Sven Gross 1, Maxim A. Olshanskii 2 and Arnold Reusken 1

Transcription

1 ESAIM: M2AN 49 (2015) DOI: /m2an/ ESAIM: Matematical Modelling and Numerical Analysis A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS Sven Gross 1, Maxim A. Olsanskii 2 and Arnold Reusken 1 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. Matematics Subject Classification. 65N30, 65N15, 76T99. Received July 5, Revised February 12, Publised online August 18, Introduction Coupled bulk-surface or bulk-interface partial differential equations arise in many applications, e.g., in multipase fluid dynamics [23] and biological applications [2]. In tis paper, we consider a coupled bulk-interface advection-diffusion 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., [8, 23, 33, 35]. 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 Keywords and prases. Finite element metod, surface PDEs, surface-bulk coupled problems, unfitted metod, transport-diffusion. Partially supported by NSF troug te Division of Matematical Sciences Grant Institut für Geometrie und Praktisce Matematik, RWTH-Aacen University, Aacen, Germany gross@igpm.rwt-aacen.de; reusken@igpm.rwt-aacen.de 2 Department of Matematics, University of Houston, Houston, TX , USA molsan@mat.u.edu Article publised by EDP Sciences c EDP Sciences, SMAI 2015

2 1304 S. GROSS ET AL. 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, 16]. In tese references finite element metods for coupled bulk-surface partial differential equations are proposed and analyzed. In [4, 16] 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[16] 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 [6] 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 [4,16] we restrict to stationary problems and a linear 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, 16]. We study a problem in wic elliptic partial differential equations in two bulk subdomains are coupled to a partial differential equation on a sarp interface wic separates te two subdomains. In te previous work only a coupling of an elliptic partial differential in one bulk domain wit a partial differential equation on te boundary of tis bulk domain is treated. Furtermore, te partial differential equations considered in tis paper 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 Poincaré 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, wic is very similar to te metod presented in [4]. In te metod treated in tis paper we do not apply te stabilization tecnique used in [4]. 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. [3, 24, 25]. For PDEs posed on surfaces, te trace finite element metod was introduced and studied in [12, 27, 29, 31]. In te metod tat we propose, te smoot interface is approximated by a piecewise smoot one, caracterized by 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 (for linear finite elements) to obtain a stiffness matrix wit conditioning properties similar to tose of a standard finite element metod. We tink

3 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1305 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. Finally we mention te recent related paper [9], in wic unfitted finite element tecniques, similar to te one used in tis paper, are applied to surface partial differential equations (witout coupling to bulk domains). 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 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) for all t [0,t 1 ]. 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). Below we write Γ instead of Γ (t). It is assumed tat te fluids are immiscible and tat tere are no pase canges. Consequently, te normal components of te fluid velocity are continuous at te interface and te interface itself is advected wit te flow, i.e., V Γ = w n olds, 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 iδu =0 inω i = Ω i (t), i =1, 2. (2.2) Here D i > 0 denotes te bulk diffusion coefficient, wic is assumed to be constant in Ω i. Mass conservation in a control area (transported by te flow field) tat is completely contained in Γ results in te surface convectiondiffusion equation (cf. [23]): v +(div Γ w)v D Γ Δ Γ v = j a j d on Γ = Γ (t), (2.3) were v = v t +w v denotes te material derivative and Δ Γ,div Γ te Laplace Beltrami and surface divergence operators, respectively. Te interface diffusion coefficient D Γ > 0 is assumed to be a constant. 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 =(w 1 ) Γ (w 2 ) Γ denotes te jump of w across Γ.

4 1306 S. GROSS ET AL. To close te system of equations, we need constitutive equations for modeling te adsorption/desorption. A standard model, cf. [33], 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) 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) fori =1, 2. Basic coices for g, f are te following: g(v) =1, f(v) =v (Henry) (2.6) g(v) =1 v, f(v) =v (Langmuir), (2.7) v were v is a constant tat quantifies te maximal concentration on Γ. Furter options are given in [33]. Note tat in te Langmuir case we ave a nonlinearity due to te term vu i. Combining (2.2) (2.5) givesa 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: u t + w u ν iδu =0 inω i (t), i =1, 2, v +(div Γ w)v ν Γ Δ 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, wit ν i = D i LW, 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. [6,13,15,35]. Te coefficients k i,a, k i,d are te dimensionless adsorption and desorption coefficients. Remark 2.1. 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, cf. also Remark 4.6 below. 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 s 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)

5 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1307 We also assume div Γ w = 0 so tat te term (div Γ w)v in te surface convection-diffusion 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 =0on Ω. Restricting te model (2.8) to an equilibrium state, we obtain te following stationary problem: ν i Δu i + w u i = f i in Ω i,, 2, ν Γ Δ 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 (3.3) i,d v on Γ, i =1, 2, n Ω u 2 =0 on Ω. In tis model, we allow source terms g L 2 (Γ )andf 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 [16]. 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 0and 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) wecanusete following transformation to reduce te number of parameters of te model (3.3): ũ i := k i,a u i, ṽ := ( k 1,a + k 2,a )v, 1 ν i := k i,a ν i, ν Γ := ν Γ /( k 1,a + k 2,a ), 1 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 will in general be discontinuous across Γ. In eac subdomain and on Γ, owever, w is regular: w Ωi = w i H 1, (Ω i ) 3, i =1, 2, and w Γ H 1 (Γ ) 3. For simplicity we omit te tilde notation in te transformed variables. Tis ten results in te following model, wic we study in te remainder of te paper: ν i Δu i + w u i = f i in Ω i, i =1, 2, ν Γ Δ 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) k i,d wit q i := k 1,a + k [0, 1]. 2,a Te data f i and g are assumed to satisfy te consistency condition (3.4). Recall tat K = LU V a fixed (scaling) constant. > 0, cf. (2.8), is

6 1308 S. GROSS ET AL. 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. Concerning te smootness of te interface, we assume tat Γ is a C 1 manifold. Tis assumption also suffices for te analysis in Section 5. We first introduce some notations. For u H 1 (Ω 1 Ω 2 )wealsowriteu =(u 1,u 2 )witu i = u Ωi H 1 (Ω i ). Furtermore: (f,g) ω := fgdx, f 2 ω := (f,f) ω, 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 =0fortevariablesusedin(3.7). Hence, we obtain te natural gauge condition ( K(1 + r)(u 1, 1) Ω1 + K 1+ 1 ) (u 2, 1) Ω2 +(v, 1) Γ =0, r := k 2,a (4.1) r k 1,a 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) bytestfunctionsfrom 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. 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

7 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1309 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) Ω2 +(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 )begiven. 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) isequivalent 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. For H 1 (Ω i )andh 1 (Γ ) te following Poincaré 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 H1 (Γ ). (4.7) For te analysis of stability of te weak formulation we need te following Poincaré type inequality in te space V. Lemma 4.1. Let r i,σ i [0, ), i =1, 2. TereexistsC P (r 1,r 2,σ 1,σ 2 ) > 0 suc tat for all (u, v) V, te following inequality olds: ) 2 (u, v) V C P ( u Ω1 Ω 2 + Γ v Γ + 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., [17]). 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 + Tx Z ) for all x X. (4.9) Ten tere exists a constant c suc tat olds. We take X = H 1 (Ω 1 ) H 1 (Ω 2 ) H 1 (Γ ) wit te norm x X c Ax Y for all x X (4.10) (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 te product norm and Z = L 2 (Ω 1 ) L 2 (Ω 2 ) L 2 (Γ ) wit te product 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).

8 1310 S. GROSS ET AL. Ten we ave A L(X, Y ). Consider A(u 1,u 2,v) = 0. Te first 9 equations yield u 1 = constant, u 2 = constant, v = constant, and substitution of tis in te last tree equations yields tat tese constants must be zero. Hence, 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 1,Γ ) 1 2 c A(u1,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] andwitasuitableα = α(q 1,q 2 ) 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). We first treat te case 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,witβ>0, and ζ = v. Tevalueofβ is cosen furter on. 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) andteviscosityν. 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. Using tis and te Poincaré 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 Poincaré s constant and ν Γ. We take β sufficiently large (depending only on c F,ĉ and K) andε>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>0isfixed. In te second case we take q 1 ε, andq 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) witu 2 =0,σ 2 =0,r 1 = K(1 + r), σ 1 = q 1,

9 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1311 we get a((u, v); (η, ζ)) = ν 1 u 1 2 Ω 1 + q 1ν Γ K Γ v 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 Γ v 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 Poincaré s constant and on ε. Wenowtakeη 1 =0,η 2 = βu 2,witβ>0and 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>0isfixed. We consider te last case, namely q 1 δ and q 2 δ. Takeη 1 = u 1, η 2 = q1 q 2 u 2, ζ = K 1 q 1 v.wetenget a((u, v); (η, ζ)) = ν 1 u 1 2 Ω 1 + ν 2 q 1 q 2 u 2 2 Ω 2 + ν Γ q 1 K Γ v 2 Γ + u 1 q 1 v 2 Γ + q 1 q 2 u 2 q 2 v 2 Γ c ( 2 u 2 Ω 1 Ω 2 + Γ v 2 Γ + u i q i v 2 ) Γ. We use (4.8) witr 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. Te case 0 q 1 q 2 1 can be treated wit almost exactly te same arguments. 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: a((u, v); (η, ζ)) inf sup C st. (4.13) (u, v) V (η, ζ) V (η,ζ) V α (u,v) Ṽ

10 1312 S. GROSS ET AL. 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 aprioriestimate (u, v) V C (f 1,f 2,g) V c( f 1 Ω1 + f 2 Ω2 + g Γ ), (4.14) 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 ),fori =1, 2, andv H 2 (Γ ). Furtermore, te solution satisfies te second a priori estimate u 1 H 2 (Ω 1) + u 2 H 2 (Ω 2) + v H 2 (Γ ) c( f 1 Ω1 + f 2 Ω2 + g Γ ). (4.15) Proof. Existence, uniqueness and te firsta prioriestimate follow from Teorem 4.2 and te Lemmas 4.3 and4.4. To sow te extra regularity of te solution, we note tat u i satisfies te weak formulation of te Poisson s equation ν i Δu i = F i := f i w u i in Ω i wit a Robin boundary condition ( 1) i n u i u i = G i := q i v. Tanks to (4.14) weavef i L 2 (Ω i ), G i H 1 2 ( Ω i )and F i Ωi + G i 1 c( f H 2 ( Ωi) 1 Ω1 + f 2 Ω2 + g Γ ). Teorem from [20] implies u i H 2 (Ω i )andteestimateforu i in (4.15). On te interface Γ, v satisfies te weak formulation of te Laplace Beltrami equation ν Γ Δ Γ v = G Γ := g w Γ v K[n u]. Tanks to (4.14), te regularity result for u i in (4.15) and te smootness of Γ,weaveG Γ L 2 (Γ )and G Γ Γ c( f 1 Ω1 + f 2 Ω2 + g Γ ). Now we apply te regularity result for te Laplace Beltrami equation on a closed C 2 -surface from Lemma 3.2 in [14]. Tis proves v H 2 (Γ )andteestimateon v H2 (Γ ) in (4.15). Remark 4.6. In [4, 16] a stationary diffusion problem on a bulk domain is linearly coupled wit a stationary diffusion equation on te boundary of tis domain. Hence tere is only one bulk domain. Well-posedness of a suitable weak formulation of tis problem is sown in [1,16]. Te analysis in [16] is significantly simpler tan te one presented above. Tis is due to te fact tat for te case of one bulk domain te coupling term is simpler and one easily verifies tat te corresponding bilinear form is elliptic. In our case, due to te coupling term 2 (u i q i v, η i Kζ) Γ in (4.2), te bilinear form is not elliptic and we ave to derive an inf-sup estimate. A furter complication, compared to te case of one bulk domain, is te Poincaré s type inequality tat we need, cf. Lemma Adjoint problem Consider te following formal adjoint problem, wit α as in Teorem 4.2. For given f L 2 (Ω) andg 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 well-posed.

11 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1313 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) isequivalentto: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 Δu i w u i = f i in Ω i, i =1, 2, ν Γ Δ 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. (5.3) In te second equation q denotes a piecewise constant function wit values q Ωi := q i. Note tat compared to te original primal problem (3.7) wenowave w instead of w and tat te roles of K and q are intercanged. Wit te same arguments as for te primal problem (cf. Tm. 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). IfΓ 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, andv H 2 (Γ ) satisfy te aprioriestimate u 1 H 2 (Ω 1) + u 2 H 2 (Ω 2) + v H 2 (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 T T diam(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). 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 C 1 smoot function in a neigborood of Γ, suc tat φ<0 in Ω 1, φ > 0 in Ω 2, and φ c 0 > 0inU δ Ω. (6.1) Here U δ Ω is a tubular neigborood of Γ of widt δ: U δ = {x R 3 :dist(x,γ) <δ}, witδ>0 a sufficiently small constant. A special coice for φ is te signed distance function to Γ.Letφ 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. For tis estimate to old, we assume tat te level set function φ as te smootness property φ C q+1 (U δ ). Clearly tis induces a similar smootness property for its zero level Γ. Ten we define Γ := { x Ω : φ (x) =0}, (6.3)

12 1314 S. GROSS ET AL. 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)=maxdist(x,γ) c q+1 (6.5) x Γ 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 (if φ C 2 (U δ )) to assume tat (6.2) olds wit q =1. Consider te space of all continuous piecewise polynomial functions of 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, for some w V bulk }, V 2, := { v C(Ω 2, ): v = w Ω2, 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 [7, 18, 24, 25]. Te trace space V Γ, is introduced in [29]. For representation of functions in tese trace spaces we use te standard nodal basis functions in te space V bulk.intespacev Γ, tese functions do not yield a basis, but form only a frame. For te case k = 1 tis linear algebra issue is studied in [27]. 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. In tis formulation we use te transformed quantities as in (3.6), but wit Ω i, Γ replaced by Ω i, and Γ, 1 respectively. For example, on Ω i, we use te transformed viscosity ν i := k i,a ν i,witν i te dimensionless viscosity as in (2.8). Similarly, te transformed velocity field w [H 1 (Ω 1, Ω 2, )] 3 H 1 (Γ ) 3 is obtained 1 after te transformation w := k i,a w on Ω i,, i =1, 2, and w := w/( k 1,a + k 2,a )onγ,witw te dimensionless smoot velocity vector field as in (2.8). As in (3.7) we omit te tilde notation in te transformed quantities. In te derivation of te skew-symmetry result (4.3) weusedtatw n =0oldsonΓ. Te property w n =0, owever, does not necessarily old on Γ. Skew-symmetry of te convection terms in a (, ) is enforced by using te skew-symmetric forms in te square brackets above. Let g L 2 (Γ ), f L 2 (Ω) be given and satisfy K(f, 1) Ω +(g, 1) Γ =0. (6.8)

13 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1315 As discrete gauge condition we introduce, cf. (4.1), ( K(1 + r)(u, 1) Ω1, + K 1+ 1 ) (u, 1) Ω2, +(v, 1) Γ =0, r 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 need te following assumption. Assumption 1. We assume tat te Poincaré Friedric s constants in (4.5) (4.8) are bounded uniformly in if Ω i is replaced by Ω i, and Γ by Γ. We did not verify tis assumption, but claim tat is a reasonable conjecture, as explained in Remark 7.2 below. In te finite element space we use te 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 ). Using te Assumption 1 we can repeat te arguments as in te proof of Teorem 4.2 and Lemmas 4.3, and4.4, 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], tereexistsα suc tat inf sup (u,v) Ṽ (η,ζ) V,α 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 for all (u, v), (η, ζ) H 1 (Ω 1, Ω 2, ) H 1 (Γ ). a ((u, v); (η, ζ)) (u, v) V (η, ζ) V C st > 0, (6.11) a ((u, v); (η, ζ)) c (u, v) V (η, ζ) V (6.12) 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 aprioriestimate (u,v ) V olds. Te constants C st and c are independent of. C 1 st (f,g ) V c( f Ω + g Γ )

14 1316 S. GROSS ET AL. 7. Error analysis For te error analysis we assume tat te level set function, wic caracterizes te interface Γ,assmootness φ C q+1 (U δ ), wit q 1. Tis implies tat te estimate (6.2) olds. In te analyis below, we also need te smootness assumption Γ C k+1,werek 1iste degree of te polynomials used in te finite element space, cf. (6.6). Concerning te velocity field w we need tat te original (unscaled) velocity w is sufficiently smoot, w H 1, (Ω). In te remainder of tis section and in Section 8 we assume tat tese smootness requirements are satisfied. Te smootness properties of Γ imply tat 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.tusfor x U δ,dist(x,γ)= d(x). Under tese conditions, for δ>0sufficiently small, but independent of, tereis 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 [11], Section 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): Γ = { (s,γ (s)) : s Γ }. From (6.5) it follows tat wit a constant c independent of s Γ Bijective mapping Ω i, Ω i γ (s) =dist ( s + γ (s)n(s),γ ) dist(γ,γ) c q+1, (7.2) 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 of [32]. For te analysis we need a tubular neigborood U δ,wit aradiusδ tat depends on. We define δ := c, witaconstantc>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 (cf. Fig. 1) 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 γe (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), witφ suc tat (6.2) olds. Te mapping Φ as te smootness properties Φ ( H 1, (Ω) ) 3, Φ ( H 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)

15 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1317 Γ Φ = I U δ Φ = I O() Γ Figure 1. Illustration to te construction of Φ : Φ gradually stretces U δ (grey region) along normal directions to Γ suc tat Φ (Γ )=Γ. 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) Proof. Te discrete surface Γ is te grap of γ (x), x Γ. Terefore, γ (x) satisfies φ (x + γ (x)n(x)) = 0, x Γ. (7.7) Since φ is continuous, piecewise polynomial and n(x) C q (U δ ) 3, q 1, te properties of an implicit function imply tat γ is continuous, piecewise C 1 and ence γ H 1, (Γ ). Te smootness properties of Φ now follow from te construction. 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) =0and(6.2) toget: Γ φ (x ) Γ (φ (x ) φ(x )) + Γ (φ(x ) φ(x)) c q. From tis and (6.5) weinfer γ L (Γ ) + Γ γ L (Γ ) c q+1. (7.8) Te following surface area transformation property can be found in, e.g., [10, 11]: 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.NotetatΦ (x) = p(x) onγ olds. From (6.2) weget 1 μ L (Γ ) c q+1. Hence, te result in (7.6) olds. For te term l(x) := δ2 d(x)2,witx δ 2 Ūδ γe (x)2,usedin(7.3) weave l L (U δ ) c and l L (U δ ) c 1.Usingtese estimates and (7.1), (7.8) weobtain id Φ L (Ω) c q+1 and I DΦ L (Ω) c q.tisproves(7.4). Te result in (7.5) immediately follows from (7.4).

16 1318 S. GROSS ET AL. Remark 7.2. From te construction of te mapping and te bounds (7.4), (7.5) it follows tat u u Φ is a omeomorpism between L 2 (Ω i ) H 1 (Ω i ) L 2 (Γ ) H 1 (Γ )andl 2 (Ω i, ) H 1 (Ω i, ) L 2 (Γ ) H 1 (Γ ) wit continuity constants uniformly bounded wit respect to. Since te proof of te Poincaré Friedric s inequalities (4.5) (4.8) is based on te compactness of te embeddings L 2 (Ω i ) H 1 (Ω i ) L 2 (Γ ) H 1 (Γ ), we conjecture tat using tis omeomorpism uniform Poincaré Friedric s estimates as in (4.5) (4.8), wit Ω i and Γ replaced by Ω i, and Γ,old,cf. Assumption 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.notetatu Φ defines an extension to Ω i,.tisextension,owever, as smootness H 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 7.3 and is more regular. As mentioned above, we make te smootness assumption Γ C k+1,werek 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 Tm. 5.4 in [36]). Tis operator satisfies E i u H m (R 3 ) c u H m (Ω i) u H k+1 (Ω i ), m =0,...,k+1, i =1, 2. (7.9) 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, E 2 (u Ω2 ) in Ω 2,. (7.10) Te next lemma quantifies in wic sense tis function u e is close to u Φ. Lemma 7.3. Te following estimates old for i =1, 2: for all u H 2 (Ω i ). u Φ u e Ωi, c q+1 u H 1 (Ω i), (7.11) ( u) Φ u e Ωi, c q+1 u H 2 (Ω i), (7.12) u Φ u e Γ c q+1 u H 2 (Ω i), (7.13) Proof. Witout loss of generality we consider i =1.Notetatu 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 =idandu = 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. (7.14) Γ δ 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.weuseteidentity (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) (7.15)

17 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1319 Due to (7.14), (7.15), te Caucy inequality and φ c 0 > 0onU δ,weget u 1 Φ u 1 2 Ω 1, γ c p s (r) r Γ δ γ c q+1 Γ δ ps(r) r r+c q+1 u 1 (s,t) 2 dt dr ds r c q+1 u 1 (s,t) 2 dt dr ds. (7.16) 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 and using tis in (7.16) yields u 1 (s,t) 2 dtdr c χ [ c q+1,c q+1 ](r t)g(t)dtdr q+1 γ c χ [ c q+1,c q+1 ] L 1 (R) g L 1 (R) c q+1 +c δ c q+1 u 1 (s,t) 2 dt, γ u 1 Φ u 1 2 Ω 1, c 2q+2 +c q+1 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 ) c2q+2 u 2 H 1 (Ω. 1) For deriving te estimate (7.12)wenotetat( 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). We continue to work in local coordinates and estimate te surface integral on te left-and side of (7.13) using te Caucy s inequality and γ L (Γ ) c q+1 : u 1 Φ u 1 2 Γ = = Γ 0 c q+1 Γ ( γ Γ (u 1 u 1 Φ 1 ) 2 r u 1 (s,t)dt μ 1 γ 0 )2 (s, 0)μ 1 ds = ds c Γ Γ [u 1 (s, 0) u 1 (s,γ (s))] 2 μ 1 ds γ γ u 1 (s,t) 2 dt ds c q+1 u 1 2 H 1 (U q+1 ). 0 u 1 (s,t) 2 dt ds Here U q+1 is a tubular neigborood of Γ of widt O( q+1 ) suc tat Γ U q+1. Now we apply te following result, proven in Lemma 4.10 in [16]: w 2 U q+1 cq+1 w 2 H 1 (R 3 ) for all w H 1 (R 3 ). We let w = u 1 (componentwise) and use E 1 (u Ω1 ) H2 (R 3 ) c u H2 (Ω 1) to prove (7.13) Approximate Galerkin ortogonality Due to te geometric errors, i.e., approximation of Ω i by Ω i, and of Γ by Γ, tere is a so-called variational crime and only an approximate Galerkin ortogonality relation olds. In tis section we derive bounds for te

18 1320 S. GROSS ET AL. deviation from ortogonality. Te analysis is rater tecnical but te approac is similar to related analyses from te literature, e.g., [4, 10, 11]. 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.Weuse(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 ) (7.17) +(f,η Φ 1 ) Ω +(g, ζ Φ 1 ) Γ (f,η) Ω (g,ζ) Γ. (7.18) 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). (7.19) To andle te tangential gradient, a more subtle approac is required because one as to relate te tangential gradient Γ to Γ.Letn (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., [11]: for w H 1 (Γ )itolds 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) (7.20) From te construction of te bijection Φ : Γ Γ it follows tat (ζ Φ 1 )e (y) =ζ(y) oldsforally Γ. Application of (7.20) yields an interface analogon of te relation (7.19): Γ (ζ Φ 1 )(x) =B(y) Γ ζ(y), x Γ, y = Φ 1 (x) Γ. (7.21) 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, we again make use of te result in Lemma 4.10 in [16]: w Uδ Ω i c 1 2 w H1 (Ω i) for all w H 1 (Ω i ), (7.22) were c depends only on Ω i. Using properties of Φ, i.e. Φ (U δ Ω i, )=U δ Ω i,and(7.19) wetusget, 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 H 1 (Ω i, ), (7.23) 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 ). For te rigt-and side in te discrete problem (6.10) weset f := J f Φ, g := μ g Φ, (7.24) wit J := det(dφ )andμ as in (7.6). We make tis coice, because ten te consistency terms in (7.18) vanis and te approximate Galerkin relation takes te form a (U e U ; Θ )=F (Θ ) for all Θ V. (7.25) Te coice (7.24) requires explicit knowledge of te transformation Φ, wic may not be practical. We comment on alternatives in Remark 7.7. In Lemma 7.4 we derive a bound for te functional F.

19 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1321 Lemma 7.4. For m =0and m =1te following estimate olds: F (Θ) = a (U e ; Θ) a(u; Θ Φ 1 ) c q+m( ) u H2 (Ω 1 Ω 2) + v H1 (Γ ))( η H 1+m (Ω 1, Ω 2, ) + ζ H1 (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 )andθ =(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 (7.19) (7.21) 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 ) (7.26) 2 ] = [(ν i u e, η) Ωi, (ν i u Φ,J (DΦ ) T η) Ωi, (7.27) 2 1 [ + (w u e,η) Ωi, ((w u) Φ,J η) Ωi, (7.28) 2 (w η, u e ) Ωi, + ( (w Φ ) (DΦ ) T ) ] η, J u Φ (7.29) Ω i, + ν Γ ( Γ v e, Γ ζ) Γ ν Γ (μ B T B Γ v e, Γ ζ) Γ (7.30) + 1 [ (w Γ v e,ζ) Γ (μ (w Φ ) B Γ v e,ζ) Γ (7.31) 2 ] (w Γ ζ,v e ) Γ +(μ (w Φ ) B Γ ζ,v e ) Γ (7.32) 2 ] + [(u e i q iv e,η i Kζ) Γ ((u i q i v) Φ,μ (η i Kζ)) Γ. (7.33) 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 (7.27). Using te estimates derived in te Lemmas 7.1, 7.3 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 + Ωi, q u e H 1 (Ω Ωi,] i,) η (7.34) c q u H2 (Ω 1 Ω 2) η H1 (Ω 1, Ω 2, ). (7.35) If η H 2 (Ω 1, Ω 2, ) we can modify te estimate (7.34) as follows. Note tat J (DΦ ) T = I on Ω i, \ U δ. Hence, using (7.23) we get ( 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, ). (7.36)

20 1322 S. GROSS ET AL. Tus, instead of (7.35) we ten obtain te upper bound c q+1 u H2 (Ω 1 Ω 2) η H2 (Ω 1, Ω 2, ). For te bulk convection term in (7.28) and(7.29) weget 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 H 1, (Ω). Using tis, te relation (7.15) 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 H2 (Ω i), were in te last step we used results from te Lemmas 7.1, 7.3. Te oter tree terms can be estimated by using 1 J L (Ω) c q, I (DΦ ) T L (Ω) c q and u e u Φ Ωi, c q+1 u H 1 (Ω i). Tusweget 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 (7.36) and obtain te following upper bound for te bulk convection term: c q+1 u H2 (Ω 1 Ω 2) η H2 (Ω 1, Ω 2, ). For te surface diffusion term in (7.30) 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 = ν Γ (A Γ v e, Γ ζ) Γ A L (Γ ) Γ v e Γ 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 (7.31) (7.32) we introduce w := P (w μ B T (w Φ )). Using te results in Lemmas 7.1, 7.3 and Pw = w it follows tat 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.

21 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1323 Using 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) weget P P L (Γ ) c q. Hence, for te surface convection term in (7.31) weobtain (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 (7.32) can be bounded in te same way. Finally we consider te adsorption-desorption term in (7.33). Using te results in Lemmas 7.1, 7.3 we get 2 (u e i q iv 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 H 2 (Ω i) + v Γ ηi Γ + ζ Γ. c q+1 ( u H2 (Ω 1 Ω 2) + v Γ )( η H1 (Ω 1, Ω 2, ) + ζ Γ ). Combining tese estimates for te terms in (7.27) (7.33) completes te proof. From te arguments in te proof above one easily sees tat te bounds derived in Lemma 7.4 also old if in a ( ; ) anda( ; ) te arguments are intercanged. Tis proves te result in te following lemma, tat we need in te L 2 -error analysis. Lemma 7.5. Te following estimate olds: a (Θ; U e ) a(θ Φ 1 ; U) c q( ) η H1 (Ω 1, Ω 2, ) + ζ H1 (Γ ))( u H2 (Ω 1 Ω 2) + v H1 (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 ). As an immediate corollary of Lemma 7.4, te definition of F and te regularity estimate in Teorem 4.5 we obtain te following result. Lemma 7.6. Assume tat te solution (u, v) of (4.2) as smootness u H 2 (Ω 1 Ω 2 ), v H 2 (Γ ). Form =0 and m =1te following olds: F (Θ) c q+m )( ) ( f Ω + g Γ η H 1+m (Ω 1, Ω 2, ) + ζ H1 (Γ ) (7.37) for all Θ =(η, ζ) H 1+m (Ω 1, Ω 2, ) H 1 (Γ ). Remark 7.7. Te consistency estimate in Lemma 7.6 is proved for te particular coice of te finite element problem rigt-and side as in (7.24). Tis coice simplifies te analysis because te terms in (7.18) vanis. Similar estimates, owever, can be proved if f and g are cosen as generic smoot extensions of f H 1 (Ω 1 Ω 2 ) and g L 2 (Γ ). For example, one may set f = f e, g = g e Γ c f,werec f is suc tat mean value condition is satisfied. In tis case, te following estimate olds for te terms in (7.18): (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

22 1324 S. GROSS ET AL. for all (η, ζ) H 1 (Ω 1, Ω 2, ) H 1 (Γ ). A proof of tis estimate is given in te preprint version of tis paper [21]. Te discretization error bounds in (7.40) and(8.3) old for tis alternative rigt-and side coice, provided f Ω is replaced by f H1 (Ω 1 Ω 2) 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,ue 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.leti 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,. 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. Here te interpolation operator I bulk is applied only on te tetraedra tat are intersected by Γ. Interpolation error bounds for tis operator are known in te literature, see, e.g., Teorem 4.2 in [34]: 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 7.8. 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) + v H k+1 (Γ )) + c q ( f Ω + g Γ ), (7.40) 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 C 1 st = Cst 1 ( c sup Θ V a (I U e U ; Θ ) sup Θ V Θ V ( a (I U e U e ; Θ ) + a (U e U ; Θ ) Θ V Θ V ) I U e U e F (Θ ) V + sup Θ V Θ V c k ( u H k+1 (Ω) + v H k+1 (Γ ))+c q ( f Ω + g Γ ). (7.41) Te desired result now follows by a triangle inequality and applying te interpolation error estimates (7.38), (7.39) oncemore. )

23 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 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 L 2 /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 L 2 (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) as te unique solution W =(w, z) H 2 (Ω 1 Ω 2 ) H 2 (Γ ), satisfying wit a constant c independent of Ẽ. w H2 (Ω 1 Ω 2) + z H2 (Γ ) c Ẽ L2 (Ω) L 2 (Γ ), (8.2) Teorem 8.1. Let te assumptions in Teorems 7.8 and 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 ( f Ω + g Γ ), (8.3) 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 ) We apply te standard duality argument and tus obtain: = (U e U γ opt (q 1,q 2, 1)) Φ 1 J 1 2 L 2 (Ω) L 2 (Γ ) c (U e U γ opt (q 1,q 2, 1)) Φ 1 L 2 (Ω) L 2 (Γ ) = c (U e U ) Φ 1 γ opt(q 1,q 2, 1) L 2 (Ω) L 2 (Γ ) = c E L 2 /R c Ẽ L 2 (Ω) L 2 (Γ ). 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 Lemmas 7.5 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 L 2 (Ω) L 2 (Γ )

24 1326 S. GROSS ET AL. 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 H2 (Ω 1 Ω 2) + z H2 (Γ )) c U e U V Ẽ L 2 (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 Ω + g Γ I W e W e V c q+1( ) f Ω + g Γ Ẽ L 2 (Ω) L 2 (Γ ). For te fourt term we use Lemma 7.6 wit m =1and(8.2): F (W e ) c q+1( ) f Ω + g Γ Ẽ L 2 (Ω) L 2 (Γ ). Combining tese results and using te bound for U e U V of Teorem 7.8 completes te proof. 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., divw =0 in Ω and w n =0onΓ. On some parts of te boundary Ω te velocity field w is pointing inwards te domain, i.e., (3.2) does not old. Hence, tere are convective fluxes on Ω and tus a Neumann boundary condition as in (3.3) is not natural. 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 =1and 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). (9.1) 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) and(6.7) witk =1,i.e., V bulk consists of piecewise linears on T. For te representation of functions in te trace finite element spaces V Γ, and V i,, cf. (6.7), we use te standard nodal basis functions in te bulk space V bulk.forv Γ, and V i, only tose basis functions are used, te support of wic as a nonzero intersection wit Γ and Ω 1,, respectively. Hence, te nodal finite element functions used for represented functions from te trace space V Γ, are a subset of te basis functions tat are used for representing functions from V i,. Te resulting coupled linear system is iteratively solved by a Generalized Conjugate Residual (GCR) metod using a block diagonal preconditioner, were te bulk and interface systems are preconditioned by te symmetric Gauss-Seidel metod. Te numerical solution u,v after 2 grid refinements and te resulting interface approximation Γ are sown in Figure 2.

25 A TRACE FINITE ELEMENT METHOD FOR A CLASS OF COUPLED BULK-INTERFACE TRANSPORT PROBLEMS 1327 Figure 2. Numerical solutions v on Γ and u visualized on a cut plane z = 0 for refinement level 2. Table 1. L 2 and H 1 errors for bulk concentration u on different refinement levels. #Ref. u u L 2 (Ω) Order u u H 1 (Ω 1, Ω 2, ) Order E E E E E E E E E E Table 2. L 2 and H 1 errors for interface concentration v and iteration numbers on different refinement levels. # Ref. # Iter. v v L 2 (Γ ) Order v v H 1 (Γ ) Order E E E E E E E E E E Te L 2 and H 1 errors for te bulk and interface concentration are given in Tables 1 and 2. For computing te discretization error on te discrete domains Ω i, and Γ we use u Ωi, := u i, i =1, 2, v Γ := v, witu i and v as in (9.1). In te column order we give te estimated p in te error beavior ansatz c p. As expected, first order convergence is obtained for te H 1 errors of bulk and interface concentration, cf. Teorem 7.8. Te respective L 2 errors are of second order, wic confirms te teoretical findings in Teorem 8.1. Te number of GCR iterations to reac a residual norm below are reported in Table 2. Altoug, due to te small areas in certain cut elements, te condition numbers of te stiffness matrices can be extremely large, we do not observe a strong deterioration in te iteration numbers of te preconditioned Krylov solver. Te iteration numbers in Table 2 sow a rater regular 1 growt beavior.