A trace finite element method for a class of coupled bulk-interface transport problems

Transcription

1 Numerical Analysis and Scientific Computing Preprint Seria A trace finite element metod for a class of coupled bulk-interface transport problems S. Gross M.A. Olsanskii A. Reusken Preprint #28 Department of Matematics University of Houston June 2014

2 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS SVEN GROSS, MAXIM A. OLSHANSKII, AND ARNOLD REUSKEN Abstract. In tis paper we study a system of advection-diffusion equations in a bulk domain coupled to an advection-diffusion equation on an embedded surface. Suc systems of coupled partial differential equations arise in, for example, te modeling of transport and diffusion of surfactants in two-pase flows. Te model considered ere accounts for adsorption-desorption of te surfactants at a sarp interface between two fluids and teir transport and diffusion in bot fluid pases and along te interface. Te paper gives a well-posedness analysis for te system of bulk-surface equations and introduces a finite element metod for its numerical solution. Te finite element metod is unfitted, i.e., te mes is not aligned to te interface. Te metod is based on taking traces of a standard finite element space bot on te bulk domains and te embedded surface. Te numerical approac allows an implicit definition of te surface as te zero level of a level-set function. Optimal order error estimates are proved for te finite element metod bot in te bulk-surface energy norm and te L 2 -norm. Te analysis is not restricted to linear finite elements and a piecewise planar reconstruction of te surface, but also covers te discretization wit iger order elements and a iger order surface reconstruction. 1. Introduction. Coupled bulk-surface or bulk-interface partial differential equations arise in many applications, e.g., in multipase fluid dynamics [17] and biological applications [2]. In tis paper, we consider a coupled bulk-interface advectiondiffusion problem. Te problem arises in models describing te beavior of soluble surface active agents (surfactants) tat are adsorbed at liquid-liquid interfaces. For a discussion of pysical penomena related to soluble surfactants in two-pase incompressible flows we refer to te literature, e.g., [17, 26, 7, 28]. Systems of partial differential equations tat couple bulk domain effects wit interface (or surface) effects pose callenges bot for te matematical analysis of equations and te development and error analysis of numerical metods. Tese callenges grow if penomena occur at different pysical scales, te coupling is nonlinear or te interface is evolving in time. To our knowledge, te analysis of numerical metods for coupled bulk-surface (convection-)diffusion as been addressed in te literature only very recently. In fact, problems related to te one studied in tis paper ave been considered only in [4, 13]. In tese references finite element metods for coupled bulk-surface partial differential equations are proposed and analyzed. In [4, 13] a stationary diffusion problem on a bulk domain is linearly coupled wit a stationary diffusion equation on te boundary of tis domain. A key difference between te metods in [4] and [13] is tat in te latter boundary fitted finite elements are used, wereas in te former unfitted finite elements are applied. Bot papers include error analyses of tese metods. In te recent paper [5] a similar coupled surface-bulk system is treated wit a different approac, based on te immersed boundary metod. In tat paper an evolving surface is considered, but only spatially two-dimensional problems are treated and no teoretical error analysis is given. In te present paper, as in [4, 13] we restrict to stationary problems and a linear Partially supported by NSF troug te Division of Matematical Sciences grant Institut für Geometrie und Praktisce Matematik, RWTH-Aacen University, D Aacen, Germany (gross@igpm.rwt-aacen.de) Department of Matematics, University of Houston, Houston, Texas (molsan@mat.u.edu). Institut für Geometrie und Praktisce Matematik, RWTH-Aacen University, D Aacen, Germany (reusken@igpm.rwt-aacen.de). 1

3 coupling. Te results obtained are a starting point for researc on oter classes of problems, e.g., wit an evolving interface, cf. te discussion in section 10. Te two main new contributions of tis paper are te following. Firstly, te class of problems considered is significantly different from te one treated in [4, 13]. We study a problem wit two bulk domains separated by a sarp interface, instead of a bulk-surface coupling. Furtermore, te partial differential equations are not pure diffusion equations, but convection-diffusion equations. Te latter model te transport by advection and diffusion of surfactants. We will briefly address some basic modeling aspects, e.g. related to adsorption (Henry and Langmuir laws), of tese coupled equations. Te first main new result is te well-posedness of a weak formulation of tis coupled system. We introduce suitable function spaces and an appropriate weak formulation of te problem. We derive a Poincare type inequality in a bulk-interface product Sobolev space and sow an inf-sup stability result for te bilinear form of te weak formulation. Tis ten leads to te well-posedness result. Te second main new contribution is te error analysis of a finite element metod. We consider an unfitted finite element metod, as in [4]. Suc an unfitted approac is particularly attractive for problems wit an evolving interface, wic will be studied in a follow-up paper. Bot interface and bulk finite element spaces are trace spaces of globally defined continuous finite element functions wit respect to a regular simplicial triangulation of te wole domain. For bulk problems, suc finite element tecniques ave been extensively studied in te literature on cut finite element metods or XFEM, cf. e.g. [18, 19, 3]. For PDEs posed on surfaces, te trace finite element metod was introduced and studied in [22, 24, 20, 10]. In te metod tat we propose, te smoot interface is approximated by a piecewise smoot one, caracterized as te zero level set of a finite element level set function. Tis introduces a geometric error in te metod. Te approac allows meses tat do not fit to tis (approximate) interface and admits implicitly defined interfaces. Te finite element formulation is sown to be well-posed. We present an error analysis tat is general in te sense tat finite element polynomials of arbitrary degree are allowed (in [4] only linear finite elements are treated) and tat te accuracy of te interface approximation can be varied. Te error analysis is rater tecnical and we aimed at a clear exposition by subdividing te analysis into several steps: te construction of a bijective mapping between te continuous bulk domains and teir numerical approximations, te definition and analysis of extensions of functions off te interface and outside te bulk domains, and te analysis of consistency terms in an approximate Galerkin ortogonality property. Tis leads to an optimal order bound for te discretization error in te energy norm of te product space. Finally, by using a suitable adjoint problem, we derive an optimal order error bound in te L 2 product norm. Results of numerical experiments are included tat illustrate te convergence beavior of te finite element metod. A point tat we do not address in tis paper is te stabilization of te discretization metod wit respect to te conditioning of te stiffness matrix. From te literature it is known tat te trace finite element approac results in a poor conditioning of te stiffness matrix. Te tecnique presented in [3, 4] can be used (at least for linear finite elements) to obtain a stiffness matrix wit conditioning properties similar to tose of a standard finite element metod. We tink tat tis tecnique is applicable also to te metod presented in tis paper, but decided not to include it, since te additional stabilization terms would lead to a furter increase of tecnicalities in te analysis. 2. Matematical model. In tis section we explain te pysical background of te coupled bulk-interface model tat we treat in tis paper. Consider a two-pase 2

4 incompressible flow system in wic two fluids occupy subdomains Ω i (t), i = 1, 2, of a given domain Ω R 3. Te corresponding velocity field of te fluids is denoted by w(x, t), x Ω, t [0, t 1 ]. For convenience, we assume tat Ω 1 (t) is simply connected and strictly contained in Ω (e.g., a rising droplet). Te outward pointing normal from Ω 1 into Ω 2 is denoted by n. Te sarp interface between te two fluids is denoted by Γ(t) and is transported by te local flow field. Te fluids are assumed to be immiscible, and ence V Γ = w n, were V Γ denotes te normal velocity of te interface. Te standard model for te fluid dynamics in suc a system consists of te Navier-Stokes equations, combined wit suitable coupling conditions at te interface. In te rest of tis paper, we assume tat te velocity field as smootness w(, t) [H 1 (Ω) H 1 (Γ)] 3 and tat w is given, i.e., we do not consider a two-way coupling between surfactant transport and fluid dynamics. Te fluid is assumed incompressible: div w = 0 in Ω. (2.1) Consider a surfactant tat is soluble in bot pases and can be adsorbed and desorbed at te interface. Te surfactant volume concentration (i.e., te one in te bulk pases) is denoted by u, u i = u Ωi, i = 1, 2. Te surfactant area concentration on Γ is denoted by v. Cange of te surfactant concentration appens due to convection by te velocity field w, diffusive fluxes in Ω i, a diffusive flux on Γ and fluxes coming from adsorption and desorption. Te net flux (per surface area) due to adsorption/desorption between Ω i and Γ is denoted by j i,a j i,d. Te total net flux is j a j d = 2 (j i,a j i,d ). Mass conservation in a control volume (transported by te flow field) tat is strictly contained in Ω i results in te bulk convection-diffusion equation u t + w u D i u = 0 in Ω i = Ω i (t), i = 1, 2. (2.2) Mass conservation in a control area (transported by te flow field) tat is completely contained in Γ results in te surface convection-diffusion equation (cf. [17]): v + (div Γ w)v D Γ Γ v = j a j d on Γ = Γ(t), (2.3) were v = v t + w v denotes te material derivative and Γ, div Γ te Laplace- Beltrami and surface divergence operators, respectively. We assume tat transport of surfactant between te two pases can only occur via adsorption/desorption. Due to V Γ = w n, te mass flux troug Γ equals te diffusive mass flux. Hence, mass conservation in a control volume (transported by te flow field) tat lies in Ω i and wit part of its boundary on Γ results in te mass balance equations ( 1) i D i n u i = j i,a j i,d, i = 1, 2. (2.4) Te sign factor ( 1) i accounts for te fact tat te normal n is outward pointing from Ω 1 into Ω 2. Summing tese relations over i = 1, 2, yields j a j d = [Dn u] Γ, were [w] Γ denotes te jump of w across Γ. To close te system of equations, we need constitutive equations for modeling te adsorption/desorption. A standard model, cf. [26], is as follows: j i,a j i,d = k i,a g i (v)u i k i,d f i (v), on Γ, (2.5) 3

5 wit k i,a, k i,d positive adsorption and desorption coefficients tat describe te kinetics. We consider two fluids aving similar adsorption/desorption beavior in te sense tat te coefficients k i,a, k i,d may depend on i, but g i (v) = g(v), f i (v) = f(v) for i = 1, 2. Basic coices for g, f are te following: g(v) = 1, f(v) = v (Henry) (2.6) g(v) = 1 v v, f(v) = v (Langmuir), (2.7) were v is a constant tat quantifies te maximal concentration on Γ. Furter options are given in [26]. Note tat in te Langmuir case we ave a nonlinearity due to te term vu i. Combining (2.2), (2.3), (2.4) and (2.5) gives a closed model. For te matematical analysis it is convenient to reformulate tese equations in dimensionless variables. Let L, W be appropriately defined lengt and velocity scales and T = L/W te corresponding time scale. Furtermore, U and V are typical reference volume and area concentrations. Te equations above can be reformulated in te dimensionless variables x = x/l, t = t/t, ũ i = u i /U, ṽ = v/v, w = w/w. Tis results in te following system of coupled bulk-interface convection-diffusion equations, were we use te notation x, t, u i, v, w also for te transformed variables: wit ν i = D i LW, u t + w u ν i u = 0 in Ω i (t), i = 1, 2, v + (div Γ w)v ν Γ Γ v = K[νn u] Γ on Γ(t), ( 1) i ν i n u i = k i,a g(v)u i k i,d v on Γ(t), i = 1, 2, ν Γ = D Γ LW, K = LU V, ki,a = T L k i,a, ki,d = T K k i,d, (2.8) and g(v) = 1 (Henry) or g(v) = 1 V v v (Langmuir). Tis model as to be complemented by suitable initial conditions for u, v and boundary conditions on Ω for u. Te resulting model is often used in te literature for describing surfactant beavior, e.g. [12, 28, 11, 5]. Te coefficients k i,a, k i,d are te dimensionless adsorption and desorption coefficients. Remark 1. Sometimes, in te literature te Robin type interface conditions ( 1) i ν i n u i = k i,a g(v)u i k i,d v in (2.8) are replaced by (simpler) Diriclet type conditions. In case of instantaneous adsorption and desorption one may assume k i,a ν i, k i,d ν i and te Robin interface conditions are approximated by k i,a g(v)u i = k i,d v, i = 1, 2. From a matematical point of view, te problem (2.8) is callenging, because convection-diffusion equations in te moving bulk pase Ω i (t) are coupled wit a convection-diffusion equation on te moving interface Γ(t). As far as we know, for tis model tere are no rigorous results on well-posedness known in te literature. 3. Simplified model. As a first step in te analysis of te problem (2.8) we consider a simplified model. We restrict to g(v) = v (Henry law), consider te stationary case and make some (reasonable) assumptions on te range of te adsorption and desorption parameters k i,a, k i,d. In te remainder we assume tat Ω i and Γ do not depend on t (e.g., an equilibrium motion of a rising droplet in a suitable frame of reference). Since te interface is passively advected by te velocity field w, tis assumption leads to te constraint w n = 0 on Γ. (3.1) 4

6 We also assume div Γ w = 0 so tat te term (div Γ w)v in te surface convectiondiffusion equation vanises. Furtermore, for te spatial part of te material derivative v we ave w v = w Γ v. We let te normal part of w to vanis on exterior boundary: w n Ω = 0 on Ω, (3.2) were n Ω denotes te outward pointing normal on Ω. From (3.2) it follows tat tere is no convective mass flux across Ω. We also assume no diffusive mass flux across Ω, i.e. te omogeneous Neumann boundary condition n Ω u 2 = 0 on Ω. Restricting te model (2.8) to an equilibrium state, we obtain te following stationary problem: ν i u i + w u i = f i in Ω i, i = 1, 2, ν Γ Γ v + w Γ v + K[νn u] Γ = g on Γ, ( 1) i ν i n u i = k i,a u i k i,d v on Γ, i = 1, 2, n Ω u 2 = 0 on Ω. (3.3) In tis model, we allow source terms g L 2 (Γ) and f i L 2 (Ω i ). Using partial integration over Ω i, i = 1, 2, and over Γ, one cecks tat tese source terms ave to satisfy te consistency condition K ( f 1 dx + f 2 dx ) + g ds = 0. (3.4) Ω 1 Ω 2 Γ A simplified version of tis model, namely wit only one bulk domain Ω 1 and wit w = 0 (only diffusion) as recently been analyzed in [13]. From pysics it is known tat for surfactants almost always te desorption rates are (muc) smaller tan te adsorption rates. Terefore, it is reasonable to assume k i,d c k i,a wit a small constant c. To simplify te presentation, we assume k i,d k 1,a + k 2,a for i = 1, 2. For te adsorption rates k i,a we exclude te singularperturbed cases k i,a 0 and k i,a. Summarizing, we consider te parameter ranges k i,a [k min, k max ], ki,d [0, k 1,a + k 2,a ], (3.5) wit fixed generic constants k min > 0, k max. Note tat unlike in previous studies we allow k i,d = 0 (i.e., only adsorption). Finally, due to te restriction on te adsorption parameter k i,a given in (3.5) we can use te following transformation to reduce te number of parameters of te model (3.3): u i k i,a u i, k 1 v ( k 1,a + k 2,a )v, ν i i,a ν i, ν Γ ν Γ /( k 1,a + k 2,a ), 1 w w i = k i,a w in Ω i, w w/( k 1,a + k 2,a ) on Γ. (3.6) Note tat after tis transformation w may be discontinuous across Γ, and so w H 1 (Ω) does not necessarily old. In eac subdomain and on Γ, owever, w is regular: w H 1 (Ω i ) 3, i = 1, 2, and w H 1 (Γ) 3. Te same notation will be used for te transformed variables. Tus te model tat we study in te remainder of te paper 5

7 is te following one: ν i u i + w u i = f i in Ω i, i = 1, 2, ν Γ Γ v + w Γ v + K[νn u] Γ = g on Γ, ( 1) i ν i n u i = u i q i v on Γ, i = 1, 2, n Ω u 2 = 0 on Ω, (3.7) wit q i := k i,d k 1,a + k 2,a [0, 1]. Te data f i and g are assumed to satisfy te consistency condition (3.4). Recall tat K = LU V > 0, cf. (2.8), is a fixed (scaling) constant. 4. Analysis of well-posedness. In tis section we derive a suitable weak formulation of te problem (3.7) and prove well-posedness of tis weak formulation. We first introduce some notations. For u H 1 (Ω 1 Ω 2 ) we also write u = (u 1, u 2 ) wit u i = u Ωi H 1 (Ω i ). Furtermore: (f, g) ω := fg, were ω is any of {Ω, Ω i, Γ}, ω ( u, w) Ω1 Ω 2 := u i w i dx, u, w H 1 (Ω 1 Ω 2 ), Ω i,2 u 2 1,Ω 1 Ω 2 := u 1 2 H 1 (Ω 1) + u 2 2 H 1 (Ω 2) = u 2 Ω + u 2 Ω 1 Ω 2. We need a suitable gauge condition. In te original dimensional variables a natural condition is conservation of total mass, i.e. (u 1, 1) Ω1 + (u 2, 1) Ω2 + (v, 1) Γ = m 0, wit m 0 > 0 te initial total mass. Due to te transformation of variables and wit an additional constant sift tis condition is transformed to K k 1,a 1 (u 1, 1) Ω1 + K k 2,a 1 (u 2, 1) Ω1 + ( k 1,a + k 2,a ) 1 (v, 1) Γ = 0 for te variables used in (3.7). Hence, we obtain te natural gauge condition K(1 + r)(u 1, 1) Ω1 + K(1 + 1 r )(u 2, 1) Ω2 + (v, 1) Γ = 0, r := k 2,a k 1,a. (4.1) Define te product spaces V = H 1 (Ω 1 Ω 2 ) H 1 (Γ), (u, v) V := ( u 2 1,Ω 1 Ω 2 + v 2 ) 1 2 1,Γ, Ṽ = { (u, v) V (u, v) satisfies (4.1) }. To obtain te weak formulation, we multiply te bulk and surface equation in (3.7) by test functions from V, integrate by parts and use interface and boundary conditions. Te resulting weak formulation reads: Find (u, v) Ṽ suc tat for all (η, ζ) V: a((u, v); (η, ζ)) = (f 1, η 1 ) Ω1 + (f 2, η 2 ) Ω2 + (g, ζ) Γ, (4.2) a((u, v); (η, ζ)) := (ν u, η) Ω1 Ω 2 + (w u, η) Ω1 Ω 2 + ν Γ ( Γ v, Γ ζ) Γ 2 + (w Γ v, ζ) Γ + (u i q i v, η i Kζ) Γ. 6

8 For te furter analysis, we note tat bot w-dependent parts of te bilinear form in (4.2) are skew-symmetric: (w u i, η i ) Ωi = (w η i, u i ) Ωi, i = 1, 2, (w Γ v, ζ) Γ = (w Γ ζ, v) Γ. (4.3) To verify te first equality in (4.3), one integrates by parts over eac subdomain Ω i : (w u 1, η 1 ) Ω1 = (w η 1, u 1 ) Ω1 ((div w)η 1, u 1 ) Ω1 + ((n w)η 1, u 1 ) Γ, (w u 2, η 2 ) Ω2 = (w η 2, u 2 ) Ω2 ((div w)η 2, u 2 ) Ω2 ((n w)η 2, u 2 ) Γ + ((n Ω w)η 2, u 2 ) Ω2 Ω. All terms wit div w, n w or n Ω w vanis due to (2.1), (3.1) and (3.2). Te variational formulation in (4.2) is te basis for te finite element metod introduced in section 6. For te analysis of well-posedness, it is convenient to introduce an equivalent formulation were te test space V is replaced by a smaller one, in wic a suitable gauge condition is used. For tis we define, for α = (α 1, α 2 ) wit α i 0, te space V α := { (u, v) V α 1 (u 1, 1) Ω1 + α 2 (u 2, 1) Ω1 + (v, 1) Γ = 0 }. Note tat Ṽ = V α for α = (K(1 + r), K(1 + 1 r )), cf. (4.1). Te data f 1, f 2, g, satisfy te consistency property (3.4). From tis and (4.3) it follows tat if a pair of trial and test functions ((u, v); (η, ζ)) satisfies (4.2) ten ((u, v); (η, ζ) + γ(k, 1)) also satisfies (4.2) for arbitrary γ R. Now let an arbitrary α = (α 1, α 2 ) be given. For every (η, ζ) V tere exists γ R and ( η, ζ) V α suc tat (η, ζ) = ( η, ζ) + γ(k, 1) olds. From tis it follows tat (4.2) is equivalent to te following problem: Find (u, v) Ṽ suc tat for all (η, ζ) V α : a((u, v); (η, ζ)) = (f 1, η 1 ) Ω1 + (f 2, η 2 ) Ω2 + (g, ζ) Γ. (4.4) For tis weak formulation we sall analyze well-posedness. On H 1 (Ω i ) and H 1 (Γ) te following Poincare-Friedric s inequalities old: u i 2 Ω i c( u i 2 Ω i + (u i, 1) 2 Ω i ) for all u i H 1 (Ω i ), (4.5) u i 2 Ω i c( u i 2 Ω i + u i 2 Γ) for all u i H 1 (Ω i ), (4.6) v 2 Γ c( Γ v 2 Γ + (v, 1) 2 Γ) for all v H 1 (Γ). (4.7) For te analysis of stability of te weak formulation we need te following Poincare type inequality in te space V. Lemma 4.1. Let r i, σ i [0, ), i = 1, 2. Tere exists C P (r 1, r 2, σ 1, σ 2 ) > 0 suc tat for all (u, v) V, te following inequality olds: (u, v) V C P ( u Ω1 Ω 2 + Γ v Γ 2 + r 1 (u 1, 1) Ω1 + r 2 (u 2, 1) Ω2 + (v, 1) Γ + (u i σ i v, 1) Γ ). (4.8) Proof. Te result follows from te Petree-Tartar Lemma (cf., [14]). For convenience, we recall te lemma: Let X, Y, Z be Banac spaces, A L(X, Y ) injective, T L(X, Z) compact and assume x X c ( ) Ax Y + T x Z for all x X. (4.9) 7

9 Ten tere exists a constant c suc tat x X c Ax Y for all x X (4.10) olds. We take X = H 1 (Ω 1 ) H 1 (Ω 2 ) H 1 (Γ) wit te norm (u 1, u 2, v) X = ( u 1 2 1,Ω 1 + u 2 2 1,Ω 2 + v 2 1,Γ) 1 2. Furtermore, Y = L 2 (Ω 1 ) 3 L 2 (Ω 2 ) 3 L 2 (Γ) 3 R 3 wit standard norm and Z = L 2 (Ω 1 ) L 2 (Ω 2 ) L 2 (Γ) wit standard norm. We introduce te bilinear forms and define te linear operators l 0 (u 1, u 2, v) := r 1 (u 1, 1) Ω1 + r 2 (u 2, 1) Ω2 + (v, 1) Γ, l 1 (u 1, u 2, v) = (u 1 σ 1 v, 1) Γ, l 2 (u 1, u 2, v) = (u 2 σ 2 v, 1) Γ, A(u 1, u 2, v) = ( u 1, u 2, Γ v, l 0 (u 1, u 2, v), l 1 (u 1, u 2, v), l 2 (u 1, u 2, v)), T (u 1, u 2, v) = (u 1, u 2, v). Ten we ave A L(X, Y ) and A is injective. Te operator T L(X, Z) is compact. Tis follows from te compactness of te embeddings H 1 (Ω i ) L 2 (Ω), H 1 (Γ) L 2 (Γ). It is easy to ceck tat te inequality (4.9) is satisfied. Te Petree-Tartar Lemma implies ( u 2 1,Ω 1 Ω 2 + v 2 Γ ) 1 2 c A(u 1, u 2, v) Y and tus te estimate (4.8) olds. Te next teorem states an inf-sup stability estimate for te bilinear form in (4.4). Teorem 4.2. Tere exists C st > 0 suc tat for all q 1, q 2 [0, 1] and wit a suitable α te following olds: inf sup (u,v) Ṽ (η,ζ) V α a((u, v); (η, ζ)) (u, v) V (η, ζ) V C st. (4.11) Proof. Let (u, v) Ṽ be given. Note tat (u, v) satisfies te gauge condition (4.1). Witout loss of generality one can assume 0 q 2 q 1 1. We consider tree cases depending on values of tese parameters. We first consider q 1, q 2 [0, ε], wit ε > 0 specified below. We take η 1 = βu 1, η 2 = βu 2, wit β > 0, and ζ = v. Tis yields a((u, v); (η, ζ)) = ν 1 β u 1 2 Ω 1 + ν 2 β u 2 2 Ω 2 + ν Γ Γ v 2 Γ + β u 1 2 Γ + β u 2 2 Γ 2 (q i β + K)(u i, v) Γ + K(q 1 + q 2 ) v 2 Γ β ν u 2 Ω 1 Ω β u 1 2 Γ β u 2 2 Γ + ν Γ Γ v 2 Γ (εβ Kβ 1 2 ) 2 v 2 Γ c F β u 2 1,Ω 1 Ω 2 + ν Γ Γ v 2 Γ (εβ Kβ 1 2 ) 2 v 2 Γ, were in te last inequality we used (4.6). Te constant c F > 0 depends only on te Friedric s constant from (4.6) and te viscosity ν. From te gauge condition we get (v, 1) 2 2K 2( (1 + r) 2 (u 1, 1) 2 Ω 1 + (1 + 1 r )2 (u 2, 1) 2 Ω 2 ) c ( u1 2 Ω 1 + u 2 2 Ω 2 ) = c u 2 Ω. 8

10 Using tis and te Poincare s inequality in (4.7) we obtain a((u, v); (η, ζ)) c F β u 2 1,Ω 1 Ω 2 + ν Γ ( Γ v 2 Γ + (v, 1) 2) ĉ u 2 Ω (εβ Kβ 1 2 ) 2 v 2 Γ c F β u 2 1,Ω 1 Ω 2 + ĉ F v 2 1,Γ ĉ u 2 Ω (εβ Kβ 1 2 ) 2 v 2 Γ. Te constant ĉ depends only on ν Γ, r, K. Te constant ĉ F > 0 depends only on a Poincare s constant and ν Γ. We take β sufficiently large (depending only on c F, ĉ and K) and ε > 0 sufficiently small suc tat te tird term can be adsorbed in te first one and te last term can be adsorbed in te second one. Tus we get a((u, v); (η, ζ)) c (u, v) 2 V c (u, v) V (η, ζ) V, wic completes te proof of (4.11) for te first case. Now ε > 0 is fixed. In te second case we take q 1 ε, and q 2 [0, δ], wit a δ (0, ε] tat will be specified below. We take η 1 = u 1, η 2 = 0, ζ = K 1 q 1 v. Using te gauge condition and (4.8) wit u 2 = 0, σ 2 = 0, r 1 = K(1 + r), σ 1 = q 1, we get a((u, v); (η, ζ)) = ν 1 u 1 2 Ω 1 + q 1ν Γ K Γv 2 Γ + u 1 q 1 v 2 Γ q 1 (u 2, v) Γ + q 2 q 1 v 2 Γ ν 1 u 1 2 Ω 1 + εν Γ K Γv 2 Γ + u 1 q 1 v 2 Γ + K(1 + r)(u 1, 1) Ω1 + (v, 1) Γ 2 K 2 (1 + 1 r )2 (u 2, 1) 2 Ω 2 q 1 (u 2, v) Γ c F ( u1 2 1,Ω 1 + v 2 1,Γ) c u2 2 Ω 2 u 2 Γ v Γ 1 2 c F ( u1 2 1,Ω 1 + v 2 1,Γ) c ( u2 2 Ω 2 + u 2 2 Γ). (4.12) Te constant c F > 0 depends on Poincare s constant and on ε. We now take η 1 = 0, η 2 = βu 2, wit β > 0 and ζ = 0. Tis yields, cf. (4.6), a((u, v); (η, ζ)) = ν 2 β u 2 2 Ω 1 + β u 2 2 Γ βq 2 (u 2, v) Γ cβ u 2 2 1,Ω βδ2 v 2 Γ. Combining tis wit (4.12) and taking β sufficiently large suc tat te last term in (4.12) can be adsorbed, we obtain for η 1 = u 1, η 2 = βu 2, ζ = K 1 q 1 v: a((u, v); (η, ζ)) 1 2 c F ( u1 2 1,Ω 1 + v 2 1,Γ) + cβ u2 2 1,Ω βδ2 v 2 Γ. Now we take δ > 0 sufficiently small suc tat te last term can be adsorbed by te second one. Hence, a((u, v); (η, ζ)) c (u, v) 2 V c (u, v) V (η, ζ) V, wic completes te proof of te inf-sup property for te second case. Now δ > 0 is fixed. We consider te last case, namely q 1 δ and q 2 δ. Take η 1 = u 1, η 2 = q1 q 2 u 2, ζ = K 1 q 1 v. We ten get a((u, v); (η, ζ)) = ν 1 u 1 2 Ω 1 + ν 2 q 1 q 2 u 2 2 Ω 2 + ν Γq 1 K Γv 2 Γ + u 1 q 1 v 2 Γ + q 1 q 2 u 2 q 2 v 2 Γ c ( u 2 Ω 1 Ω 2 + Γ v 2 Γ u i q i v 2 Γ).

11 We use (4.8) wit r 1 = K(1 + r), r 2 = K(1 + 1 r ), wit r from (4.1), and σ i = q i. Tis yields a((u, v); (η, ζ)) c (u, v) 2 V c (u, v) V (η, ζ) V, wit a constant c > 0 tat depends on δ, but is independent of (u, v). In all tree cases, since (u, v) obeys te gauge condition (4.1), we get (η, ζ) V α, for suitable α = (α 1, α 2 ) wit α i > 0. Note tat te α used in Teorem 4.2 may depend on q i. In te remainder, for given problem parameters q i [0, 1], i = 1, 2, we take α as in Teorem 4.2 and use tis α in te weak formulation (4.2). For te analysis of a dual problem, we also need te stability of te adjoint bilinear form given in te next lemma. Lemma 4.3. Tere exists C st > 0 suc tat for all q 1, q 2 [0, 1] and wit α as in Teorem 4.2 te following olds: inf sup (η,ζ) V α (u,v) Ṽ a((u, v); (η, ζ)) (u, v) V (η, ζ) V C st. (4.13) Proof. Take (η, ζ) V α, (η, ζ) (0, 0). Te arguments of te proof of Teorem 4.2 sow tat for (η, ζ) V α tere exists (u, v) Ṽ suc tat a((u, v); (η, ζ)) C st (u, v) V (η, ζ) V olds, wit te same constant as (4.11). Finally, we give a result on continuity of te bilinear form. Lemma 4.4. Tere exists a constant c suc tat for all q 1, q 2 [0, 1] te following olds: a((u, v); (η, ζ)) c (u, v) V (η, ζ) V for all (u, v), (η, ζ) V. Proof. Te continuity estimate is a direct consequence of Caucy-Scwarz inequalities and boundedness of te trace operator. We obtain te following well-posedness and regularity results. Teorem 4.5. For any f i L 2 (Ω i ), i = 1, 2, g L 2 (Γ) suc tat (3.4) olds, tere exists a unique solution (u, v) Ṽ of (4.2), wic is also te unique solution to (4.4). Tis solution satisfies te a-priori estimate (u, v) V C (f 1, f 2, g) V c( f 1 Ω1 + f 2 Ω2 + g Γ ), wit constants C, c independent of f i, g and q 1, q 2 [0, 1]. If in addition Γ is a C 2 - manifold and Ω is convex or Ω is C 2 smoot, ten u i H 2 (Ω i ), for i = 1, 2, and v H 2 (Γ). Furtermore, te solution satisfies te second a-priori estimate u 1 H2 (Ω 1) + u 2 H2 (Ω 2) + v H2 (Γ) c( f 1 Ω1 + f 2 Ω2 + g Γ ). Proof. Existence, uniqueness and te first a-priori estimate follow from Teorem 4.2 and te Lemmas 4.3 and 4.4. Te extra regularity and second a-priori estimate follow from regularity results for te Poisson problem wit a Neumann boundary condition [16] and for te Laplace-Beltrami equation on a smoot closed surface [1]. 10

12 5. Adjoint problem. Consider te following formal adjoint problem, wit α as in Teorem 4.2. For given f L 2 (Ω) and g L 2 (Γ) find (u, v) V α suc tat for all (η, ζ) Ṽ: a((η, ζ); (u, v)) = (f 1, η 1 ) Ω1 + (f 2, η 2 ) Ω2 + (g, ζ) Γ. (5.1) Due to te results in Teorem 4.2 and Lemmas 4.3, 4.4, te problem (5.1) is wellposed. Now we look for te corresponding strong formulation of tis adjoint problem. We introduce an appropriate gauge condition for te rigt-and side: q 1 (f 1, 1) Ω1 + q 2 (f 2, 1) Ω2 + (g, 1) Γ = 0. (5.2) For any (η 1, η 2, ζ) V tere is a γ R suc tat (η 1, η 2, ζ) + γ(q 1, q 2, 1) Ṽ olds. From te definition of te bilinear form it follows tat a((q 1, q 2, 1); (u, v)) = 0 olds. Hence, if te rigt-and side satisfies condition (5.2), te formulation (5.1) is equivalent to: Find (u, v) V α suc tat a((η, ζ); (u, v)) = (f 1, η 1 ) Ω1 + (f 2, η 2 ) Ω2 + (g, ζ) Γ for all (η, ζ) V. Varying (η, ζ) we find te strong formulation of te dual problem to (3.7): ν i u i w u i = f i in Ω i, i = 1, 2, ν Γ Γ v w Γ v + [qνn u] Γ = g on Γ, ( 1) i ν i n u i = u i Kv on Γ, i = 1, 2, n Ω u 2 = 0 on Ω, wit q = (q 1, q 2 ). (5.3) Note tat compared to te original primal problem (3.7) we now ave w instead of w and tat te roles of K and q are intercanged. Wit te same arguments as for te primal problem, cf. Teorem 4.5, te following H 2 -regularity result for te dual problem can be derived. Teorem 5.1. For any f i L 2 (Ω i ), i = 1, 2, g L 2 (Γ) suc tat (5.2) olds, tere exists a unique weak solution (u, v) V α of (5.3). If Γ is a C 2 -manifold and Ω is convex or Ω is C 2 smoot, ten u i H 2 (Ω i ), for i = 1, 2, and v H 2 (Γ) satisfy te a-priori estimate u 1 H2 (Ω 1) + u 2 H2 (Ω 2) + v H2 (Γ) c( f 1 Ω1 + f 2 Ω2 + g Γ ). 6. Unfitted finite element metod. Let te domain Ω R 3 be polyedral and {T } >0 a family of tetraedral triangulations of Ω suc tat max diam(t ). T T Tese triangulations are assumed to be regular, consistent and stable. It is computationally convenient to allow triangulations tat are not fitted to te interface Γ. We use a discrete interface Γ, wic approximates Γ (as specified below), and can intersect tetraedra from T. To tis end, assume tat te surface Γ is implicitly defined as te zero set of a non-degenerate level set function φ: Γ = {x Ω : φ(x) = 0}, were φ is a sufficiently smoot function, suc tat φ < 0 in Ω 1, φ > 0 in Ω 2, and φ c 0 > 0 in U δ Ω. (6.1) 11

13 Here U δ Ω is a tubular neigborood of Γ of widt δ: U δ = {x R 3 : dist(x, Γ) < δ}, wit δ > 0 a sufficiently small constant. A special coice for φ is te signed distance function to Γ. Let φ be a given continuous piecewise polynomial approximation (w.r.t. T ) of te level set function φ wic satisfies φ φ L (U δ ) + (φ φ ) L (U δ ) c q+1, (6.2) wit some q 1. Ten we define Γ := { x Ω : φ (x) = 0 }, (6.3) and assume tat is sufficiently small suc tat Γ U δ olds. Furtermore Ω 1, := { x Ω : φ (x) < 0 }, Ω 2, := { x Ω : φ (x) > 0 }. (6.4) From (6.1) and (6.2) it follows tat dist(γ, Γ) c q+1 (6.5) olds. In many applications only suc a finite element approximation φ (e.g, resulting from te level set metod) to te level set φ is known. For suc a situation te finite element metod formulated below is particularly well suited. In cases were φ is known, one can take φ := I (φ), were I is a suitable piecewise polynomial interpolation operator. If φ is a P 1 continuous finite element function, ten Γ is a piecewise planar closed surface. In tis practically convenient case, it is reasonable to assume tat (6.2) olds wit q = 1. Consider te space of all continuous piecewise polynomial functions of a degree k 1 wit respect to T : V bulk := {v C(Ω) : v T P k (T ) T T }. (6.6) We now define tree trace spaces of finite element functions: V Γ, := {v C(Γ ) : v = w Γ for some w V bulk }, V 1, := {v C(Ω 1, ) : v = w Ω1, V 2, := {v C(Ω 2, ) : v = w Ω2, for some w V bulk }, for some w V bulk }. (6.7) We need te spaces V Ω, = V 1, V 2, and V = V Ω, V Γ, H 1 (Ω 1, Ω 2, ) H 1 (Γ ). Te space V Ω, is studied in many papers on te so-called cut finite element metod or XFEM [18, 19, 6, 15]. Te trace space V Γ, is introduced in [22]. We consider te finite element bilinear form on V V, wic results from te bilinear form of te differential problem using integration by parts in advection terms and furter replacing Ω i by Ω i, and Γ by Γ : a ((u, v); (η, ζ)) = 2 { (ν i u, η) Ωi, + 1 [ (w u, η) 2 (w ] } Ωi, η, u) Ωi, + ν Γ ( Γ v, Γ ζ) Γ [(w Γ v, ζ) Γ (w Γ ζ, v) Γ ] + 2 (u i q i v, η i Kζ) Γ. 12

14 In tis formulation we use te transformed quantities as in (3.6), but wit Ω i, Γ replaced by Ω i, and Γ, respectively. For example, on Ω i, we use te transformed 1 viscosity k i,a ν i, wit ν i te dimensionless viscosity as in (2.8). Similarly, te transformed velocity field w [H 1 (Ω 1, Ω 2, )] 3 is obtained after te transformation 1 w k i,a w =: w on Ω i,, i = 1, 2, and w w/( k 1,a + k 2,a ) =: w on Γ, wit w te dimensionless velocity as in (2.8). We use te skew-symmetric form of te advection term because w n = 0 olds on Γ but not necessarily on Γ. Let g L 2 (Γ ), f L 2 (Ω) be given and satisfy As discrete gauge condition we introduce, cf. (4.1), K(f, 1) Ω + (g, 1) Γ = 0. (6.8) K(1 + r)(u, 1) Ω1, + K(1 + 1 r )(u, 1) Ω2, + (v, 1) Γ = 0, r := k 2,a k 1,a. (6.9) Furtermore, define V,α := { (η, ζ) V : α 1 (η, 1) Ω1, + α 2 (η, 1) Ω2, + (ζ, 1) Γ = 0}, for arbitrary (but fixed) α 1, α 2 0, and Ṽ := V,α, wit α 1 = K(1 + r), α 2 = K(1 + 1 r ). Te finite element metod is as follows: Find (u, v ) Ṽ suc tat a ((u, v ); (η, ζ)) = (f, η) Ω + (g, ζ) Γ for all (η, ζ) V. (6.10) Wit te same arguments as for te continuous problem, cf. (4.4), based on te consistency condition (6.8) we obtain an equivalent discrete problem if te test space V is replaced by V,α. Te latter formulation is used in te analysis below. We sall use te Poincare and Friedric s inequalities (4.5) (4.8) wit Ω i replaced by Ω i, and Γ by Γ. We assume tat te corresponding Poincare-Friedric s constants are bounded uniformly in. In te finite element space we use te approximate V-norm given by (η, ζ) 2 V := η 2 H 1 (Ω 1, Ω 2, ) + ζ 2 H 1 (Γ ), (η, ζ) H1 (Ω 1, Ω 2, ) H 1 (Γ ). Repeating te same arguments as in te proof of Teorem 4.13 and Lemmas 4.3, and 4.4, we obtain an inf-sup stability result for te discrete bilinear form and its dual as well as a continuity estimate. Teorem 6.1. (i) For any q 1, q 2 [0, 1], tere exists α suc tat inf sup (u,v) Ṽ (η,ζ) V,α a ((u, v); (η, ζ)) (u, v) V (η, ζ) V C st > 0, (6.11) wit a positive constant C st independent of and of q 1, q 2 [0, 1]. (ii) Tere is a constant c independent of suc tat a ((u, v); (η, ζ)) c (u, v) V (η, ζ) V (6.12) for all (u, v), (η, ζ) H 1 (Ω 1, Ω 2, ) H 1 (Γ ). As a corollary of tis teorem we obtain te well-posedness result for te discrete problem. Teorem 6.2. For any f L 2 (Ω ), g L 2 (Γ ) suc tat (6.8) olds, tere exists a unique solution (u, v ) V of (6.10). For tis solution te a-priori estimate (u, v ) V C 1 st (f, g ) V c( f Ω + g Γ ) olds. Te constants C st and c are independent of. 13

15 7. Error analysis. We assumed tat Γ is a closed, C 2 surface embedded in R 3. Hence, tere exists a C 2 signed distance function d : U δ R suc tat Γ = {x U δ : d(x) = 0}. We assume tat d is negative on Ω 1 U δ and positive on Ω 2 U δ. Tus for x U δ, dist(x, Γ) = d(x). Under tese conditions, for δ > 0 sufficiently small, but independent of, tere is an ortogonal projection p : U δ Γ given by p(x) = x d(x)n(x), were n(x) = d(x). Let H = D 2 d = n be te Weingarten map. More details of te present formalism can be found in [9], 2.1. Given v H 1 (Γ), we denote by v e H 1 (U δ ) its extension from Γ along normals, i.e. te function defined by v e (x) = v(p(x)); v e is constant in te direction normal to Γ. Te following olds: v e (x) = (I d(x)h(x)) Γ v(p(x)) for x U δ. (7.1) We need some furter (mild) assumption on ow well te mes resolves te geometry of te (discrete) interface. We assume tat Γ U δ is te grap of a function γ (s), s Γ in te local coordinate system (s, r), s Γ, r [ δ, δ], wit x = s + rn(s): From (6.5) it follows tat Γ = { (s, γ (s)) : s Γ }. γ (s) = dist ( s + γ (s)n(s), Γ ) dist(γ, Γ) c q+1, (7.2) wit a constant c independent of s Γ Bijective mapping Ω i, Ω i. For te analysis of te consistency error we need a bijective mapping Ω i, Ω i, i = 1, 2. We use a mapping tat is similar to te one given in Lemma 5.1 in [25]. For te analysis we need a tubular neigborood U δ, wit a radius δ tat depends on. We define δ := c, wit a constant c > 0 tat is fixed in te remainder. We assume tat is sufficiently small suc tat Γ U δ U δ olds, cf. (6.5). Define Φ : Ω Ω as x n(x) δ2 d(x)2 Φ (x) = δ 2 γe (x)2 γe (x) if x Ūδ, (7.3) x if x Ω \ U δ. We assume tat is sufficiently small suc tat for all x Ūδ te estimate δ 2 γ e(x)2 > c 2 olds wit a mes independent constant c > 0. Using tis and te definition in (7.3), we conclude tat Φ is a bijection on Ω wit te properties: Φ (Ω i, ) = Ω i, Φ (x) = p(x) for x Γ, p(φ (x)) = p(x) for x U δ. Some furter properties of tis mapping are derived in te following lemma. Lemma 7.1. Consider Γ as defined in (6.3), wit φ suc tat (6.2) olds. Te mapping Φ as te smootness properties Φ ( W 1, (Ω) ) 3, Φ ( W 1, (Γ ) ) 3. Furtermore, for sufficiently small te estimates id Φ L (Ω) + I DΦ L (Ω) c q+1 (7.4) 1 det(dφ ) L (Ω) c q (7.5) old, were DΦ is te Jacobian matrix. For surface area elements we ave ds(φ (x)) = µ ds (x), x Γ, wit 1 µ L (Γ ) c q+1. (7.6) 14

16 Proof. Te smootness properties follow from te construction of Φ and te fact tat γ W 1, (Γ). Note tat γ (x) is an implicit function given by φ (x + γ (x)n(x)) = 0, x Γ. (7.7) To compute te surface gradient of γ, we differentiate tis identity and using te cain rule we obtain te relation: Γ γ (x) = (I + γ (x)h(x)) Γ φ (x ) n(x) φ (x, x = x + γ (x)n(x), x Γ. ) For te denominator in tis expression we get, using x x dist(γ, Γ) c q+1, (6.1), (6.2) and taking sufficiently small: n(x) φ (x ) = n(x) ( φ (x ) φ(x )) + n(x) ( φ(x ) φ(x)) + φ(x) c 0 c q 1 2 c 0. For te nominator we use Γ φ(x) = 0 and (6.2) to get: Γ φ (x ) Γ (φ (x ) φ(x )) + Γ (φ(x ) φ(x)) c q. From tis and (6.5) we infer γ L (Γ) + Γ γ L (Γ) c q+1. (7.8) Te following surface area transformation property can be found in, e.g., [8, 9]: µ (x)ds (x) = ds(p(x)), x Γ, µ (x) := (1 d(x)κ 1 (x))(1 d(x)κ 2 (x))n(x) T n (x), wit κ 1, κ 2 te nonzero eigenvalues of te Weingarten map and n te unit normal on Γ. Note tat Φ (x) = p(x) on Γ olds. From (6.2) we get 1 µ L (Γ ) c q+1. Hence, te result in (7.6) olds. For te term g(x) := δ2 d(x)2 δ 2 γe (x)2, wit x Ūδ, used in (7.3) we ave g L (U δ ) c and g L (U δ ) c 1. Using tese estimates and (7.1), (7.8) we obtain id Φ L (Ω) c q+1 and I DΦ L (Ω) c q. Tis proves (7.4). Te result in (7.5) immediately follows from (7.4) Smoot extensions. For functions v on Γ we ave introduced above te smoot constant extension along normals, denoted by v e. Below we also need a smoot extension to Ω i, of functions u defined on Ω i. Tis extension will also be denoted by u e. Note tat u Φ defines an extension to Ω i,. Tis extension, owever, as smootness W 1, (Ω i, ), wic is not sufficient for te interpolation estimates tat we use furter on. Hence, we introduce an extension u e, wic is close to u Φ in te sense as specified in Lemma 7.2 and is more regular. We make te smootness assumption Γ C k+1, were k is te degree of te polynomials used in te finite element space, cf. (6.6). We denote by E i a linear bounded extension operator H k+1 (Ω i ) H k+1 (R 3 ) (see Teorem 5.4 in [29]). Tis operator satisfies E i u Hm (R 3 ) c u Hm (Ω i) u H k+1 (Ω i ), m = 0,..., k + 1, i = 1, 2. (7.9) 15

17 For a piecewise smoot function u H k+1 (Ω 1 Ω 2 ), we denote by u e its transformation to a piecewise smoot function u e H k+1 (Ω 1, Ω 2, ) defined by { u e E1 (u = Ω1 ) in Ω 1, (7.10) E 2 (u Ω2 ) in Ω 2,. Te next lemma quantifies in wic sense tis function u e is close to u Φ. Lemma 7.2. Te following estimates old for i = 1, 2: u Φ u e Ωi, c q+1 u H 1 (Ω i), (7.11) ( u) Φ u e Ωi, c q+1 u H2 (Ω i), (7.12) for all u H 2 (Ω i ). Proof. Witout loss of generality we consider i = 1. Note tat u Φ = E 1 (u Ω1 ) Φ in Ω 1, and u e = E 1 (u Ω1 ) in Ω 1,. To simplify te notation, we write u 1 = E 1 (u Ω1 ) H 1 (R 3 ). We use tat Φ = id and u = u e on Ω \ U δ and transform to local coordinates in U δ using te co-area formula: u Φ u e 2 Ω 1, = u 1 Φ u 1 2 Ω 1, = u 1 Φ u 1 2 Ω 1, U δ γ = (u 1 Φ u 1 ) 2 φ 1 dr ds. Γ δ (7.13) In local coordinates te mapping Φ can be represented as Φ (s, r) = (s, p s (r)), wit p s (r) = r δ2 r2 δ 2 γ (s) 2 γ (s). Te function p s satisfies p s (r) r c q+1. We use te identity (u 1 Φ u 1 )(s, r) = ps(r) r r u 1 (s, t)dt, r = Φ (s, r) (s, r) Φ (s, r) (s, r). (7.14) Due to (7.13), (7.14), te Caucy inequality and φ c 0 > 0 on U δ, we get γ ps(r) u 1 Φ u 1 2 Ω 1, c p s (r) r u 1 (s, t) 2 dt dr ds Γ δ r γ r+c q+1 (7.15) c q+1 u 1 (s, t) 2 dt dr ds. Γ δ r c q+1 Let χ [ c q+1,c q+1 ] be te caracteristic function on [ c q+1, c q+1 ] and define g(t) = u 1 (s, t) 2 for t [ δ c q+1, γ + c q+1 ], g(t) = 0, t / [ δ c q+1, γ + c q+1 ]. Applying te L 1 -convolution inequality we get γ r+c q+1 δ r c q+1 u 1 (s, t) 2 dtdr c q+1 γ +c c χ [ c q+1,c q+1 ] L1 (R) g L1 (R) c q+1 χ [ c q+1,c q+1 ](r t)g(t) dt dr δ c q+1 u 1 (s, t) 2 dt, and using tis in (7.15) yields γ +c q+1 u 1 Φ u 1 2 Ω 1, c 2q+2 E 1 (u Ω1 )(s, t) 2 dr ds Γ δ c q+1 c 2q+2 E 1 (u Ω1 ) 2 H 1 (R 3 ) c 2q+2 u 2 H 1 (Ω. 1) 16

18 For deriving te estimate (7.12) we note tat ( u) Φ = (E i (( u) Ωi ) ) Φ = ( (E i (u Ωi )) ) Φ = ( u 1 ) Φ in Ω i,. Hence we ave ( u) Φ u e Ω1, = ( u 1 ) Φ u 1 Ω1,. We can repeat te arguments used above, wit u 1 replaced by u1 x j, j = 1, 2, 3, and tus obtain te estimate (7.12) Approximate Galerkin ortogonality. Due to te geometric errors, i.e., approximation of Ω i by Ω i, and of Γ by Γ, only an approximate Galerkin ortogonality relation olds. In tis section we derive bounds for te deviation from ortogonality. Te analysis is rater tecnical but in te same spirit as in [9, 8, 4]. For te rigt-and side of te differential problem (3.7) we assume f H 1 (Ω 1 Ω 2 ) and g L 2 (Γ) and for te rigt-and side in te discrete problem (6.10) we take f := f e and g := g e Γ c f, were c f R is suc tat te mean value condition (6.8) is satisfied. Let (u, v) Ṽ be te solution of te weak formulation (4.2) and (u, v ) Ṽ te discrete solution of (6.10). We take an arbitrary finite element test function (η, ζ) V. We use (η, ζ) Φ 1 V as a test function in (4.2) and ten obtain te approximate Galerkin relation: a ((u e u, v e v ); (η, ζ)) = a ((u e, v e ); (η, ζ)) a((u, v); (η, ζ) Φ 1 ) (7.16) + (f, η Φ 1 ) Ω + (g, ζ Φ 1 ) Γ (f e, η) Ω (g e, ζ) Γ + (c f, ζ) Γ. (7.17) In te analysis of te rigt-and side of tis relation we ave to deal wit full and tangential gradients (η Φ 1 ), Γ(ζ Φ 1 ). For te full gradient in te bulk domains one finds (η Φ 1 )(x) = DΦ (y) T η(y), x Ω, y := Φ 1 (x). (7.18) To andle te tangential gradient, a more subtle approac is required because one as to relate te tangential gradient Γ to Γ. Let n (y), y Γ, denote te unit normal on Γ (defined a.e. on Γ ). Furtermore, P(x) = I n(x) n(x) (x U δ ), P (y) = I n (y) n (y) (y Γ ). Recall tat Γ u(x) = P(x) u(x), Γ u(y) = P (y) u(y). We use te following relation, given in, e.g., [9]: for w H 1 (Γ) it olds Γ w(p(y)) = B(y) Γ w e (y) a.e. on Γ, B(y) = (I d(y)h(y)) 1 P (y), P (y) := I n (y) n(y) n (y) n(y). (7.19) From te construction of te bijection Φ : Γ Γ it follows tat (ζ Φ 1 )e (y) = ζ(y) olds for all y Γ. Application of (7.19) yields an interface analogon of te relation (7.18): Γ (ζ Φ 1 )(x) = B(y) Γ ζ(y), x Γ, y = Φ 1 (x) Γ. (7.20) Te mapping Φ equals te identity outside te (small) tubular neigborood U δ. In te analysis we want to make use of te fact tat te widt beaves like δ = c. For tis, te following result, proven in Lemma 4.1 in [13] is crucial: w Uδ Ω i c 1 2 w H1 (Ω i) for all w H 1 (Ω i ). (7.21) 17

19 Using properties of Φ and (7.18) we tus get, wit J = det(dφ ): w Uδ Ω i, = w Φ 1 J 1 2 U δ Ω i c 1 2 w Φ 1 H 1 (Ω i) c 1 2 w H1 (Ω i, ), (7.22) for all w H 1 (Ω i, ), wit a constant c independent of w and. We introduce a convenient compact notation for te approximate Galerkin relation. We use U := (u, v) = (u 1, u 2, v), and similarly U e = (u e, v e ), U := (u, v ) V, Θ = (η, ζ) H 1 (Ω 1, Ω 2, ) H 1 (Γ ). Furtermore F (Θ) := a (U e ; Θ) a(u; Θ Φ 1 ) + (f, η Φ 1 ) Ω + (g, ζ Φ 1 (f e, η) Ω (g e, ζ) Γ + (c f, ζ) Γ. Wit tis notation te approximate Galerkin relation can be represented as a (U e U ; Θ ) = F (Θ ) for all Θ V. (7.23) In Lemma 7.3 and Lemma 7.5 we derive bounds for te two parts (7.16) and (7.17) tat togeter form te functional F. We always assume tat Γ and te velocity field w in (3.3) are sufficiently smoot. Lemma 7.3. For m = 0 and m = 1 te following estimate olds: a (U e ; Θ) a(u; Θ Φ 1 ) c q+m( ) u H 2 (Ω 1 Ω 2) + v H (Γ))( 1 η H 1+m (Ω 1, Ω 2, ) + ζ H 1 (Γ ) for all U = (u, v) H 2 (Ω 1 Ω 2 ) H 1 (Γ), Θ = (η, ζ) H 1+m (Ω 1, Ω 2, ) H 1 (Γ ). Proof. Take U = (u, v) H 2 (Ω 1 Ω 2 ) H 1 (Γ) and Θ = (η, ζ) H 1+m (Ω 1, Ω 2, ) H 1 (Γ ). Denote J = det(dφ ). Using te definitions of te bilinear forms, te relations (7.18), (7.19), (7.20) and an integral transformation rule we get a (U e ; Θ) a(u; Θ Φ 1 ) = a ((u e, v e ); (η, ζ)) a((u, v); (η, ζ) Φ 1 ) (7.24) 2 ] = [(ν i u e, η) Ωi, (ν i u Φ, J (DΦ ) T η) Ωi, (7.25) [ (w u e, η) Ωi, ((w u) Φ, J η) Ωi, (7.26) (w η, u e ) Ωi, + ( (w Φ ) (DΦ ) T η, J u Φ ) Ω i, ] ) Γ (7.27) + ν Γ ( Γ v e, Γ ζ) Γ ν Γ (µ B T B Γ v e, Γ ζ) Γ (7.28) + 1 [ (w Γ v e, ζ) Γ (µ (w Φ ) B Γ v e, ζ) Γ (7.29) 2 ] (w Γ ζ, v e ) Γ + (µ (w Φ ) B Γ ζ, v e ) Γ (7.30) + 2 [(u e i q i v e, η i Kζ) Γ ((u i q i v) Φ, µ (η i Kζ)) Γ ]. (7.31) In tis expression te different terms correspond to bulk diffusion, bulk convection, surface diffusion, surface convection and adsorption, respectively. We derive bounds for tese terms. We start wit te bulk diffusion term in (7.25). Using te estimates 18

20 derived in te Lemmas 7.1, 7.2 we get 2 (ν i u e, η) Ωi, (ν i u Φ, J (DΦ ) T η) Ωi, c 2 [ ν i ( (u e u Φ ), J (DΦ ) T η) Ωi, + ( u e, (I J (DΦ ) T ) η) Ωi, ] 2 [ q+1 u H 2 (Ω i) η + q Ωi, u e H 1 (Ω Ωi,] i,) η (7.32) c q u H2 (Ω 1 Ω 2) η H1 (Ω 1, Ω 2, ). (7.33) If η H 2 (Ω 1, Ω 2, ) we can modify te estimate (7.32) as follows. J (DΦ ) T = I on Ω i, \ U δ. Hence, using (7.22) we get Note tat ( u e, (I J (DΦ ) T ) η) Ωi, = ( u e, (I J (DΦ ) T ) η) Uδ Ω i, c q u e Uδ Ω i, η Uδ Ω i, c q+1 u e H 2 (Ω i, ) η H 2 (Ω i, ) c q+1 u H2 (Ω i) η H2 (Ω i, ). (7.34) Tus, instead of (7.33) we ten obtain te upper bound c q+1 u H 2 (Ω 1 Ω 2) η H 2 (Ω 1, Ω 2, ). For te bulk convection term in (7.26)-(7.27) we get 2 1 (w u e, η) Ωi, ((w u) Φ, J η) Ωi, (w η, u e ) Ωi, ( (w Φ ) (DΦ ) T η, J u Φ 2 )Ω i, 2 1 (w u e (w u) Φ, J η) Ωi, 1 + (w u e, (1 J )η) Ωi, ( w (w Φ ) (DΦ ) T ) η, J u Φ ) Ωi, 1 + (w η, u e J u Φ ) Ωi, 2 2 Te difference w w Φ can be bounded using te assumption tat te original (unscaled) velocity w is sufficiently smoot, w W 1, (Ω). Using tis, te relation (7.14) and te definition of w we get w w Φ L (Ω i, ) c q+1. Te first term on te rigt-and side above can be bounded using w u e (w u) Φ Ωi, = w u e (w Φ ) ( u Φ ) Ωi, (w w Φ ) u e Ωi, + (w Φ ) ( u e u Φ ) Ωi, c q+1 u H 2 (Ω i), were in te last step we used results from te Lemmas 7.1, 7.2. Te oter tree terms can be estimated by using 1 J L (Ω) c q, I (DΦ ) T L (Ω) c q 19

21 and u e u Φ Ωi, c q+1 u H 1 (Ω i). Tus we get a bound 2 1 (w u e, η) Ωi, ((w u) Φ, J η) Ωi, (w η, u e ) Ωi, ( (w Φ ) (DΦ ) T η, J u Φ 2 )Ω i, c q u H 2 (Ω 1 Ω 2) η H 1 (Ω 1, Ω 2, ). If η H 2 (Ω 1, Ω 2, ) we can apply an argument very similar to te one in (7.34) and obtain te following upper bound for te bulk convection term: c q+1 u H2 (Ω 1 Ω 2) η H 2 (Ω 1, Ω 2, ). For te surface diffusion term in (7.28) we introduce, for y Γ, te matrix A(y) := P (y) ( I µ (y)b(y) T B(y) ) P (y). Using d(y) c q+1, 1 µ (y) c q+1 and P P = P we get A L (Γ ) c q+1 and tus: ν Γ ( Γ v e, Γ ζ) Γ (µ B T B Γ v e, Γ ζ) Γ = νγ (A Γ v e, Γ ζ) Γ A L (Γ ) Γ v e Γ ζ Γ c q+1 v H 1 (Γ) ζ H 1 (Γ ). For te derivation of a bound for te surface convection term in (7.29)-(7.30) we introduce w := P (w µ B T (w Φ )). Using te results in Lemmas 7.1, 7.2 and Pw = w it follows tat Using w L (Γ ) P (w w Φ ) L (Γ ) + P (I µ B T )(w Φ ) L (Γ ) c P (I P T )P L (Γ ) + c q+1 c P n L (Γ ) Pn L (Γ ) + c q+1 = c (P P)n L (Γ ) (P P )n L (Γ ) + c q+1 c P P 2 L (Γ ) + cq+1. n (y) n(y) = n (y) n(p(y)) = φ (y) φ (y) φ(p(y)) φ(p(y)) in combination wit y p(y) c q+1 and te approximation error bound (6.2) we get P P L (Γ ) c q. Hence, for te surface convection term in (7.29) we obtain (w Γ v e, ζ) Γ (µ (w Φ ) B Γ v e, ζ) Γ = ( w Γ v e, ζ) Γ w L (Γ ) Γ v e Γ ζ Γ c q+1 v H 1 (Γ) ζ H 1 (Γ ). Te term in (7.30) can be bounded in te same way. Finally we consider te adsorption term in (7.31). Using te results in Lemmas 7.1, 7.2 we get 2 (u e i q i v e, η i Kζ) Γ ((u i q i v) Φ, µ (η i Kζ)) Γ 2 (u e i µ u i Φ, η i Kζ) Γ + qi ((1 µ )v e, η i Kζ) Γ c q+1 2 ( )( ) ui H1 (Ω i) + v Γ ηi Γ + ζ Γ. c q+1 ( u H 1 (Ω 1 Ω 2) + v Γ )( η H 1 (Ω 1, Ω 2, ) + ζ Γ ). 20

22 Combining tese estimates for te terms in (7.25)-(7.31) completes te proof. From te arguments in te proof above one easily sees tat te bounds derived in Lemma 7.3 also old if in a ( ; ) and a( ; ) te arguments are intercanged. Tis proves te result in te following lemma, tat we need in te L 2 -error analysis. Lemma 7.4. Te following estimate olds: a (Θ; U e ) a(θ Φ 1 ; U) c q( ) η H1 (Ω 1, Ω 2, ) + ζ H 1 (Γ ))( u H 2 (Ω 1 Ω 2) + v H 1 (Γ) for all U = (u, v) H 2 (Ω 1 Ω 2 ) H 1 (Γ), Θ = (η, ζ) H 1 (Ω 1, Ω 2, ) H 1 (Γ ). Lemma 7.5. Assume tat te data as smootness f H 1 (Ω 1 Ω 2 ), g L 2 (Γ). For te term in (7.17) te following estimate olds: (f, η Φ 1 ) Ω + (g, ζ Φ 1 ) Γ (f e, η) Ω (g e, ζ) Γ + (c f, ζ) Γ c q+1( )( ) f H1 (Ω 1 Ω 2) + g Γ η H1 (Ω 1, Ω 2, ) + ζ Γ for all (η, ζ) H 1 (Ω 1, Ω 2, ) H 1 (Γ ). Proof. Define J = det(dφ ). Using integral transformation rules we obtain (f, η Φ 1 ) Ω + (g, ζ Φ 1 ) Γ (f e, η) Ω (g e, ζ) Γ + (c f, ζ) Γ = (f Φ, J η) Ω + (g e, µ ζ) Γ (f e, η) Ω (g e, ζ) Γ + (c f, ζ) Γ (7.35) (J f Φ f e, η) Ω + ((1 µ )g e, ζ) Γ + c cf ζ Γ. Te first term on te rigt-and side can be estimated using (7.11) and (7.22): (J f Φ f e, η) Ω (J 1)f Φ, η) Ω + (f Φ f e, η) Ω c q (f Φ, η) Uδ + c q+1 f H1 (Ω 1 Ω 2) η Ω c q+1 f Φ H 1 (Ω 1, Ω 2, ) η H 1 (Ω 1, Ω 2, ) + c q+1 f H 1 (Ω 1 Ω 2) η Ω c q+1 f H1 (Ω 1 Ω 2) η H1 (Ω 1, Ω 2, ). For te second term we ave: ((1 µ )g e, ζ) Γ ) c q+1 g Γ ζ Γ. It remains to estimate c f = Γ 1 (K(f f e, 1) Ω + (g, 1) Γ (g e, 1) Γ ). Note tat (f f e, 1) Ω = 2 (f, 1) Ω i (f e, 1) Ωi,. Witout loss of generality we consider i = 1. Te extension of f 1 = f Ω1 is denoted by ˆf 1 := E 1 (f 1 ). In U δ we use te local coordinate system (s, r) wit x = s + rn(s), s = p(x). Note tat 21

23 ˆf 1 (s, 0) = f 1 (s, 0) olds. We tus obtain γ (f, 1) Ω1 (f e, 1) Ω1, ( E 1 (f 1 ), 1) Ω1 Ω 1, = ˆf 1 (s, r) φ 1 dr ds (7.36) Γ 0 γ r = Γ 0 f 1(s, 0) + ˆf 1 (s, t)dt 0 r φ 1 dr ds γ r γ L (Γ) f 1 L 1 (Γ) + c ˆf 1 (s, t) dt dr ds r Finally, te estimate Γ 0 γ γ c q+1 f 1 L1 (Γ) + c ˆf 1 dt dr ds Γ 0 0 r c q+1 f 1 H1 (Ω 1) + c q+1 E 1 (f 1 ) H1 (Ω 1 Ω 1, ) c q+1 f 1 H 1 (Ω 1). (g, 1) Γ (g e, 1) Γ c q+1 g Γ follows immediately from (7.6). Hence, for te tird term we ave c c f ζ Γ c q+1 ( f H1 (Ω 1 Ω 2) + g Γ ) ζ Γ. Combining tese results completes te proof. As an immediate corollary of te previous two lemmas, te definition of F and te regularity estimate in Teorem 4.5 we obtain te following result. Lemma 7.6. Let f H 1 (Ω 1 Ω 2 ), g L 2 (Γ). Assume tat te solution (u, v) of (4.2) as smootness u H 2 (Ω 1 Ω 2 ), v H 2 (Γ). For m = 0 and m = 1 te following olds: F (Θ) c q+m )( ) ( f H 1 (Ω 1 Ω 2) + g Γ η H 1+m (Ω 1, Ω 2, ) + ζ H 1 (Γ ) (7.37) for all Θ = (η, ζ) H 1+m (Ω 1, Ω 2, ) H 1 (Γ ) Discretization error bound in te V -norm. Based on te stability, continuity and approximate Galerkin properties presented in te previous sections, we derive a discretization error bound in te V norm. Due to te approximation of te interface te discrete solution U = (u, v ) as a domain tat differs from tat of te solution U = (u, v) to te continuous problem. Terefore, it is not appropriate to define te error as U U. It is natural to define te discretization error eiter as U e U, wit functions defined on te domain corresponding to te discrete problem, or as U U Φ 1, wit functions defined on te domain corresponding to te continuous problem. We use te former definition. In te analysis we need suitable interpolation operators, applicable to U e. Te function u e consists of te pair u e = (E 1 (u Ω1 ) Ω1,, E 2 (u Ω2 ) Ω2, ) =: (u e 1, u e 2), cf. (7.10). As is standard in analyses of XFEM (or unfitted FEM) we define an interpolation based on te standard nodal interpolation of te smoot extension in te bulk space V bulk. Let I bulk denote te nodal interpolation in V bulk (wic consists of finite elements of degree k). We define I u e V Ω, as follows: I u e = ( [I bulk E 1 (u Ω1 )] Ω1,, [I bulk ) E 2 (u Ω2 )] Ω2,. 22

24 Te construction of tis operator and interpolation error bounds for I bulk immediately yield I u e u e H 1 (Ω 1, Ω 2, ) c k u H k+1 (Ω 1 Ω 2). (7.38) For te interpolation of v e we use a similar approac, namely I v e := [I bulk v e ] Γ. Interpolation error bounds for tis operator are known in te literature, see, e.g., Teorem 4.2 in [27]: I v e v e H 1 (Γ ) c k v H k+1 (Γ). (7.39) Using tese interpolation error bounds we obtain te following main teorem. Teorem 7.7. Assume f H 1 (Ω 1 Ω 2 ). Let te solution (u, v) Ṽ of (4.4) be sufficiently smoot. For te finite element solution (u, v ) Ṽ te following error estimate olds: (u e u, v e v ) V c k( ) u H k+1 (Ω 1 Ω 2) + v H k+1 (Γ) + c q( ) (7.40) f H 1 (Ω 1 Ω 2) + g Γ, were k is te degree of te finite element polynomials and q te geometry approximation order defined in (6.2). Proof. We use arguments similar to te second Strang s lemma. Recalling te stability and continuity results from (6.11), (6.12), te consistency error bound in Lemma 7.6 and te interpolation error bounds in (7.38), (7.39) we get, wit I U e = (I u e, I v e ): I U e U V Cst 1 a (I U e U ; Θ ) sup Θ V Θ V ( = Cst 1 a (I U e U e ; Θ ) sup + a (U e U ; Θ ) ) Θ V Θ V Θ V ( c I U e U e F (Θ ) ) V + sup Θ V Θ V c k ( u H k+1 (Ω 1 Ω 2) + v H k+1 (Γ)) + c q ( f H 1 (Ω 1 Ω 2) + g Γ ). (7.41) Te desired result now follows by a triangle inequality and applying te interpolation error estimates (7.38), (7.39) once more. 8. Error estimate in L 2 -norm. In tis section we use a duality argument to sow iger order convergence of te unfitted finite element metod in te L 2 product norm. As typical in te analysis of elliptic PDEs wit Neumann boundary conditions, one considers te L 2 norm in a factor space: U L2 /R = inf γ R U γ(q 1, q 2, 1) L 2 (Ω) L 2 (Γ), for U L 2 (Ω) L 2 (Γ), and q 1, q 2 [0, 1] from (3.7). A similar norm can be defined on L 2 (Ω) L 2 (Γ ). Define te error E := (U e U ) Φ 1 L2 (Ω) L 2 (Γ). Tere is a constant γ R suc tat Ẽ := E γ(q 1, q 2, 1) satisfies te consistency condition (5.2). According to Teorem 5.1 te dual problem: Find W V α suc tat a(θ; W ) = (Ẽ, Θ) L2 (Ω) L 2 (Γ) for all Θ V, (8.1) 23

25 as te unique solution W = (w, z) H 2 (Ω 1 Ω 2 ) H 2 (Γ), satisfying w H2 (Ω 1 Ω 2) + z H 2 (Γ) c Ẽ L 2 (Ω) L 2 (Γ), (8.2) wit a constant c independent of Ẽ. Teorem 8.1. Let te assumptions in Teorem 7.7 and Teorem 5.1 be fulfilled. For te finite element solution (u, v ) Ṽ te following error estimate olds: u e u, v e v L2 /R c k+1( ) u H k+1 (Ω 1 Ω 2) + v H k+1 (Γ) + c q+1( ) (8.3) f H 1 (Ω 1 Ω 2) + g Γ, were k is te degree of te finite element polynomials and q te geometry approximation order defined in (6.2). Proof. First, let γ opt := arg inf γ R E γ(q 1, q 2, 1) L2 (Ω) L 2 (Γ). Observe te cain of estimates: U e U L 2 /R U e U γ opt (q 1, q 2, 1) L 2 (Ω) L 2 (Γ ) = (U e U γ opt (q 1, q 2, 1)) Φ 1 c (U e U γ opt (q 1, q 2, 1)) Φ 1 J 1 2 L 2 (Ω) L 2 (Γ) L 2 (Ω) L 2 (Γ) = c (U e U ) Φ 1 γ opt(q 1, q 2, 1) L2 (Ω) L 2 (Γ) = c E L 2 /R c Ẽ L 2 (Ω) L 2 (Γ). We apply te standard duality argument and tus obtain: Ẽ 2 L 2 (Ω) L 2 (Γ) = a(ẽ, W ) = a(e, W ) = a(e ; W ) a (U e U ; W e ) + a (U e U ; W e I W e ) a (U e U ; I W e ) = [ a(e ; W ) a (U e U ; W e ) ] + a (U e U ; W e I W e ) + F (I W e ) = [ a(e ; W ) a (U e U ; W e ) ] + a (U e U ; W e I W e ) + F (I W e W e ) + F (W e ). Tese terms can be estimated as follows. For te term between square brackets we use Lemma 7.4 and (8.2): a(e ; W ) a (U e U ; W e ) c q U e U V ( w H 2 (Ω 1 Ω 2) + z H 1 (Γ)) c q U e U V Ẽ L2 (Ω) L 2 (Γ) For te second term we use continuity, te interpolation error bound and (8.2): a (U e U ; W e I W e ) c U e U V ( w H 2 (Ω 1 Ω 2) + z H 2 (Γ)) c U e U V Ẽ L2 (Ω) L 2 (Γ). For te tird term we use Lemma 7.6 wit m = 0, te interpolation error bound and (8.2): F (I W e W e ) c q( f H1 (Ω 1 Ω 2) + g Γ ) I W e W e V c q+1( f H 1 (Ω 1 Ω 2) + g Γ ) Ẽ L 2 (Ω) L 2 (Γ). For te fourt term we use Lemma 7.6 wit m = 1 and (8.2): F (W e ) c q+1( f H 1 (Ω 1 Ω 2) + g Γ ) Ẽ L 2 (Ω) L 2 (Γ). Combining tese results and using te bound for U e U V of Teorem 7.7 completes te proof. 24

26 9. Numerical results. We consider te stationary coupled bulk-interface convection diffusion problem (3.3) in te domain Ω = [ 1.5, 1.5] 3 and wit te unit spere Γ = {x Ω : x 2 = 1} as interface. For te velocity field we take a rotating field in te x-z plane: w = 1 10 (z, 0, x). Tis w satisfies te conditions (2.1) and (3.1), i.e., div w = 0 in Ω and w n = 0 on Γ. On some parts of te boundary Ω te velocity field w is pointing inwards te domain, so natural boundary conditions as in (3.3) are not suitable ere. For tis reason, and to simplify te implementation, we use Diriclet boundary conditions on Ω. Note tat in tis case we do not need te additional condition (4.1) to obtain well-posedness. For te scaling constant we take K = Convergence study. In tis experiment, te material parameters are cosen as ν 1 = 0.5, ν 2 = 1, ν Γ = 1 and k 1,a = 0.5, k 2,a = 2, k 1,d = 2, k 2,d = 1. Te source terms f i L 2 (Ω), i = 1, 2, and g L 2 (Γ) in (3.3) and te Diriclet boundary data are taken suc tat te exact solution of te coupled system is given by v(x, y, z) = 3x 2 y y 3, u 1 (x, y, z) = 2u 2 (x, y, z), u 2 (x, y, z) = e 1 x2 y 2 z 2 v(x, y, z). Note tat te gauge condition (3.4) is satisfied for tis coice of f and g. For te initial triangulation, Ω is divided into sub-cubes eac consisting of 6 tetraedra. Tis initial mes is uniformly refined up to 4 times, yielding T. Te discrete interface Γ is obtained by linear interpolation of te signed distance function corresponding to Γ. We use te finite element spaces in (6.6)-(6.7) wit k = 1, i.e., V bulk consists of piecewise linears on T. Te resulting coupled linear system is iteratively solved by a GCR metod using a block diagonal preconditioner, were te bulk and interface systems are preconditioned by te SSOR metod. Te numerical solution u, v after 2 grid refinements and te resulting interface approximation Γ are sown in Figure 9.1. Te L 2 and H 1 errors for te bulk and interface concentration are given in Tables 9.1 and 9.2. As expected, first order convergence is obtained for te H 1 errors of bulk and interface concentration, cf. Teorem 7.7. Te respective L 2 errors are of second order, wic confirms te teoretical findings in Teorem 8.1. # ref. u u L2 (Ω) order u u H 1 (Ω 1, Ω 2, ) order E E E E E E E E E E Table 9.1 L 2 and H 1 errors for bulk concentration u on different refinement levels Effect of small desorption. Te teory presented indicates tat bot te model and te discretization are stable if te desorption coefficients tend to zero. Related to tis we performed an experiment wit a small or even vanising desorption 25

27 Fig Numerical solutions v on Γ and u visualized on a cut plane z = 0 for refinement level 2. # ref. v v L 2 (Γ ) order v v H 1 (Γ ) order E E E E E E E E E E Table 9.2 L 2 and H 1 errors for interface concentration v on different refinement levels. coefficient. For te bulk concentration, omogeneous Diriclet boundary data on Ω are cosen. Te source terms are set to f i = 0, i = 1, 2 and g = 1, so bulk concentration can only be generated by desorption of interface concentration from Γ. Te material parameters are cosen as ν 1 = 0.5, ν 2 = 1, ν Γ = 1 and k 1,a = k 2,a = k 2,d = 1, k 1,d = ε wit ε 0. We use te same initial triangulation as before. Tis initial mes is uniformly refined 3 times, and te discrete problem is solved on tis mes for different values of ε, yielding solutions u ε i, V i,, v ε V Γ,. Table 9.3 sows te mean bulk concentration of u ε 1, in Ω 1,, ū 1, (ε) := Ω 1, 1 Ω 1, u ε 1, dx, for different values of te desorption coefficient k 1,d = ε. We clearly observe a linear beavior, wic can be expected, based on te relation k 1,a u 1 ds = u 1 ds = k 1,d v ds, Γ Γ Γ 26