satisfy h v h ds = 0. We assume that in Vh a discrete analogon of the Friedrich s inequality (2.6) holds uniformly with respect to h, i.e., there exists a constant C F independent of h such that v h 2 L 2 ( h ) C F h v h 2 L 2 ( h ) for all v h V h. (3.27) Now we reduce the parameter domain ε (0, 1], c > 0 as follows. For a given generic constant c 0 with 0 < c 0 < 1, in the remainder we restrict to the parameter set For c > 0 we then have ε (0, 1], c {0} [c 0 ε, ). (3.28) c 0 v h 2 L 2 ( h ) 2c 0 1 c c c 1 2 vh 2 L 2 ( h ) 2 c/c 0 + c v h 2 2 ε + c v h 2, and combing this with the result in (3.27) for the case c = 0 we get v h L2 ( h ) 1 ε + c v h for all v h V h (3.29) and arbitrary ε (0, 1], c {0} [c 0 ε, ). In the proof of heorem 3.8 below we need a bound for v h L 2 ( h ) in terms of v h. Such a result is derived with the help of the following lemma. Lemma 3.6. Assume the outer tetrahedra mesh size satisfies h h 0, with some sufficiently small h 0 1, depending only on the constant c 2 from (3.8). he following holds: vh 2 ds h 1 v h 2 L 2 ( h ) + h h v h 2 L 2 ( h ) for all v h Vh (3.30) h Proof. Let h be an edge of a triangle F h and S h is the corresponding tetrahedron of the outer triangulation. Consider the patch ω(s ) of all S h touching S. Denote ω(s ) = ω(s ) h. Let v h be an arbitrary fixed function from Vh. We shall prove the bound vh 2 ds h 1 v h 2 L 2 (ω(s )) + h h v h 2 L 2 (ω(s )). (3.31) hen summing over all h and using that ω(s ) consists of a uniformly bounded number of tetrahedra (due to the regularity of the outer mesh), we obtain (3.30). Let P be a plane containing. We can define for sufficiently small h an injective mapping ϕ : ω(s ) P such that ϕ 1 and (ϕ 1 ) 1. For example, ϕ can be build by the orthogonal projection on P. hen ϕ 1 and (ϕ 1 )(x) (sin α) 1, where α is the angle between P and n h (ϕ 1 (x)). Due to assumption (3.8) we have 1 sin α for sufficiently small h. If ϕ is the orthogonal projection on P, then ϕ() =

12 12 M. A. OLSHANSKII, A. RUSKN, AND X. XU. hus we get vhds 2 = (v h ϕ 1 ) 2 ds, ϕ() v h L2 (ω(s )) v h ϕ 1 L2 (ϕ(ω(s ))), h v h L2 (ω(s )) P (v h ϕ 1 ) L2 (ϕ(ω(s ))). (3.32) Due to the shape regularity of S ω(s ) we have h dist(, ω(s )) dist(, ω(s )). Hence, from ϕ 1 1 it follows that h dist(ϕ(), ϕ(ω(s ))). hus, we may consider a rectangle Q ϕ(ω(s )) such that = ϕ() is a side of Q and Q h. By the standard trace theorem and scaling argument we get ϕ() (v h ϕ 1 ) 2 ds h 1 v h ϕ 1 2 L 2 (Q) + h P(v h ϕ 1 ) 2 L 2 (Q). his together with (3.32) and Q ϕ(ω(s )) implies (3.31). An immediate consequence of the lemma and (3.29) is the following corollary. Corollary 3.7. he following estimate holds: h h vh 2 ds ( 1 ε + c + h2 ) vh 2 for all v h Vh. (3.33) ε We are now in position to prove a continuity result for the surface finite element bilinear form. Lemma 3.8. For any u H 2 () and v h Vh, we have a h (u e (I h u e ) h, v h ) (ε 1/2 + h 1/2 + c 12 h + h 2 ε + c + h3 ε ) h u H2 () v h. (3.34) Proof. Define ϕ = u e (I h u e ) h, then a h (ϕ, v h ) = ε h ϕ h v h ds h 1( + (we h ϕ)v h (w e h v h )ϕ ) + c ϕv h ds h 2 + ( ε h ϕ + w e h ϕ + cϕ) w e h v h ds. F h δ (3.35) We estimate a h (ϕ, v h ) term by term. Due to (3.20), (3.21), we get for the first term on the righthand side of (3.35): ε h ϕ h v h ds εh u H2 () h v h L2 ( h ) ε 1 2 h u H2 () v h. (3.36) h

13 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 13 o the second term on the righthand side of (3.35) we apply integration by parts: 1( (we h ϕ)v h (w e h v h )ϕ ) + cϕv h ds h 2 = cϕv h ds (w e h v h )ϕds h h + 1 (w e m h )ϕv h ds 1 (3.37) (div h w e )ϕv h ds 2 2 h F h =: I 1 + I 2 + I 3 + I 4. he term I 1 can be estimated by I 1 c 1 2 ϕ L2 ( h ) cv h L2 ( h ) h 2 c 1 2 u H2 () v h. o estimate I 2, we consider the advection-dominated case and the diffusion-dominated case separately. If Pe > 1, we have ( ) 1/2 (w e h v h )ϕds δ 1/2 ϕ L2 ( ) δ (w e h v h ) 2 ds ( 1/2 max(h 1/2, c 1/2 ) ϕ L2 ( ) δ (w e h v h ) ds) 2, and if Pe 1: Summing over F h we obtain (w e h v h )ϕds w e L ( ) h v h L2 ( ) ϕ L2 ( ) ε 1/2 h 1 ε 1/2 h v h L 2 ( ) ϕ L 2 ( ). I 2 (h 1/2 + ε 1/2 h 1 ) ϕ L2 ( h ) v h h(c 1/2 h + h 1/2 + ε 1/2 ) u H2 () v h. he term I 3 is estimated using Pw e = w e, Lemmas 3.4, 3.5, and Corollary 3.7: I 3 (w e [m h ])ϕv h ds h ( h h 3 u H2 () ϕ 2 ds ( ) 1/2 ( h h h v 2 h ds) 1/2 P[m h ] 2 v 2 h ds ) 1/2 ( h 3 ε + c + h4 ε ) u H2 () v h. (3.38) he term I 4 in (3.37) can be bounded due to Lemma 3.1, the interpolation bounds and (3.29): I div h w e L ( h ) ϕ L2 ( h ) v h L2 ( h ) h 3 u H2 () v h L2 ( h ) h3 ε + c u H 2 () v h.

14 14 M. A. OLSHANSKII, A. RUSKN, AND X. XU Finally we treat the third term on the righthand side of (3.35). Using δ w e L ( ) h, δ ε 1, δ c 1 and the interpolation estimates (3.20) and (3.21) we obtain: F h δ ( F h δ ( ε h ϕ + w e h ϕ + cϕ)w e h v h ds ( ε 2 h u e 2 L 2 ( ) + we h ϕ 2 L 2 ( ) + c2 ϕ 2 ) ) 1/2 L 2 ( ) v h (ε 1/2 + h 1/2 + c 1 2 h)h u H 2 () v h. Combing the inequalities (3.36) (3.39) proves the result of the lemma. (3.39) 3.5. Consistency estimate. he consistency error of the finite element method (2.12) is due to geometric errors resulting from the approximation of by h. o estimate this geometric errors we need a few additional definitions and results, which can be found in, for example, [13]. For x h define P h (x) = I n h (x)n(x) /(n h (x) n(x)). One can represent the surface gradient of u H 1 () in terms of h u e as follows u(p(x)) = (I d(x)h(x)) 1 Ph (x) h u e (x) a.e. x h. (3.40) Due to (3.9), we get u v ds = A h h u e h v e ds for all v H 1 (), (3.41) h with A h (x) = µ h (x) P h (x)(i d(x)h(x)) 2 Ph (x). From w n = 0 on and w e (x) = w(p(x)), n(x) = n(p(x)) it follows that n(x) w e (x) = 0 and thus w(p(x)) = P h (x)w e (x) holds. Using this, we get the relation (w u)v ds = (B h w e h u e )v e ds, (3.42) h with B h = µ h (x) P h (I dh) 1 Ph. In the proof we use the lifting procedure h given by v l h(p(x)) := v h (x) for x h. (3.43) It is easy to see that v l h H1 (). he following lemma estimates the consistency error of the finite element method (2.12). Lemma 3.9. Let u H 2 () be the solution of (2.5), then we have sup v h V h f h (v h ) a h (u e, v h ) v h ( h c 1 2 h + h c + ε ) h( u H2 () + f L2 ()). (3.44) Proof. he residual is decomposed as f h (v h ) a h (u e, v h ) = f h (v h ) f(v l h) + a(u, v l h) a h (u e, v h ). (3.45)

15 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 15 he following holds: f(vh) l = fvh l ds = µ h f e v h ds, h a(u, vh) l = ε u vh l ds + (w u)vhds l + cuvh l ds = ε u vh l ds + 1 (w u)vh l (w v l 2 h)u ds + = ε A h h u e h v h ds + 1 (B h w e h u e )v h ds h 2 h 1 (B h w e h v h )u e ds + µ h cu e v h ds. 2 h h Substituting these relations into (3.45) and using (2.13), (2.14) results in cuv l h ds f h (v h ) a h (u e, v h ) = (1 µ h )f e v h ds + ε (A h P h ) h u e h v h ds h h + 1 ((B h P h )w e h u e )v h ds 1 ((B h P h )w e h v h )u e ds 2 h 2 h + (µ h 1)cu e v h ds + ( δ f e + ε h u e w e h u e cu e) w e h v h ds h F h =: I 1 + I 2 + I 3 + I 4 + I 5 + Π 1. (3.46) We estimate the I i terms separately. Applying (3.10) and (3.29) we get I 1 h 2 f e L2 ( h ) v h L2 ( h ) h2 c + ε f L2 () v h, (3.47) I 5 h 2 c 1 2 u e L2 ( h ) cv h L2 ( h ) h 2 c 1 2 u L2 () v h. (3.48) One can show, cf. (3.43) in [8], the bound Using this we obtain P h A h = P h (I A h ) h 2. I 2 εh 2 h u e L2 ( h ) h v h L2 ( h ) ε 1/2 h 2 u e H2 () v h. (3.49) Since (I dh) 1 = I + O(h 2 ), we also estimate his yields B h P h h 2 + A h P h h 2. I 3 h 2 h u e L2 ( h ) v h L2 ( h ) h2 c + ε u H2 () v h. (3.50) o estimate I 4 we use the definition (2.15) of δ. If Pe 1, then ε 1 2 w e L ( ) 2 w e L ( ) h 2 S holds. If Pe > 1, then δ 1 2 max(c 1 2, δ w e L ( ) h 2 S ) holds.

16 16 M. A. OLSHANSKII, A. RUSKN, AND X. XU Using the assumption that the outer triangulation is quasi-uniform we get h 1 S h 1. hus, we obtain min{ε 1 2 w e L ( ), δ 1 2 } max(c 1 2, h 1 2 ) c h 1 2, and I 4 max B h P h (ε 1 2 h v h L2 ( ) + δ 1 2 w e x h h v h L2 ( )) h F h min{ε 1 2 w e L ( ), δ 1 2 } ue L2 ( ) h(c 1 2 h + h 1 2 ) vh u L2 (). (3.51) Now we estimate Π 1. Using the equation ε u + w u + cu = f on we get Π 1 = ( f p + ε h u e w e h u e cu e) w e h v h ds F h δ = F h δ + F h δ + F h δ ( ε( u) p + ε h u e) w e h v h ds ( (we u) p w e h u e) w e h v h ds ( cu p cu e ) w e h v h ds =: Π Π Π 3 1. (3.52) From u e = u p it follows that Π1 3 = 0 holds. For Π1 2 we obtain, using (3.40) and µ 1 h B h P h µ h 1 µ h 1 B h + B h P h h 2, Π1 2 = ( (µ 1 h B h P h )w e h u e) w e h v h ds F h δ h 2 ( F h δ w e 2 L h u e 2 L 2 ( ) ) 1 2 ( F h δ w e h v h 2 L 2 ( ) h 5 2 u H2 () v h. (3.53) Since δ ε h 2, we get Π 1 2 ( ) 1 2 ( ε 2 δ ( u) p h u e 2 L 2 ( ) δ w e h v h 2 L 2 ( ) F h F h ( ) 1 2 hε 1 2 ( u) p h u e 2 L 2 ( ) v h. (3.54) F h We finally consider the term between brackets in (3.54). Using the identity div f = tr( f) and n u e (p(x)) = 0 we obtain for x, with 2 :=. u(p(x)) = div u(p(x)) = tr(p P u e (p(x))) = tr(p 2 u e (p(x))p). (3.55) ) 1 2 ) 1 2

17 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 17 From the same arguments it follows that h u e (x) = tr(p h P h u e (x)) = tr(p h 2 u e (x)p h ) (3.56) holds. Using (3.14) and d(x) h 2, P P h h, H 1, H 1 we obtain P h 2 u e (x)p h = P 2 u e (p(x))p + R, hus, using (3.55) and (3.56), we get R h ( 2 u e (p(x)) + u e (p(x)) ). u(p(x)) h u e (x) tr(p 2 u e (p(x))p P h 2 u e (x)p h ) and combining this with (3.54) yields Π 1 2 hε 1 2 ( F h ( u) p h u e 2 L 2 ( ) h ( 2 u e (p(x)) + u e (p(x)) ), ) 1 2 v h ε 1 2 h 2 u H 2 () v h. Combining this with the results (3.46)-(3.51) and (3.53) proves the lemma Main theorem. Now we put together the results derived in the previous sections to prove the main result of the paper. heorem Let Assumption 3.1 be satisfied. We consider problem parameters ε and c as in (3.28). Assume that the solution u of (2.5) has regularity u H 2 (). Let u h be the discrete solution of the SUPG finite element method (2.12). hen the following holds: u e u h h ( h 1/2 + ε 1/2 + c 1 2 h + h ε + c + h3 ε ) ( u H 2 () + f L 2 ()). (3.57) Proof. he triangle inequality yields u e u h u e (I h u e ) h + (I h u e ) h u h. (3.58) he second term in the upper bound can be estimated using Lemmas 3.2, 3.8, 3.9: 1 2 (I hu e ) h u h 2 a h ((I h u e ) h u h, (I h u e ) h u h ) = a h ((I h u e ) h u e, (I h u e ) h u h ) + a h (u e u h, (I h u e ) h u h ) h ( h 1/2 + ε 1/2 + c 1 2 h + h 2 ε + c + h3 ε ) u H2 () (I h u e ) h u h + a h (u e, (I h u e ) h u h ) f h ((I h u e ) h u h ) h ( h 1/2 + ε 1/2 + c 1 2 h + his results in h ε + c + h3 ε )( u H2 () + f L2 ()) (Ih u e ) h u h. (I h u e ) h u h h ( h 1/2 +ε 1/2 +c 1 h 2 h+ + h3 ) ( u H ε + c ε 2 ()+ f L 2 ()). (3.59) he error estimate (3.57) follows from (3.22) and (3.59).

18 18 M. A. OLSHANSKII, A. RUSKN, AND X. XU 3.7. Further discussion. We comment on some aspects related to the main theorem. Concerning the analysis we note that the norm, which measures the error on the left-hand side of (3.57), is the standard SUPG norm as found in standard analyses of planar streamline-diffusion finite element methods. he analysis in this paper contains new ingredients compared to the planar case. o control the geometric errors (approximation of by h ) we derived a consistency error bound in Lemma 3.9. o derive a continuity result (Lemma 3.8), as in the planar case, we apply partial integration to the term h (w e h ϕ)v h ds, cf. (3.37). However, different from the planar case, this results in jumps across the edges h which have to be controlled, cf. (3.38). For this the new results in the Lemmas 3.5 and 3.6 are derived. hese jump terms across the edges cause the term h4 ε in the error bound in (3.57). Consider the error reduction factor h 3/2 + ε 1/2 h + c 1 2 h 2 + h2 ε+c + h4 ε on the righthand side of (3.57). he first three terms of it are typical for the error analysis of planar SUPG finite element methods for P 1 elements. In the standard literature for the planar case, cf. [3], one typically only considers the case c > 0. Our analysis also applies to the case c = 0, cf. (3.28). Furthermore, the estimates are uniform w.r.t. the size of the parameter c. For a fixed c > 0 and ε h the first four terms can be estimated by h 3/2, a bound similar to the standard one for the planar case. he only suboptimal term is the last one, which is caused by (our analysis of) the jump terms. Note, however, that h4 ε h 3/2 if h 5 ε holds, which is a very mild condition. he norm provides a robust control of streamline derivatives of the solution. Cross-wind oscillations, however, are not completely suppressed. It is well known that nonlinear stabilization methods can be used to get control over cross-wind derivatives as well. xtending such methods to surface PDs is not within the scope of the present paper. he error estimates in this paper are in terms of the maximum mesh size over tetrahedra in ω h, denoted by h. In practice, the stabilization parameter δ is based on the local Peclet number and the stabilization is switched off or reduced in the regions, where the mesh is sufficiently fine. o prove error estimates accounting for local smoothness of the solution u and the local mesh size (as available for planar SUPG F method), one needs local interpolation properties of finite element spaces, instead of (3.20), (3.21). Since our finite element space is based on traces of piecewise linear functions, such local estimates are not immediately available. he extension of our analysis to this non-quasi-uniform case is left for future research. Finally, we remark on the case of a varying reaction term coefficient c. If the coefficient c in the third term of (2.4) varies, the above analysis is valid with minor modifications. We briefly explain these modifications. he stabilization parameter δ should be based on elementwise values c = max x c(x). For the well-possedness of (2.4), it is sufficient to assume c to be strictly positive on a subset of with positive measure: A := meas{x : c(x) c 0 } > 0, with some c 0 > 0. If this is satisfied, the Friedrich s type inequality [12] (see, also Lemma 3.1 in [8]) v 2 L 2 () C F ( v 2 L 2 () + cv 2 L 2 () ) for all v V holds, with a constant C F depending on c 0 and A. Using this, all arguments in the analysis can be generalized to the case of a varying c(x) with obvious modifications.

19 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 19 With c min := ess inf x c(x) and c max := ess sup x c(x), the final error estimate takes the form u e u h h ( h 1/2 + ε 1/2 + c 1 2 max h + h ε + cmin + h3 ε ) ( u H2 () + f L2 ()). 4. Numerical experiments. In this section we show results of a few numerical experiments which illustrate the performance of the method. xample 1. he stationary problem (2.4) is solved on the unit sphere, with the velocity field w(x) = ( x 2 1 x 2 3, x 1 1 x 2 3, 0), which is tangential to the sphere. We set ε = 10 6, c 1 and consider the solution u(x) = x ( ) 1x 2 π arctan x3. ε Note that u has a sharp internal layer along the equator of the sphere. he corresponding right-hand side function f is given by f(x) = 8ε3/2 (2 + ε + 2x 2 3)x 1 x 2 x 3 π(ε + 4x 2 + 6εx 1x 2 + x x 2 2 (x2 1 x 2 ( ) 2) x3 arctan + u. 3 )2 π ε We consider the standard (unstabilized) finite element method in (2.11) and the stabilized method (2.12). A sequence of meshes was obtained by the gradual refinement of the outer triangulation. he induced surface finite element spaces have dimensions N = 448, 1864, 7552, he resulting algebraic systems are solved by a direct sparse solver. Finite element errors are computed outside the layer: he variation of the quantities err L 2 = u u h L2 (D), err H 1 = u u h H1 (D), and err L inf = u u h L (D), with D = {x : x 3 > 0.3}, are shown in Figure 4.1. he results clearly show that the stabilized method performs much better than the standard one. he results for the stabilized method indicate a O(h 2 ) convergence in the L 2 (D)-norm and L (D)-norm. In the H 1 (D)-norm a first order convergence is observed. Note that the analysis predicts (only) O(h 3 2 ) convergence order in the (global) L 2 -norm L 2 errors slope 1 stablized FM standard FM slope 1/2 log 10 (err H1 ) H 1 errors stablized FM standard FM slope 1/2 log 10 (err inf ) slope 1 L errors stablized FM standard FM log 10 (N) log 10 (N) log 10 (N) Fig Discretization errors for xample 1

20 20 M. A. OLSHANSKII, A. RUSKN, AND X. XU Figure 4.2 shows the computed solutions with the two methods. Since the layer is unresolved, the finite element method (2.11) produces globally oscillating solution. he stabilized method gives a much better approximation, although the layer is slightly smeared, as is typical for the SUPG method. (a) (b) Fig xample 1: solutions using the stabilized method and the standard method (2.11). xample 2. Now we consider the stationary problem (2.4) with c 0. he problem is posed on the unit sphere, with this same velocity field w as in xample 1. We set ε = 10 6, and consider the solution u(x) = x ( ) 1x 2 π arctan x3. ε he corresponding right-hand side function f is now given by f(x) = 8ε3/2 (2 + ε + 2x 2 3)x 1 x 2 x 3 π(ε + 4x 2 + 6εx 1x 2 + x x 2 2 (x2 1 x 2 ( ) 2) x3 arctan. 3 )2 π ε Note that for c = 0 one looses explicit control of the L 2 norm in. hus we consider the streamline diffusion error : err SD = w (u u h ) L2 (D). Results for this error quantity and for err L inf = u u h L (D) are shown in Figure 4.3. We observe a O(h) behavior for the streamline diffusion error, which is consistent with our theoretical analysis. he L -norm of the error also shows a first order of convergence. xample 3. We show how this stabilization can be applied to a time dependent problem and illustrate its stabilizing effect. We consider a non-stationary problem (2.3) posed on the torus = {(x 1, x 2, x 3 ) ( x x2 2 1)2 + x 2 3 = 1 }. (4.1) 16 We set ε = 10 6 and define the advection field w(x) = 1 x x 2 2 ( x 2, x 1, 0),

21 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS streamline diffusion errors stablized FM standard FM L errors stablized FM standard FM log 10 (err SD ) log 10 (err inf ) slope 1/2 0.7 slope 1/ log 10 (N) log 10 (N) Fig Discretization errors for xample 2 which is divergence free and satisfies w n = 0. he initial condition is u 0 (x) = x ( ) 1x 2 π arctan x3. ε he function u 0 possesses an internal layer, as shown in Figure 4.4(a). he stabilized spatial semi-discretization of (2.3) reads: determine u h = u h (t) such that V h m( t u h, v h ) + â h (u h, v h ) = 0 for all v h V h. (4.2) with m( t u, v) := t uv ds + δ t u(w e h v) ds, h F h â h (u, v) :=ε h u h v ds + 1 [ ] (w e h u)v ds (w e h v)u ds h 2 h h + ( ε h u + w e h u)w e h v ds. F h δ Note that â h (, ) is the same as a h (, ) in (2.13) with c 0. he resulting system of ordinary differential equations is discretized in time by the Crank-Nicolson scheme. For ε = 0 the exact solution is the transport of u 0 (x) by a rotation around the x 3 axis. hus the inner layer remains the same for all t > 0. For ε = 10 6, the exact solution is similar, unless t is large enough for dissipation to play a noticeable role. he space Vh is constructed in the same way as in the previous examples. he spatial discretization has 5638 degrees of freedom. he fully discrete problem is obtained by combining the SUPG method in (4.2) and the Crank-Nicolson method with time step δt = 0.1. he evolution of the solution is illustrated in Figure 4.4 demonstrating a smoothly rotated pattern. We repeated this experiment with δ = 0 in the bilinear forms m(, ) and â h (, ) in (4.2), i.e. the method without stabilization. As expected, we obtain (on the same grid) much less smooth discrete solutions (Figure 4.5).

22 22 M. A. OLSHANSKII, A. RUSKN, AND X. XU (a) (b) (c) (d) Fig xample 3: solutions for t = 0, 0.6, 1.2, 1.8 using the SUPG stabilized FM. (a) (b) Fig xample 3: solutions for t = 1.2, 1.8 using the standard FM. Remark 4.1. With respect to mass conservation of the scheme we note the following. For vh 1 in (4.2) we get, with Mh (t) := h uh (x, t) ds, d Mh (t) = dt 1 e w h uh ds h = we [m]uh ds + divh we uh ds 2 2 h h 2 uh ds + Ch uh ds..h h h

23 A FM FOR ADVCION-DIFFUSION QUAIONS ON SURFACS 23 Here we used estimates from Lemmas 3.1 and 3.5. Using Lemma 3.6 we get ( ) u h ds h u 2 2 hds h h h 1( u h L2 ( h ) + h h u h L2 ( h )). Assume that for the discrete solution we have a bound u h L2 ( h )+h h u h L2 ( h ) c with c independent of h. hen we obtain d dt M(t) h and thus M h(t) M h (0) cth, with a constant c independent of h and t. Hence, with respect to mass conservation we have an error that is (only) first order in h. Concerning mass conservation it would be better to use a discretization in which in the discrete bilinear form in (2.13) one replaces 1 2 [ ] (w e h u)v ds (w e h v)u ds h h by (w e h v)u ds. h (4.3) d his method results in optimal mass conservation: dt M h(t) = 0. It turns out, however, that (with our approach) it is more difficult to analyze. In particular, it is not clear how to derive a satisfactory coercivity bound. In numerical experiments we observed that the behavior of the two methods (i.e, with the two variants given in (4.3)) is very similar. In particular, the mass conservation error bound M(t) M(0) cth for the first method seems to be too pessimistic (in many cases). o illustrate this, we show results for the problem described above, but with initial condition u 0 (x) = ( ) π arctan x3, u 0 ds = π ε Figure 4.6 shows the quantity M h (t) for several mesh sizes h. For t = 0 we have, due to interpolation of the initial condition u 0, a difference between M h (0) and u 0ds that is of order h 2. For t > 0 we see, except for the very coarse mesh with h = 1/4 a very good mass conservation. xample 4. As a final illustration we show results for the non-stationary problem (2.3), but now on a surface with a less regular shape. We take the surface given in [4]: = {(x 1, x 2, x 3 ) (x 1 x 2 3) 2 + x x 2 3 = 1}. (4.4) We set ε = 10 6 and define the advection field as the -tangential part of w = ( 1, 0, 0). his velocity field does not satisfy the divergence free condition. he initial condition is taken as u 0 (x) = 1. We apply the same method as in xample 3. he mesh size is 1/8 and the time step is δt = 0.1. Figure 4.7 shows the solution for several t values. We observe that as time evolves mass is transported from the two poles on the right to the left pole, just as expected. Our discretization yields for this strongly convection-dominated transport problem a qualitatively good discrete result, even with a (very) low grid resolution. Acknowledgments. his work has been supported in part by the DFG through grant R1461/4-1, by the Russian Foundation for Basic Research through grants , , and by NSFC project We thank J. Grande for his support with the implementation of the methods and C. Lehrenfeld for useful discussions.

24 24 M. A. OLSHANSKII, A. RUSKN, AND X. XU mass meshsize=1/4 meshsize=1/8 meshsize=1/ t Fig otal mass variation for xample 3 (a) (b) (c) (d) Fig xample 4: solutions for t = 0, 0.5, 1.0, 2.0 using the SUPG stabilized FM. RFRNCS [1] S. Gross and A. Reusken. Numerical Methods for wo-phase Incompressible Flows, volume 40 of Springer Series in Computational Mathematics. Springer, Heidelberg, [2] M. L. Agrawal and R. Neumann. Surface diffusion in monomolecular films II. experiments and theory. J. Coll. Interface Sci, 121: , 1988.

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