Key words. surface PDE, finite element method, transport equations, advection-diffusion equation, SUPG stabilization

Transcription

1 A SABILIZED FINIE ELEMEN MEHOD FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES MAXIM A. OLSHANSKII, ARNOLD REUSKEN, AND XIANMIN XU Abstract. A recently developed Eulerian finite element metod is applied to solve advectiondiffusion equations posed on ypersurfaces. Wen transport processes on a surface dominate over diffusion, finite element metods tend to be unstable unless te mes is sufficiently fine. e paper introduces a stabilized finite element formulation based on te SUPG tecnique. An error analysis of te metod is given. Results of numerical experiments are presented tat illustrate te performance of te stabilized metod. Key words. surface PDE, finite element metod, transport equations, advection-diffusion equation, SUPG stabilization AMS subject classifications. 58J32, 65N12, 65N30, 76D45, Introduction. Matematical models involving partial differential equations posed on ypersurfaces occur in many applications. Often surface equations are coupled wit oter equations tat are formulated in a (fixed) domain wic contains te surface. is appens, for example, in common models of multipase fluids dynamics if one takes so-called surface active agents into account [1]. In tis application, depending on te bulk flow, interface processes may be dominated by transport penomena, resulting in advection-diffusion equations on te surface wit dominating advection terms. e 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 tat finite element discretization metods for advection-diffusion problems need an additional stabilization mecanism, unless te mes size is sufficiently small to resolve boundary and internal layers in te solution of te differential equation. For te planar case, tis topic as been extensively studied in te literature and a variety of stabilization metods as been developed, see, e.g., [2]. We are, owever, not aware of any studies of stable finite element metods for advection-diffusion equations posed on surfaces. In te past decade te study of numerical metods for PDEs on surfaces as been a rapidly growing researc area. e development of finite element metods for solving elliptic equations on surfaces can be traced back to te paper [3], wic considers a piecewise polygonal surface and uses a finite element space on a triangulation of tis discrete surface. is approac as been furter analyzed and extended in several directions, see, e.g., [4] and te references terein. Anoter approac as been introduced in [5] and builds on te ideas of [6]. e metod in tat paper applies to cases in wic te surface is given implicitly by some level set function and te key idea is to solve te partial differential equation on a narrow band around te surface. Unfitted finite element spaces on tis narrow band are used for discretization. Anoter surface finite element metod based on an outer (bulk) mes as been introduced in [7] and Department of Matematics, University of Houston, Houston, exas and Department of Mecanics and Matematics, Moscow State University, Moscow, Russia, (Maxim.Olsanskii@mtu-net.ru). Institut für Geometrie und Praktisce Matematik, RWH-Aacen University, D Aacen, Germany (reusken@igpm.rwt-aacen.de,xu@igpm.rwt-aacen.de). LSEC, Institute of Computational Matematics and Scientific/Engineering Computing, NCMIS, AMSS, Cinese Academy of Sciences, Beijing , Cina 1

2 2 M. A. OLSHANSKII, A. REUSKEN, AND X. XU furter studied in [8, 9]. e main idea of tis metod is to use finite element spaces tat are induced by triangulations of an outer domain to discretize te partial differential equation on te surface by considering traces of te bulk finite element space on te surface, instead of extending te PDE off te surface, as in [6, 5]. e metod is particularly suitable for problems in wic te surface is given implicitly by a level set or VOF function and in wic tere is a coupling wit a differential equation in a fixed outer domain. If in suc problems one uses finite element tecniques for te discetization of equations in te outer domain, tis setting immediately results in an easy to implement discretization metod for te surface equation. e approac does not require additional surface elements. In tis paper we reconsider te volume mes finite element metod from [7] and study a new aspect, tat as not been studied in te literature so far, namely te stabilization of advection-dominated problems. We restrict ourselves to te case of a stationary surface. o stabilize te discrete problem for te case of large mes Peclet numbers, we introduce a surface variant of te SUPG metod. For a class of stationary advection-diffusion equation an error analysis is presented. Altoug te convergence of te metod is studied using a SUPG norm similar to te planar case [2], te analysis is not standard and contains new ingredients: Some new approximation properties for te traces of finite elements are needed and geometric errors require special control. e main teoretical result is given in eorem It yields an error estimate in te SUPG norm wic is almost robust in te sense tat te dependence on te Peclet number is mild. is dependence is due to some insufficiently controlled geometric errors, as will be explained in section 3.7. e remainder of te paper is organized as follows. In section 2, we recall equations for transport-diffusion processes on surfaces and present te stabilized finite element metod. Section 3 contains te teoretical results of te paper concerning te approximation properties of te finite element space and discretization error bounds for te finite element metod. Finally, in section 4 results of numerical experiments are given for bot stationary and time-dependent advection-dominated surface transport-diffusion equations, wic sow tat te stabilization performs well and tat numerical results are consistent wit wat is expected from te SUPG metod in te planar case. 2. Advection-diffusion equations on surfaces. Let Ω be an open domain in R 3 and be a connected C 2 compact yper-surface contained in Ω. For a sufficiently smoot function g : Ω R te tangential derivative at is defined by g = g ( g n )n, (2.1) were n denotes te unit normal to. Denote by te Laplace-Beltrami operator on. Let w : Ω R 3 be a given divergence-free (div w = 0) velocity field in Ω. If te surface evolves wit a normal velocity of w n, ten te conservation of a scalar quantity u wit a diffusive flux on (t) leads to te surface PDE: u + (div w)u ε u = 0 on (t), (2.2) were u denotes te advective material derivative, ε is te diffusion coefficient. In [10] te problem (2.2) was sown to be well-posed in a suitable weak sense. In tis paper, we study a finite element metod for an advection-dominated problem on a steady surface. erefore, we assume w n = 0, i.e. te advection velocity is everywere tangential to te surface. is and div w = 0 implies div w = 0, and

3 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 3 te surface advection-diffusion equation takes te form: u t + w u ε u = 0 on. (2.3) Altoug te metodology and numerical examples of te paper are applied to te equations (2.3), te error analysis will be presented for te stationary problem ε u + w u + c(x)u = f on, (2.4) wit f L 2 () and c(x) 0. o simplify te presentation we assume c( ) to be constant, i.e. c(x) = c 0. e analysis, owever, also applies to non-constant c, cf. section 3.7. Note tat (2.3) and (2.4) can be written in intrinsic surface quantities, since w u = w u, wit te tangential velocity w = w (w n )n. We assume w H 1, () L () and scale te equation so tat w L () = 1 olds. Furtermore, since we are interested in te advection-dominated case we take ε (0, 1]. Introduce te bilinear form and te functional: a(u, v) := ε u v ds + (w u)v ds + c uv ds, f(v) := fv ds. e weak formulation of (2.4) is as follows: Find u V suc tat wit 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 te Lax-Milgram lemma, tere exists a unique solution of (2.5). For te case c = 0 te following Friedric s inequality [11] olds: v 2 L 2 () C F v 2 L 2 () v V. (2.6) 2.1. e stabilized volume mes FEM. In tis section we recall te volume mes FEM introduced in [7] and describe its SUPG type stabilization. Let { } >0 be a family of tetraedral triangulations of te domain Ω. ese triangulations are assumed to be regular, consistent and stable. o simplify te presentation we assume tat tis family of triangulations is quasi-uniform. e latter assumption, owever, is not essential for our analysis. We assume tat for eac a polygonal approximation of, denoted by, is given: is a C 0,1 surface witout boundary and can be partitioned in planar triangular segments. It is important to note tat is not a triangulation of in te usual sense (an O( 2 ) approximation of, consisting of regular triangles). Instead, we (only) assume tat is consistent wit te outer triangulation in te following sense. For any tetraedron S suc tat meas 2 (S ) > 0, define = S. We assume tat every is a planar segment and tus it is eiter a triangle or a quadrilateral. Eac quadrilateral segment can be divided into two triangles, so we may assume tat every is a triangle. An illustration of suc a triangulation is given in Figure 2.1. e results sown in tis figure are obtained by representing a spere implicitly by its signed distance

4 4 M. A. OLSHANSKII, A. REUSKEN, AND X. XU Fig Approximate interface for a spere, resulting from a coarse tetraedral triangulation (left) and after one refinement (rigt). function, constructing te piecewise linear nodal interpolation of tis distance function on a uniform tetraedral triangulation of Ω and ten considering te zero level of tis interpolant. Let F be te set of all triangular segments, ten can be decomposed as = F. (2.7) Note tat te triangulation F is not necessarily regular, i.e. elements from may ave very small internal angles and te size of neigboring triangles can vary strongly, cf. Figure 2.1. In applications wit level set functions (tat represent implicitly), te approximation can be obtained as te zero level of a piecewise linear finite element approximation of te level set function on te tetraedral triangulation. e surface finite element space is te space of traces on of all piecewise linear continuous functions wit respect to te outer triangulation. is can be formally defined as follows. We define a subdomain tat contains : ω = S, (2.8) F an a corresponding volume mes finite element space V := {v C(ω ) v S P 1 F }, (2.9) were P 1 is te space of polynomials of degree one. V induces te following space on : V := {ψ H 1 ( ) v V suc tat ψ = v }. (2.10) Wen c = 0, we require tat any function v from V satisfies v ds = 0. Given te surface finite element space V, te finite element discretization of (2.5) is as follows: Find u V suc tat ε u v ds + (w e u)v ds + c uv ds = f e v ds (2.11)

5 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 5 for all v V. Here we and f e are te extensions of w and f, respectively, along normals to (te precise definition is given in te next section). Similar to te plain Galerkin finite element for advection-diffusion equations te metod (2.11) is unstable unless te mes is sufficiently fine suc tat te mes Peclet number is less tan one. We introduce te following stabilized finite element metod based on te standard SUPG approac, cf. [2]: Find u V suc tat wit a (u, v ) = f (v ) v V, (2.12) a (u, v) :=ε u v ds + c uvds + 1 [ ] (w e u)v ds (w e v)u ds 2 + ( ε u + w e u + c u)w e v ds, F δ f (v) := f e vds + δ F (2.13) f e (w e v) ds. (2.14) e stabilization parameter δ depends on S. e diameter of te tetraedron S is denoted by S. Let Pe := S w e L ( ) be te cell Peclet number. We 2ε take δ 0 S w δ e if Pe > 1, L ( ) = and δ δ 1 2 = min{ δ, c 1 }, (2.15) S if Pe 1, ε wit some given positive constants δ 0, δ 1 0. Since u V is linear on every we ave u = 0 on, and tus a (u, v ) simplifies to a (u, v ) = ε u v ds + 1 [ ] (w e u )v (w e v )u ds 2 + c u (v + δ(x)w e v ) ds + δ(x)(w e u )(w e v ) ds, (2.16) were δ(x) = δ for x. 3. Error analysis. e analysis in tis 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 tese analysis we obtain te finite element error bound given in section 3.6. In te error analysis we use te following mes-dependent norm: ( ) 1 u := ε u 2 ds + δ(x) w e u 2 ds + c u 2 2 ds. (3.1) Here and in te remainder denotes te Euclidean norm for vectors and te corresponding spectral norm for matrices.

6 6 M. A. OLSHANSKII, A. REUSKEN, AND X. XU 3.1. Preliminaries. For te ypersurface, we define its -neigborood: U := {x R 3 dist(x, ) < c 0 }, (3.2) and assume tat c 0 is sufficiently large suc tat ω U and sufficiently small suc tat 5c 0 < ( max i=1,2 κ i L ()) 1 (3.3) olds, wit κ i being te principal curvatures of. Here and in wat follows denotes te maximum diameter for tetraedra of outer triangulation: = max S ω diam(s). Let d : U R be te signed distance function, d(x) = dist(x, ) for all x U. us is te zero level set of d. We assume d < 0 in te interior of and d > 0 in te exterior and define n(x) := d(x) for all x U. Hence, n = n on and n(x) = 1 for all x U. e Hessian of d is denoted by H(x) := 2 d(x) R 3 3, x U. (3.4) e eigenvalues of H(x) are denoted by κ 1 (x), κ 2 (x), and 0. For x te eigenvalues κ i, i = 1, 2, are te principal curvatures. For eac x U, define te projection p : U by p(x) = x d(x)n(x). (3.5) Due to te assumption (3.3), te decomposition x = p(x) + d(x)n(x) is unique. We will need te ortogonal projector P(x) := I n(x)n(x), for x U. Note tat n(x) = n(p(x)) and P(x) = P(p(x)) for x U olds. e tangential derivative can be written as g(x) = P g(x) for x. One can verify tat for tis projection and for te Hessian H te relation HP = PH = H olds. Similarly, define P (x) := I n (x)n (x), for x, x is not on an edge, (3.6) were n is te unit (outward pointing) normal at x (not on an edge). e tangential derivative along is given by g(x) = P (x) g(x) (not on an edge). Assumption 3.1. In tis paper, we assume tat for all F : ess sup x d(x) c 1 2 S, (3.7) ess sup x n(x) n (x) c 2 S, (3.8) were S denotes te diameter of te tetraedron S tat contains, i.e., = S and constants c 1, c 2 are independent of,. e assumptions (3.7) and (3.8) describe ow accurate te piecewise planar approximation of is. If is constructed as te zero level of a piecewise linear interpolation of a level set function tat caracterizes (as in Fig. 2.1) ten tese assumptions are fulfilled, cf. Sect. 7.3 in [1]. In te remainder, A B means A cb for some positive constant c independent of and of te problem parameters ε and c. A B means tat bot A B and B A.

7 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 7 For x, define µ (x) = (1 d(x)κ 1 (x))(1 d(x)κ 2 (x))n (x)n (x). e surface measures ds and ds on and, respectively, are related by e assumptions (3.7) and (3.8) imply tat µ (x)ds (x) = ds(p(x)), x. (3.9) ess sup x (1 µ ) 2, (3.10) cf. (3.37) in [7]. e solution of (2.4) is defined on, wile its finite element approximation u V is defined on. We need a suitable extension of a function from to its neigborood. For a function v on we define v e (x) := v(p(x)) for all x U. (3.11) e following formula for tis lifting function are known (cf. section 2.3 in [12]): u e (x) = (I d(x)h) u(p(x)) a.e. on U, (3.12) u e (x) = P (x)(i d(x)h) u(p(x)) a.e. on, (3.13) wit H = H(x). By direct computation one derives te 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 smoot u and µ 2, using tis relation one obtains te estimate D µ u e (x) D µ u(p(x)) + u(p(x)) a.e. on U, (3.15) µ =2 (cf. Lemma 3 in [3]). is furter leads to (cf. Lemma 3.2 in [7]): D µ u e L2 (U ) u H2 (), µ 2. (3.16) e next lemma is needed for te analysis in te following section. Lemma 3.1. e following olds: div w e L ( ) w L (). Proof. We use te following representation for te tangential divergence: div w(x) = tr( w(x)) = tr(p w(x)). (3.17) ake x, not lying on an edge. Using (3.12) we obtain div w e (x) = tr(p w e (x)) = tr (P (I d(x)h) w(p(x))) = tr (P w(p(x))) + tr ((P P) w(p(x))) d(x) tr (P H w(p(x))). e first term vanises due to tr (P w(p(x))) = div w(p(x)) = 0 e second and te tird term can be bounded using (3.7), (3.8): is proves te lemma. P P, d(x)p H 2.

8 8 M. A. OLSHANSKII, A. REUSKEN, AND X. XU 3.2. Coercivity analysis. In te next lemma we present a coercivity result. We use te norm introduced in (3.1). Lemma 3.2. e following olds: a (v, v ) 1 2 v 2 for all v V. (3.18) Proof. For any v V, we ave a (v, v ) = v 2 + c δ(x)v (w e v ) ds. (3.19) e coice of δ, cf. (2.15), implies c δ(x) 1. Hence te last term in (3.19) can be estimated as follows: c δ(x)v (w e v ) ds 1 c v 2 ds + 1 c δ(x) 2 (w e v ) 2 ds v 2. is yields (3.18). As a consequence of tis result we obtain te well-posedness of te discrete problem (2.12) Interpolation error bounds. Let I : C( ω ) V be te nodal interpolation operator. For any u H 2 () te surface finite element function (I u e ) V is an interpolant of u e in V. For any u H 2 () te following estimates old [7]: u e (I u e ) L2 ( ) 2 u H2 (), (3.20) u e (I u e ) L2 ( ) u H2 (). (3.21) Using tese results we easily obtain an interpolation error estimate in te - norm: Lemma 3.3. For any u H 2 () te following olds: u e (I u e ) (ε 1/2 + 1/2 + c 1 2 ) u H2 (). (3.22) Proof. Define φ := u e (I u e ) H 1 ( ). Using te definition (2.15) of δ(x), we get δ(x) w e φ 2 ds = δ w e φ 2 ds F (3.23) φ 2 L 2 ( ) 3 u 2 H 2 (). e remaining two terms in u e (I u e ) are estimated in a straigtforward way using (3.20) and (3.21). is and (3.23) imply te inequality (3.22). e next lemma estimates te interpolation error on te edges of te surface triangulation. In te remainder, E denotes te set of all edges in te interface triangulation F.

9 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 9 Lemma 3.4. For all u H 2 () te following olds: ( E E E (u e I u e ) 2 ds) 1/2 3/2 u H 2 (). (3.24) Proof. Define ϕ := u e I u e H 1 (ω ). ake E E and let F be a corresponding planar segment of wic E is an edge. Let W be a side of te tetraedron S suc tat E W. From Lemma 3 in [13] we ave From te standard trace inequality ϕ 2 L 2 (E) 1 ϕ 2 L 2 (W ) + ϕ 2 H 1 (W ). w 2 L 2 ( S ) 1 w 2 L 2 (S ) + w 2 H 1 (S ) for all w H 1 (S ), applied to ϕ and xi ϕ, i = 1, 2, 3, we obtain 1 ϕ 2 L 2 (W ) 2 ϕ 2 L 2 (S ) + ϕ 2 H 1 (S ), ϕ 2 H 1 (W ) ϕ 2 H 1 (S ) + 2 u e 2 H 2 (S ). From standard error bounds for te nodal interpolation operator I we get ϕ 2 L 2 (E) 2 ϕ 2 L 2 (S ) + ϕ 2 H 1 (S ) + 2 u e 2 H 2 (S ) 2 u e 2 H 2 (S ). Summing over E E and using u e H2 (ω ) 1 2 u H2 (), cf. (3.16), results in E E ϕ 2 L 2 (E) 2 u e 2 H 2 (ω ) 3 u 2 H 2 (), wic completes te proof Continuity estimates. In tis section we derive a continuity estimate for te bilinear form a (, ). If one applies partial integration to te integrals tat occur in a (, ) ten jumps across te edges E E occur. We start wit a lemma tat yields bounds for suc jump terms. Related to tese jump terms we introduce te following notation. For eac F, denote by m E te outer normal to an edge E in te plane wic contains element. Let [m ] E = m + + m be te jump of te outer normals to te edge in two neigboring elements, c.f. Figure 3.1. Lemma 3.5. e following olds: P(x)[m ](x) 2 a.e. x E. (3.25) Proof. Let E be te common side of two elements 1 and 2 in F, and n +, n, m+ and m are te unit normals as illustrated in Figure 3.1. Denote by s te unit (constant) vector along te common side E, wic can be represented as s = n + m+ = m n. e jump across E is given by [m ] = s (n + n ). For eac x E and p(x), let n = n(p(x)) be te unit normal to at p(x) and P = P(x) = I nn te corresponding ortogonal projection. Using (3.8), we get n n+ n+ n + n n S 1 + S2.

10 10 M. A. OLSHANSKII, A. REUSKEN, AND X. XU n 2 n 1 E m m Fig Since n = n+ = n = 1, te above estimate implies We also ave n + n = c2 n + e 1, e 1 n, e 1. s = n + m+ = (n + (n+ n)) m+ = n m+ + e 2, e 2. We use te decomposition P[m ] = P [ (n m + + e 2) ( c 2 n + e 1 )]. Since e 1 n we ave (n m + ) e 1 n and tus P ( (n m + ) e 1) = 0. erefore, we get i.e., te result (3.25) olds. P[m ] 2 + e 1 e 2 2, (3.26) In te analysis below we need an inequality of te form v L2 ( ) v for all v V. is result can be obtained as follows. First we consider te case c = 0. en te functions v V satisfy v ds = 0. We assume tat in V a discrete analogon of te Friedric s inequality (2.6) olds uniformly wit respect to, i.e., tere exists a constant C F independent of suc tat v 2 L 2 ( ) C F v 2 L 2 ( ) for all v V. (3.27) We reduce te parameter domain ε (0, 1], c > 0 as follows. For a given generic constant c 0 wit 0 < c 0 < 1, in te remainder we restrict to te parameter set For c > 0 we ten ave ε (0, 1], c {0} [c 0 ε, ). (3.28) c 0 v 2 L 2 ( ) 2c 0 1 c c c 1 2 v 2 L 2 ( ) 2 c/c 0 + c v 2 2 ε + c v 2, and combing tis wit te result in (3.27) for te case c = 0 we get v L 2 ( ) 1 ε + c v for all v V (3.29)

11 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 11 and arbitrary ε (0, 1], c {0} [c 0 ε, ). In te proof of eorem 3.8 below we need a bound for v L2 (E) in terms of v. Suc a result is derived wit te elp of te following lemma. Lemma 3.6. Assume te outer tetraedra mes size satisfies 0, wit some sufficiently small 0 1, depending only on te constant c 2 from (3.8). e following olds: v 2 ds 1 v 2 L 2 ( ) + v 2 L 2 ( ) for all v V (3.30) E E E Proof. Let E E be an edge of a triangle F and S is te corresponding tetraedron of te outer triangulation. Consider te patc ω(s ) of all S toucing S. Denote ω(s ) = ω(s ). Let v be an arbitrary fixed function from V. We sall prove te bound v 2 ds 1 v 2 L 2 (ω(s )) + v 2 L 2 (ω(s )). (3.31) E en summing over all E E and using tat ω(s ) consists of a uniformly bounded number of tetraedra (due to te regularity of te outer mes), we obtain (3.30). Let P be a plane containing. We can define for sufficiently small an injective mapping ϕ : ω(s ) P suc tat ϕ 1 and (ϕ 1 ) 1. For example, ϕ can be build by te ortogonal projection on P. en ϕ 1 and (ϕ 1 )(x) (sin α) 1, were α is te angle between P and n (ϕ 1 (x)). Due to assumption (3.8) we ave 1 sin α for sufficiently small. If ϕ is te ortogonal projection on P, ten ϕ(e) = E. us we get v ds 2 = (v ϕ 1 ) 2 ds, E ϕ(e) v L2 (ω(s )) v ϕ 1 (3.32) L2 (ϕ(ω(s ))), v L2 (ω(s )) P (v ϕ 1 ) L2 (ϕ(ω(s ))). Due to te sape regularity of S ω(s ) we ave dist(e, ω(s )) dist(e, ω(s )). Hence, from ϕ 1 1 it follows tat dist(ϕ(e), ϕ(ω(s ))). us, we may consider a rectangle Q ϕ(ω(s )) suc tat E = ϕ(e) is a side of Q and Q E. By te standard trace teorem and scaling argument we get (v ϕ 1 ) 2 ds 1 v ϕ 1 2 L 2 (Q) + P(v ϕ 1 ) 2 L 2 (Q). ϕ(e) is togeter wit (3.32) and Q ϕ(ω(s )) implies (3.31). An immediate consequence of te lemma and (3.29) is te following corollary. Corollary 3.7. e following estimate olds: E E E v 2 ds ( 1 ε + c + 2 ) v 2 for all v V. (3.33) ε

12 12 M. A. OLSHANSKII, A. REUSKEN, AND X. XU We are now in position to prove a continuity result for te surface finite element bilinear form. Lemma 3.8. For any u H 2 () and v V, we ave a (u e (I u e ), v ) (ε 1/2 + 1/2 + c 12 2 ) u H ε + c ε 2 () v. (3.34) Proof. Define ϕ = u e (I u e ), ten a (ϕ, v ) = ε ϕ v ds 1( + (we ϕ)v (w e v )ϕ ) + c ϕv ds 2 + ( ε ϕ + w e ϕ + cϕ) w e v ds. F δ (3.35) We estimate a (ϕ, v ) term by term. Due to (3.20), (3.21), we get for te first term on te rigtand side of (3.35): ε ϕ v ds ε u H2 () v L2 ( ) ε 1 2 u H2 () v. (3.36) o te second term on te rigtand side of (3.35) we apply integration by parts: 1( (we ϕ)v (w e v )ϕ ) + cϕv ds 2 = cϕv ds (w e v )ϕds + 1 (w e m )ϕv ds 1 (3.37) (div w e )ϕv ds 2 2 F =: I 1 + I 2 + I 3 + I 4. e term I 1 can be estimated by I 1 c 1 2 ϕ L2 ( ) cv L2 ( ) 2 c 1 2 u H2 ( ) v. o estimate I 2, we consider te advection-dominated case and te diffusion-dominated case separately. If Pe > 1, we ave ( ) 1/2 (w e v )ϕds δ 1/2 ϕ L2 ( ) δ (w e v ) 2 ds and if Pe 1: max( 1/2, c 1/2 ) ϕ L2 ( ) ( δ (w e v ) 2 ds) 1/2, (w e v )ϕds w e L ( ) v L2 ( ) ϕ L2 ( ) ε 1/2 1 ε 1/2 v L 2 ( ) ϕ L 2 ( ).

13 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 13 Summing over F we obtain I 2 ( 1/2 + ε 1/2 1 ) ϕ L2 ( ) v (c 1/2 + 1/2 + ε 1/2 ) u H2 () v. e term I 3 is estimated using Pw e = w e, Lemmas 3.4, 3.5, and Corollary 3.7: I 3 (w e [m ])ϕv ds E E ( E E E E 3 u H2 () ϕ 2 ds ( ) 1/2 ( E E E E E E v 2 ds) 1/2 P[m ] 2 v 2 ds ) 1/2 ( 3 ε + c + 4 ε ) u H2 () v. (3.38) e term I 4 in (3.37) can be bounded due to Lemma 3.1, te interpolation bounds and (3.29): I div w e L ( ) ϕ L2 ( ) v L2 ( ) 3 u H2 () v L2 ( ) 3 ε + c u H 2 () v. Finally we treat te tird term on te rigtand side of (3.35). Using δ w e L ( ), δ ε 1, δ c 1 and te interpolation estimates (3.20) and (3.21) we obtain: ( ε ϕ + w e ϕ + cϕ)w e v ds F δ ( F δ ( ε 2 u e 2 L 2 ( ) + we ϕ 2 L 2 ( ) + c2 ϕ 2 ) ) 1/2 L 2 ( ) v (ε 1/2 + 1/2 + c 1 2 ) u H2 () v. Combing te inequalities (3.36) (3.39) proves te result of te lemma. (3.39) 3.5. Consistency estimate. e consistency error of te finite element metod (2.12) is due to geometric errors resulting from te approximation of by. o estimate tis geometric errors we need a few additional definitions and results, wic can be found in, for example, [12]. For x define P (x) = I n (x)n(x) /(n (x) n(x)). One can represent te surface gradient of u H 1 () in terms of u e as follows u(p(x)) = (I d(x)h(x)) 1 P (x) u e (x) a.e. x. (3.40) Due to (3.9), we get u v ds = A u e v e ds for all v H 1 (), (3.41) wit A (x) = µ (x) P (x)(i d(x)h(x)) 2 P (x). From w n = 0 on and w e (x) = w(p(x)), n(x) = n(p(x)) it follows tat n(x) w e (x) = 0 and tus w(p(x)) = P (x)w e (x) olds. Using tis, we get te relation (w u)v ds = (B w e u e )v e ds, (3.42)

14 14 M. A. OLSHANSKII, A. REUSKEN, AND X. XU wit B = µ (x) P (I dh) 1 P. In te proof we use te lifting procedure given by v l (p(x)) := v (x) for x. (3.43) It is easy to see tat v l H1 (). e following lemma estimates te consistency error of te finite element metod (2.12). Lemma 3.9. Let u H 2 () be te solution of (2.5), ten we ave sup v V f (v ) a (u e, v ) v ( c c + ε ) ( u H2 () + f L2 ()). (3.44) Proof. e residual is decomposed as f (v ) a (u e, v ) = f (v ) f(v l ) + a(u, v l ) a (u e, v ). (3.45) e following olds: f(v) l = fv l ds = µ f e v ds, a(u, v) l = ε u v l ds + (w u)vds l + cuv l ds = ε u v l ds + 1 (w u)v l (w v l 2 )u ds + = ε A u e v ds + 1 (B w e u e )v ds 2 1 (B w e v )u e ds + µ cu e v ds. 2 Substituting tese relations into (3.45) and using (2.13), (2.14) results in cuv l ds f (v ) a (u e, v ) = (1 µ )f e v ds + ε (A P ) u e v ds + 1 ((B P )w e u e )v ds 1 ((B P )w e v )u e ds (µ 1)cu e v ds + ( δ f e + ε u e w e u e cu e) w e v ds F =: I 1 + I 2 + I 3 + I 4 + I 5 + Π 1. (3.46) We estimate te I i terms separately. Applying (3.10) and (3.29) we get I 1 2 f e L 2 ( ) v L 2 ( ) 2 c + ε f L 2 () v, (3.47) I 5 2 c 1 2 u e L2 ( ) cv L2 ( ) 2 c 1 2 u L2 () v. (3.48) One can sow, cf. (3.43) in [7], te bound P A = P (I A ) 2.

15 Using tis we obtain A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 15 I 2 ε 2 u e L2 ( ) v L2 ( ) ε 1/2 2 u e H2 () v. (3.49) Since (I dh) 1 = I + O( 2 ), we also estimate is yields B P 2 + A P 2. I 3 2 u e L 2 ( ) v L 2 ( ) 2 c + ε u H 2 () v. (3.50) o estimate I 4 we use te definition (2.15) of δ. If Pe 1 ten ε 1 2 w e L ( ) 2 w e L ( ) 2 S olds. If Pe > 1 ten δ 1 2 max(c 1 2, δ w e L ( ) 2 S ) olds. Using te assumption tat te outer triangulation is quasi-uniform we get 1 S 1. us we obtain and min{ε 1 2 w e L ( ), δ 1 2 } max(c 1 2, 1 2 ) c , I 4 max B P (ε 1 2 v L x 2 ( ) + δ 1 2 w e v L 2 ( )) F min{ε 1 2 w e L ( ), δ 1 2 } ue L2 ( ) (c ) v u L2 (). (3.51) Now we estimate Π 1. Using te equation ε u + w u + cu = f on we get Π 1 = ( f p + ε u e w e u e cu e) w e v ds F δ = F δ + F δ + F δ ( ε (u p) + ε u e) w e v ds ( we (u p) w e u e) w e v ds ( cu p cu e ) w e v ds =: Π Π Π 3 1. (3.52) From u e = u p it follows tat Π1 3 = 0 olds. For Π1 2 we obtain, using (3.40) and µ 1 B P µ 1 µ 1 B + B P 2, Π1 2 = ( (µ 1 B P )w e u e) w e v ds F δ 2 ( F δ w e u e 2 L 2 ( ) ) 1 2 ( F δ w e v 2 L 2 ( ) 5 2 u H2 () v. (3.53) ) 1 2

16 16 M. A. OLSHANSKII, A. REUSKEN, AND X. XU Since δ ε 2, we get Π 1 2 ( ) 1 2 ( ε 2 δ (u p) u e 2 L 2 ( ) δ w e v 2 L 2 ( ) F F ( ) 1 2 ε 1 2 (u p) u e 2 L 2 ( ) v. (3.54) F We finally consider te term between brackets in (3.54). Using te identity div f = tr( f) and n u e (p(x)) = 0 we obtain for x, wit 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) From te same arguments it follows tat u e (x) = tr(p P u e (x)) = tr(p 2 u e (x)p ) (3.56) olds. Using (3.14) and d(x) 2, P P, H 1, H 1 we obtain P 2 u e (x)p = P 2 u e (p(x))p + R, us, using (3.55) and (3.56), we get R ( 2 u e (p(x)) + u e (p(x)) ). u(p(x)) u e (x) tr(p 2 u e (p(x))p P 2 u e (x)p ) and combining tis wit (3.54) yields Π 1 2 ε 1 2 ( F (u p) u e 2 L 2 ( ) ( 2 u e (p(x)) + u e (p(x)) ), ) 1 2 ) 1 2 v ε u H 2 () v. Combining tis wit te results (3.46)-(3.51) and (3.53) proves te lemma Main teorem. Now we combine te results derived in te previous sections to prove te main result of te paper. eorem Let Assumption 3.1 be satisfied. We consider problem parameters ε and c as in (3.28). Assume tat te solution u of (2.5) as regularity u H 2 (). Let u be te discrete solution of te SUPG finite element metod (2.12). en te following olds: u e u ( 1/2 + ε 1/2 + c ε + c + 3 ε ) ( u H2 () + f L2 ()). (3.57) Proof. e triangle inequality yields u e u u e (I u e ) + (I u e ) u. (3.58)

17 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 17 e second term in te upper bound can be estimated using Lemmas 3.2, 3.8, 3.9: 1 2 (I u e ) u 2 a ((I u e ) u, (I u e ) u ) = a ((I u e ) u e, (I u e ) u ) + a (u e u, (I u e ) u ) ( 1/2 + ε 1/2 + c ε + c + 3 ε ) u H2 () (I u e ) u + a (u e, (I u e ) u ) f ((I u e ) u ) ( 1/2 + ε 1/2 + c is results in ε + c + 3 ε )( u H2 () + f L2 ()) (I u e ) u. (I u e ) u ( 1/2 +ε 1/2 +c ) ( u H2 () + f L2 ()). (3.59) ε + c ε e error estimate (3.57) follows from (3.22) and (3.59) Furter discussion. We comment on some aspects related to te main teorem. Concerning te analysis we note tat te norm, wic measures te error on te left-and side of (3.57), is te standard SUPG norm as found in standard analyses of planar streamline-diffusion finite element metods. e analysis in tis paper contains new ingredients compared to te planar case. o control te geometric errors (approximation of by ) we derived a consistency error bound in Lemma 3.9. o derive a continuity result (Lemma 3.8), as in te planar case, we apply partial integration to te term (w e ϕ)v ds, cf. (3.37). However, different from te planar case, tis results in jumps across te edges E E wic ave to be controlled, cf. (3.38). For tis te new results in te Lemmas 3.5 and 3.6 are derived. ese jump terms across te edges cause te term 4 ε in te error bound in (3.57). Consider te error reduction factor 3/2 + ε 1/2 + c ε+c + 4 ε on te rigtand side of (3.57). e first tree terms of it are typical for te error analysis of planar SUPG finite element metods for P 1 elements. In te standard literature for te planar case, cf. [2], one typically only considers te case c > 0. Our analysis also applies to te case c = 0, cf. (3.28). Furtermore, te estimates are uniform w.r.t. te size of te parameter c. For a fixed c > 0 and ε te first four terms can be estimated by 3/2, a bound similar to te standard one for te planar case. e only suboptimal term is te last one, wic is caused by (our analysis of) te jump terms. Note, owever, tat 4 ε 3/2 if 5 ε olds, wic is a very mild condition. e norm provides a robust control of streamline derivatives of te solution. Cross-wind oscillations, owever, are not completely suppressed. It is well known tat nonlinear stabilization metods can be used to get control over cross-wind derivatives as well. Extending suc metods to surface PDEs is not witin te scope of te present paper. e error estimates in tis paper are in terms of te maximum mes size over tetraedra in ω, denoted by. In practice, te stabilization parameter δ is based on te local Peclet number and te stabilization is switced off or reduced in te regions, were te mes is sufficiently fine. o prove error estimates accounting for local smootness of te solution u and te local mes size (as available for planar SUPG

18 18 M. A. OLSHANSKII, A. REUSKEN, AND X. XU FE metod), 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, suc local estimates are not immediately available. e extension of our analysis to tis non-quasi-uniform case is left for future researc. Finally, we remark on te case of a varying reaction term coefficient c. If te coefficient c in te tird term of (2.4) varies, te above analysis is valid wit minor modifications. We briefly explain tese modifications. e stabilization parameter δ sould be based on elementwise values c = max x c(x). For te well-possedness of (2.4), it is sufficient to assume c to be strictly positive on a subset of wit positive measure: A := meas{x : c(x) c 0 } > 0, wit some c 0 > 0. If tis is satisfied, te Friedric s type inequality [11] (see, also Lemma 3.1 in [7]) v 2 L 2 () C F ( v 2 L 2 () + cv 2 L 2 () ) for all v V olds, wit a constant C F depending on c 0 and A. Using tis, all arguments in te analysis can be generalized to te case of a varying c(x) wit obvious modifications. Wit c min := ess inf x c(x) and c max := ess sup x c(x), te final error estimate takes te form u e u ( 1/2 + ε 1/2 + c 1 2 max + ε + cmin + 3 ε ) ( u H2 () + f L2 ()). 4. Numerical experiments. In tis section we sow results of a few numerical experiments wic illustrate te performance of te metod. Example 1. e stationary problem (2.4) is solved on te unit spere, wit te velocity field w(x) = ( x 2 1 x 2 3, x 1 1 x 2 3, 0), wic is tangential to te spere. We set ε = 10 6, c 1 and consider te solution u(x) = x ( ) 1x 2 π arctan x3. ε Note tat u as a sarp internal layer along te equator of te spere. e corresponding rigt-and 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 te standard (unstabilized) finite element metod in (2.11) and te stabilized metod (2.12). A sequence of meses was obtained by te gradual refinement of te outer triangulation. e induced surface finite element spaces ave dimensions N = 448, 1864, 7552, e resulting algebraic systems are solved by a direct sparse solver. Finite element errors are computed outside te layer: e variation of te quantities err L 2 = u u L2 (D), err H 1 = u u H1 (D), and err L inf = u u L (D), wit D = {x : x 3 > 0.3}, are sown in Figure 4.1. e results clearly sow tat te stabilized metod performs muc better tan te

19 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES L 2 errors slope 1 stablized FEM standard FEM slope 1/2 log 10 (err H1 ) H 1 errors stablized FEM standard FEM slope 1/2 log 10 (err inf ) slope 1 L errors stablized FEM standard FEM log 10 (N) log 10 (N) log 10 (N) Fig Discretization errors for Example 1 standard one. e results for te stabilized metod indicate a O( 2 ) convergence in te L 2 (D)-norm and L (D)-norm. In te H 1 (D)-norm a first order convergence is observed. Note tat te analysis predicts (only) O( 3 2 ) convergence order in te (global) L 2 -norm. Figure 4.2 sows te computed solutions wit te two metods. Since te layer is unresolved, te finite element metod (2.11) produces globally oscillating solution. e stabilized metod gives a muc better approximation, altoug te layer is sligtly smeared, as is typical for te SUPG metod. (a) (b) Fig Example 1: solutions using te stabilized metod and te standard metod (2.11). Example 2. Now we consider te stationary problem (2.4) wit c 0. e problem is posed on te unit spere, wit tis same velocity field w as in Example 1. We set ε = 10 6, and consider te solution u(x) = x ( ) 1x 2 π arctan x3. ε e corresponding rigt-and 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 tat for c = 0 one looses explicit control of te L 2 norm in. us we

20 20 M. A. OLSHANSKII, A. REUSKEN, AND X. XU consider te streamline diffusion error : err SD = w (u u ) L2 (D). Results for tis error quantity and for err L inf = u u L (D) are sown in Figure 4.3. We observe a O() beavior for te streamline diffusion error, wic is consistent wit our teoretical analysis. e L -norm of te error also sows a first order of convergence streamline diffusion errors stablized FEM standard FEM L errors stablized FEM standard FEM 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 Example 2 Example 3. Finally we sow ow tis stabilization can be applied to a time dependent problem and illustrate its stabilizing effect. We consider a non-stationary problem (2.3) posed on te 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 te advection field 1 w(x) = ( x x x 2 2, x 1, 0), 2 wic satisfies w n = 0. e initial condition is u 0 (x) = x ( ) 1x 2 π arctan x3. ε e function u 0 possesses an internal layer, as sown in Figure 4.4(a). e stabilized spatial discretization of (2.3) reads: m( t u, v ) + â (u, v ) = 0, (4.2) wit m( t u, v) := t uv ds + δ t u(w e v) ds, F â (u, v) :=ε u v ds + 1 [ ] (w e u)v ds (w e v)u ds 2 + ( ε u + w e u)w e v ds. F δ

21 A FEM FOR ADVECION-DIFFUSION EQUAIONS ON SURFACES 21 Note tat â (, ) is te same as a (, ) in (2.13) wit c 0. e resulting system of ordinary differential equations is discretized in time by te Crank-Nicolson sceme. For ε = 0 te exact solution is te transport of u 0 (x) by a rotation around te x 3 axis. us te inner layer remains te same for all t > 0. For ε = 10 6, te exact solution is similar, unless t is large enoug for dissipation to play a noticeable role. e space V is constructed in te same way as in te previous examples. e spatial discretization as 5638 degrees of freedom. e fully discrete problem is obtained by combining te SUPG metod in (4.2) and te Crank-Nicolson metod wit time step δt = 0.1. e evolution of te solution is illustrated in Figure 4.4 demonstrating a smootly rotated pattern. (a) (b) (c) (d) Fig Example 3: solutions for t = 0, 0.6, 1.2, 1.8 using te SUPG stabilized FEM. We repeated tis experiment wit δ = 0 in te bilinear forms m(, ) and â (, ) in (4.2), i.e. te metod witout stabilization. As expected, we ten obtain (on te same grid) muc less smoot discrete solutions(figure 4.5). Acknowledgments. is work as been supported in part by te DFG troug grant RE1461/4-1 and by te Russian Foundation for Basic Researc troug grants , We tank J. Grande for is support wit te implementation of te metods and C. Lerenfeld for elpful discussions on te skew-symetric variational formula. REFERENCES [1] S. Gross and A. Reusken. Numerical Metods for wo-pase Incompressible Flows, volume 40 of Springer Series in Computational Matematics. Springer, Heidelberg, 2011.

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