INF-SUP STABILITY OF GEOMETRICALLY UNFITTED STOKES FINITE ELEMENTS

Transcription

1 IN-SUP SABILIY O GEOMERICALLY UNIED SOKES INIE ELEMENS JOHNNY GUZMÁN AND MAXIM OLSHANSKII Abstract. e paper sows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations wic are not fitted to a given smoot or polygonal domain. e property implies stability and optimal error estimates for a class of unfitted finite element metods for te Stokes and Stokes interface problems, suc as Nitsce-XEM or cutem. e error analysis is presented for te Stokes problem. All assumptions made in te paper are satisfied once te background mes is sape-regular and fine enoug. XEM, cutem, Stokes problem, LBB condition, finite elements [2010]65N30, 65N12, 76D07, 65N85 1. Introduction Unfitted finite element (E) metods incorporate geometrical information about te domain were te problem is posed witout fitting te mes to lower dimensional structures suc as pysical boundaries or internal interfaces. is is opposite to fitted desretizations suc as (isoparametric) traditional E and isogeometric analysis. e advantage of te unfitted approac is a relative ease of andling prorogating interface and geometries defined implicitly, i.e. wen a surface parametrization is not readily available. Prominent classes of unfitted E are given by XEM [22] and cutem [13] also known as Nitsce-XEM metods or trace E in te case of embedded surfaces. In cutem, one considers background mes and E spaces not tailored to te problem geometry, wile numerical integration in E bilinear forms is performed over te pysical domains Ω and/or Ω wic cut troug te background mes in an arbitrary way. Effectively, tis leads to traces of te ambient E spaces on te pysical domain, were te original problem is posed, and integration over arbitrary cut simplexes. e idea of unfitted E can be followed back at least to te works of Barrett and Elliott [2, 3, 4], were a cut E metod was studied for te planar elliptic problems and elliptic interface problems. Over te last decades, unfitted E metods emerge in a powerful discretization approac tat as been applying to te wide range of problems, including problems wit interfaces, fluid equations, PDEs posed on surfaces, surface-bulk coupled problems, equations posed on evolving domains, etc, see, e.g., [5, 10, 14, 19, 20, 24, 30, 31, 35, 36, 38]. Among important enabling tecniques used in unfitted EM are te Nitsce metod for enforcing essential boundary and interface conditions [26], gost penalty stabilization [12], and te properties of trace E spaces on embedded surfaces [34]. We note tat many of tese developments are accomplised wit rigorous stability and convergence analysis of te unfitted E, wic demonstrate bot utility and reliability of te approac. Partially supported by NS troug te Division of Matematical Sciences grant

2 2 J. GUZMÁN AND M.A. OLSHANSKII One important application of unfitted E metods is te numerical simulation of fluid problems wit evolving interfaces as occurs in fluid-structure interaction problems and two-pase flows. If te fluid is treated as incompressible, ten te prototypical model suitable for numerical analysis is te stationary (interface) Stokes problem. is paper addresses te question of numerical stability of a certain class of geometrically unfitted Stokes finite elements. Unfitted E metods for te Stokes problem received recently a closer attention in te literature. In [15] optimal order convergence results were sown for te unfitted inf-sup stable pressure-velocity 2D E wit Nitsce treatment of te boundary conditions and gost-penalty stabilization for triangles cut by Ω. is analysis was extended to te Stokes interface problem and P 1 isop 2 P 1 elements in [27]. Optimal order convergence in te energy norm for P bubble 1 P 1 unfitted E using sligtly different pressure stabilization over cut triangles was sown for te Stokes interface problem in [16]. In [28] te P 2 P 1 elements were analysed for te Stokes interface problem, wen te pressure element is enriced to allow for te jump over unfitted interface, wile te velocity element is globally continuous. Globally stabilized unfitted Stokes finite elements, P 1 P 1 and P 1 P1 disc, were studied in [16, 33, 40]. Oter related work on geometrically unfitted E for te Stokes problem can be found in [1, 25, 29, 37]. e analysis of inf-sup stable unfitted Stokes elements, owever, is not a straigtforward extension of te standard results for saddle point problems. In particular, it essentially relies on a certain uniform stability property of te finite element velocity-pressure pair. is property can be found as an assumption (explicitly or implicitly made) in [15, 16, 27]. Loosely speaking te following condition on E velocity-pressure spaces is required: Assume a family of sape-regular triangulations { } >0 of R 2, and let Ω R 2 be a bounded domain wit smoot boundary. Consider te family of domains Ω, were eac Ω consists of all triangles from wic are strictly inside Ω. en one requires tat te LBB constants (optimal constants from te E pressure-velocity inf-sup stability condition) for te domains Ω are uniformly in bounded away from zero. In te same way te property is formulated in 3D. In section 2 we discuss wat sort of difficulties one encounters trying to employ common tecniques to verify tis property. Recently, in [28] te required uniform stability condition was proved for P 2 P 1, te lowest order aylor-hood element. In tis paper, we sow te uniform inf-sup stability result for a wider class of elements, including P k+1 P k and P k+1 Pk 1 disc for k 1 and several oter elements, see Section 6. ollowing [28] we employ te argument from [39]. is elps us to formulate more local condition on E spaces wic are sufficient for te uniform inf-sup stability, but easier to ceck. urter we sow tat tis condition is satisfied by a number of popular LBB-stable E pairs. e paper also applies te acquired uniform stability result to sow te optimal order error estimates of te unfitted E metod for te Stokes problem. e analysis improves over te available in te literature by eliminating certain assumptions on ow te surface Ω (or an interface in te two-pase fluid case) intersects te background mes. Instead, we impose certain assumptions, wic are always satisfied once te background mes is sape-regular and te mes size is not too coarse wit respect to te problem geometry, see section 4 and for te assumptions and furter discussion in Remark 1. e remainder of te paper is organized as follows. In section 2 we define te problem of interest and formulate te central question we address in tis paper about uniform inf-sup stability. Section 3 collects necessary preliminaries and auxiliary results. Here we present te unfitted finite element metod for te Stokes problem. urter we formulate assumptions

3 IN-SUP SABILIY O UNIED SOKES ELEMENS 3 sufficient for te main uniform stability result. Section 5 sows ow te well posedness and optimal order error estimates for te unfitted E metod follow from our assumptions. In section 6 we give te examples of velocity and pressure spaces satisfying te assumptions. 2. Problem setting Consider te Stokes problem posed on a bounded smoot domain Ω R d, d = 2, 3, u + p = f in Ω, (2.1) div u = 0 in Ω, u = 0 on Ω. Vector function u [ H0 1(Ω)] d and p L 2 0 (Ω) := {q L 2 (Ω) : Ω q dx = 0} are te weak solution to (2.1), aving te pysical meaning of fluid velocity and normalized kinematic pressure. Assume tere is a domain S Ω, and we let { } >0 be an admissible family of triangulations of S. We are interested in a finite element metod for (2.1) using spaces of piecewise polynomial functions wit respect to. Note tat we make no assumption on ow Ω overlaps wit, i.e. Ω may cut troug tetraedra or triangles from in an arbitrary way. In te next section we give details of te finite element metod. Now we formulate te stability condition, wic is crucial for te analysis of tis metod (and likely many oter unfitted E metods for (2.1)). Consider te set of all strictly internal simplexes and define te corresponding subdomain of Ω: i := { : Ω}, Ω i := Int ( or background finite element velocity and pressure spaces V and Q, consider teir restrictions on Ω i, tat is V i = V [ H0 1(Ωi )] d and Q i = Q L 2 0 (Ωi ), and define θ := inf sup q Q i v V i We are interested in te following condition: (2.2) 0 < inf < 0 θ, i (div v, q) v H 1 (Ω i ) q L 2 (Ω i ). for some positive 0. Note tat standard arguments based on te Nečas inequality and ortin s projection operator, cf. [9], cannot be applied in a straigtforward way to yield (2.2) for inf-sup stable elements (e.g., for aylor-hood element). or te reference purpose recall te Nečas inequality: ). (2.3) C N (Ω i ) q L 2 (Ω i ) sup v [H 1 0 (Ωi )]d (div v, q) v H 1 (Ω i ) q L 2 0(Ω i ). Since Ω i is Lipscitz for any given, te inequality olds wit some domain dependent constant C N (Ω i ) > 0, see, e.g., [11, 23]. However, we are not aware of a result in te literature wic imply tat C N (Ω ) are uniformly bounded from below by a positive constant independent of. or example, te well-known argument for proving (2.3) is based on te decomposition of a Lipscitz domain into a finite number of strictly star saped domains and applying te result

4 4 J. GUZMÁN AND M.A. OLSHANSKII of Bogovskii [11] in eac of te star domains. However, te number of te star domains in te decomposition of Ω i may infinitely grow for 0 even if Ω is smoot and { } >0 is saperegular, wic would drive te lower bound for C N (Ω i ) to zero. Alternatively, te recent analysis from [7] provides a lower bound for C N (Ω i ) if tere exist diffeomorpisms Φ : Ω i Ω wit uniformly bounded W 1, (Ω i ) norms. We do not see ow to construct suc diffeomorpisms (note tat Ω i is not necessarily a grap of a function in te natural coordinates of Ω). Additional difficulty stems from te observation tat i does not necessarily inerit a macroelement structure tat may possess. is said, we sall look for a different approac to verify (2.2). We end tis section noting tat te finite element metod and te analysis of te paper can be easily extended to te Stokes interface problem, a prototypical model of two-pase incompressible fluid flow. However, we are not adding tese extra details to te present report. 3. inite element metod 3.1. Preliminaries. We adopt te convention tat elements and element edges (also faces in 3D) are open sets. We use over-line symbol to refer to teir closure. or eac simplex, let denote its diameter and define te global parameter of te triangulation by = max. We assume tat is sape regular, i.e. tere exists κ > 0 suc tat for every te radius ρ of its inscribed spere satisfies (3.1) ρ > /κ. e set of elements cutting te interface Γ Ω, and restricted to Ω are also of interest. ey are defined by: Γ := { : Γ }, e := { : i or Γ }. In particular for Γ we denote Γ = Γ. Observe tat te definition of Γ guarantees tat Γ Γ = Γ. Under tese definitions we define te discrete domains ( Ω Γ := Int ( ), Ω e := Int ). Γ e Note tat Ω e = Ωi ΩΓ and tat Ωi Ω Ωe. or tese domains define sets of faces: i Γ e := { : is an interior face of i }, Γ := { : is a face of, Ωe }, e := { : is an interior face of }. Now we can define finite element spaces. A space of continuous functions on Ω e wic are polynomials of degree k on eac e is denoted by W k. e spaces of discontinuous and continuous pressure spaces are given by Q d = {q L2 (Ω e ) : q P kp ( ), e }, Q c = Qd H1 (Ω e ).

5 IN-SUP SABILIY O UNIED SOKES ELEMENS 5 rougout tis paper we will consider eiter Q = Q d (for k p 0) or Q = Q c (for k p 1). We will denote te finite element velocity space by V [H 1 (Ω e )]d, and we will assume ( W k u ) d V ( W) s d for some integer s k u 1. In section 4, we introduce a more tecnical assumption 3 tat our pair of spaces {Q, V } as to satisfy. en, later we give examples of pairs tat satisfy all necessary assumptions. or example, if Q = Q d ten V can be te space of continuous piecewise polynomials of degree k p + 2; and if Q = Q c ten V can be te space of continuous piecewise polynomials of degree k p + 1. We give more examples of spaces satisfying our assumptions in Section inite element metod. We will use te notation (v, w) = Ω vw dx. Introduce te mes-dependent bilinear forms wit and wit a (u, v ) := ( u, v ) + s (u, v ) + j (u, v ) + ηj (u, v ), s (u, v) = {( u n) v + ( v n) u}ds, Γ j (u, v) = 1 u vds, Γ Γ j (u, v) = s Γ l=1 [ ] [ ] 1+2l nu l nv l ds; b (p, v ) := (p, div v ) + r (p, v ), r (p, v) = Γ p v n ds. Here and furter nq l on face denotes te derivative of order l of p in direction n, were n is normal to ; and [φ] denotes te jump of a quantity φ over a face. We can now define te numerical metod : ind u V and p Q suc tat { a (u, v ) + b(p, v ) = (f, v ), (3.2) for all v V, q Q, were J (q, p) = b(q, u ) J (p, q ) = 0, k p Γ l=0 [ ] [ ] 1+2l nq l np l ds. Before proceeding wit te analysis, we briefly discuss te role of different terms in te finite element formulation (3.2). irst note tat all integrals in (3.2) are computed over pysical domains Ω and Γ rater tan computational domain Ω e. e gradient and div-terms appear due to te integration by parts in a standard weak formulation of te Stokes problem. Since finite element velocity trial and test functions do not satisfy omogenous Diriclet conditions

6 6 J. GUZMÁN AND M.A. OLSHANSKII strongly on Γ, te integration by parts brings te s and r terms to te formulation. e Γ ( v n) uds integral in s is added to make formulation symmetric. It vanises for u, te Stokes equations solution. e same is true for te r term in te continuity equation in (3.2). e penalty term j (u, v ) weakly enforces te Diriclet boundary conditions for u, as common for te Nitsce metod, wit a parameter η = O(1). e terms j (u, v ) and J (p, q ) are added for te numerical stability of te metod: we need j (u, v ) to gain control over normal velocity derivatives in s, and we need J for pressure stability over cut triangles. In practice, bot j and J can be scaled by additional stabilization parameters of O(1) order; we omit tis detail ere. We note tat te unfitted EM analysed in te paper is closely related to te extended finite element metod (XEM). Indeed, te trace space of background finite element functions on te domain Ω can be alternatively described as a E space spanned over nodal sape functions from Ω and furter enriced by certain degrees of freedom tailored to Ω. Hence te results of tis paper can be as well considered as te analysis of a certain class of XEM metods for te Stokes problem. Next section proves te key result for getting numerical stability and optimal order error estimates for te unfitted finite element metod (3.2). 4. Stability We need to define some norms and semi-norms. irst we define te mes-dependent norm for te velocity u 2 V = u 2 H 1 (Ω) + j (u, u) + j (u, u). Note tat due to te boundary term j, te functional u V defines a norm on V equivalent to te H 1 (Ω) norm, u H 1 (Ω) u V 1 u H 1 (Ω). We need a set of all tetraedra intersected by Γ togeter wit all tetraedra from Ω toucing tose: and also Γ = { : Γ Ω Γ := Int ( or Ω, ΩΓ }, or a generic set of tetraedra denote ω( ) te set of all tetraedra aving at least one vertex in. We need te following assumptions on ow well te geometry is resolved by te mes. Assumption 1. or any Γ we assume tat te set W ( ) = i ω (ω( )) is not empty. Wit any Γ we associate one tetraedra K W ( ), cosen as follows: If i ω( ), ten we pick arbitrary K i ω( ) and set K := K; oterwise K is arbitrary, but fixed tetraedra from W ( ). Note tat K Γ i. Let 1, 2 Γ and 1 sares a face wit 2, ten we assume tat K 2 can be reaced from K 1 by crossing a finite, independent of, number of faces of tetraedra from i. Γ ). Since Γ is smoot, tere exists a finite atlas {(U α, φ α )} of Γ consisting of C 2 -carts. Assumption 2. or eac Γ, we assume tat Γ U α for some α.

7 IN-SUP SABILIY O UNIED SOKES ELEMENS 7 Remark 1. One can ceck tat te assumptions 1 2 are always satisfied if is sufficiently small and te minimal angle condition (3.1) olds. is is an improvement of te available analysis of unfitted finite elements wic commonly imposes a furter restriction on ow interface intersects. In 2D tis extra assumption is formulated as following: Ω does not intersect any edge from e more tan one time, see, e.g. [26]. An analogous restriction was commonly assumed in 3D. One easily builds an example sowing tat tis extra assumption is not necessarily true for arbitrary fine mes and smoot Ω, wile enforcing it by eliminating illegible elements introduces O( 2 ) geometrical error diminising possible benefits of using iger order elements. We do not need tis extra assumption. e assumptions 1 2 also allow local mes refinement. One can sow te following stability result. Lemma 1. or η sufficiently large and 0 for sufficiently small 0, tere exists a mesindependent constant c 0 > 0 suc tat (4.1) c 0 v 2 V a (v, v ) v V. Proof. o sow (4.1) we need te following estimate, see Lemma 5.1 in [32]: or any 1, 2 from Γ saring a face = 1 2 it olds ( m (4.2) q 2 L 2 ( 1 ) c q 2 L 2 ( 2 ) + ) [ ] 2 1+2l nq l ds, q P m ( 1 ) P m ( 2 ), l=0 wit a constant C depending only on te sape regularity of and polynomial degree m. anks to E inverse inequality, (4.2) and Poincare inequality, we ave for any 1, 2 from Γ saring a face = 1 2 te following estimate ( s ) [ ] 2 v 2 L 2 ( 1 ) C 2 v α 2 L 2 ( 1 ) C 2 2 v α 2 L 2 ( 2 ) + 1+2l nv l l=1 (4.3) ( s ) [ ] 2 c v 2 L 2 ( 2 ) + 1+2l nv l, v V, l=1 were we take α = v ds. Note tat tis inequality is found in Proposition 5.1 in [32] in te case V = [W 1 ]2. anks to assumption 1 te estimate (4.3) implies (4.4) v 2 L 2 (Ω e ) C( v 2 L 2 (Ω) + j (v, v ) ). One makes use of te trace inequality: (4.5) v L 2 ( Γ) C( 1 2 v L 2 ( ) v L 2 ( )) Γ, v H1 ( ). wit a constant C independent of v, and ow Γ intersects. Under assumption 2, te result in (4.5) is proved in [17]. Witout any special notice we sall also use te particular case of (4.5), were te norm on te left-and side is replaced by v L 2 ( ). urter, one uses te Caucy-Scwarz inequality, trace inequality (4.5) and te E

8 8 J. GUZMÁN AND M.A. OLSHANSKII inverse inequality to estimate s (v, v ) = ( v n) v ds v L 2 ( Γ ) v L 2 ( Γ ) Γ 1 2η C 2η Γ Γ v 2 L 2 ( Γ ) + η 2 j (v, v ) ( v 2 L 2 ( ) v 2 L 2 ( ) ) + η 2 j (v, v ) Γ C 2η v 2 L 2 (Ω e ) + η 2 j (v, v ). Combining tis wit (4.4) and coosing η sufficiently large, but independent of, proves te lemma. We need to define te scaled semi-norms for te pressure: p 2 H = 2,i 1 p 2 L 2 ( ) + [p] 2 L 2 ( ), p 2 H 1,e i = i 2 p 2 L 2 ( ) + [p] 2 L 2 ( ). e e Assumption 3. Assume tat tere exists a constant β > 0 independent of and only depending on polynomial degree of finite element spaces and te sape regularity of suc tat (div v, q) (4.6) β q H 1 sup q Q,i, v V i v H 1 (Ω i ) were V i = V [ H 1 0 (Ωi )] d. We also need te following extension result. A proof of tis result is given in te appendix. Lemma 2. or every q Q tere exists a E q Q suc and E q = q on Ω i (4.7) E q H 1,e C q H 1,i. Using te degrees of freedom of piecewise linear functions one can sow te following result found in [28]. Lemma 3. or every v W 1 tere exists a unique decomposition (4.8) v = π 1 v + π 2 v, were π i v W 1 for i = 1, 2, π 2v is supported in Ω Γ (4.9) π 2 v = v on Ω Γ and suc tat

9 and (4.10) IN-SUP SABILIY O UNIED SOKES ELEMENS 9 Γ 1 2 π 2 v 2 L 2 ( ) C Γ 1 2 v 2 L 2 ( ). e constant C is independent of v and and only depends on te sape regularity of te mes. In particular, note tat tis implies π 1 v V i. e following lemma sows te LBB stability result for te internal domain Ω i and so proves te desired uniform bound (2.2). Note again tat Ω i is not an O(2 ) approximation of a smoot domain and tere is no uniform in result concerning decomposition of Ω i into a union of a finite number of star-saped domains. e latter is a standard assumption for proving te differential counterpart of tis finite element condition, see, e.g., [23]. is result is te key for te stability and convergence analysis of te unfitted E metod (3.2). or te lowest order aylor-hood element, te proof of te following result is found in [28]. We follow a similar argument, but extend te result so it can be applied to iger order elements in two and tree dimensions. Lemma 4. Suppose Assumption 3 olds. en, tere exists a constant θ > 0 and a constant 0 suc tat for all q Q wit Ω qdx = 0 we ave te following result for i 0 (4.11) θ q L 2 (Ω i ) sup (div v, q). v V i v H 1 (Ω i ) e constant θ > 0 is independent of q and. Proof. Let ψ = E q and let c = 1 Ω Ω E q. ere exists a v [H0 1(Ω)]2 wit te following properties, cf. [11]: (4.12) div v = ψ c on Ω and (4.13) v H 1 (Ω) C ψ c L 2 (Ω). Extend v by zero to all of Ω e. Let v be te Scott-Zang interpolant of v. We will write (v, w) e = Ω vwdx. Wit te elp of (4.12) and te decomposition (4.8), we obtain e (4.14) ψ c 2 L 2 (Ω) = (div v, ψ) = (div π 1v, ψ) e Integration by parts over eac e gives (div(v v ), ψ) e = (v v ) ψdx e + (div(v v ), ψ) e + (div π 2 v, ψ) e + (div v, ψ) e. e [ψ](v v ) nds. We proceed by applying te Caucy-Scwarz inequality, elementwise trace inequality, and te definition of te H,e 1 norm. is gives te bound (div(v v ), ψ) e C 1/2 ( 1 2 v v 2 L 2 ( ) + (v v ) 2 L 2 ( ) ) ψ H 1.,e e

10 10 J. GUZMÁN AND M.A. OLSHANSKII Using te approximation properties of te Scott-Zang interpolant, (4.13) and (4.7), we ave (4.15) (div(v v ), ψ) e C ψ c L 2 (Ω) ψ H 1,e C ψ c L 2 (Ω) q H 1,i. In a similar fasion, but now using inverse E estimates instead of trace inequalities, and recalling tat supp(π 2 v ) Ω Γ, we sow (4.16) 1/2 1/2 1 (div π 2 v, ψ) C π 2 v 2 1 L 2 ( ) ψ H 1 C,e v 2 L 2 ( ) ψ H 1.,e Γ Note te following riedric s type E inequality: 2 2 v 2 L 2 ( ) + 1 v 2 L 2 ( ) C( v 2 L 2 ( ) + 1 v 2 L 2 ( ) ), is a face of. We apply te above inequality elementwise and use v = 0 on Ω e to sow tat (4.17) 1/2 1 v 2 L 2 ( ) C v L 2 (Ω Γ ) C v L 2 (Ω e ) = C v L 2 (Ω). Γ 2 In te last inequality we used te stability of te Scott-Zang interpolant. Hence, using (4.13) we get from (4.16) (4.17) te estimate Γ (4.18) (div π 2 v, ψ) e C ψ c L 2 (Ω) q H 1,i. e last term on te rigt and side of (4.14) we andle as follows: (div π 1 v, ψ) e = (div π 1 v, ψ) L 2 (Ω i ) π 1v H 1 (Ω i ) sup (div w, q). w V i w H 1 (Ω i ) Now we bound π 1 v H 1 (Ω), π 1 v H 1 (Ω) ( π 2 v H 1 ( Ω Γ ) + v H 1 (Ω)). Using inverse estimates, (4.10) and (4.17) we get π 2 v H 1 ( Ω Γ ) C v H 1 (Ω e ) = C v H 1 (Ω) Hence, te stability of te Scott-Zang interpolant and (4.13) imply (4.19) π 1 v H 1 (Ω) C ψ c L 2 (Ω). erefore, we get from (4.15), (4.18), (4.19) and (4.14) te upper bound ( ) (div w, q) ψ c L 2 (Ω) C + q w H 1. H 1 (Ω i ),i Using assumption 3 we get sup w V i (div w, q) (4.20) ψ c L 2 (Ω) C sup. w V i w H 1 (Ω i ) 2

11 inally, note tat IN-SUP SABILIY O UNIED SOKES ELEMENS 11 c L 2 (Ω) Ω 1/2 c = Ω 1/2 Ω ψdx = Ω 1/2 Ω\Ω i ψdx. e last equality olds since ψ = q in Ω i and Ω q dx = 0. After applying Caucy-Scwarz i inequality and using tat Ω \ Ω i 1/2 1/2 we ave tat c L 2 (Ω) C 1/2 ψ L 2 (Ω). Hence, using te triangle inequality in (4.20) and assuming is sufficiently small we ave (div w, q) ψ L 2 (Ω) C sup. w V i w H 1 (Ω i ) We note tat te constant C is independent of and q. e result now follows after noting tat q L 2 (Ω i ) ψ L 2 (Ω) and letting θ = 1 C. Corollary 1. If assumptions 1 3 old true, te following stability condition is satisfied by te b and J forms of te finite element metod (3.2), (4.21) c b q L 2 (Ω) sup v V b (v, q) v V + J 1 2 (q, q), q Q. e constant c b > 0 is independent of q and. Proof. Extending functions from V i by zero on Ωe \Ωi we ave te natural embedding V i V. By te E inverse inequalities we ave for v V i, j (v, v) = s ( ) 2 1+2l nv l ds C v 2 L 2 ( ) ds C v 2 H 1 (Ω i ). Ω i l=1 e i is and j (v, v) = 0 for v V i yield te uniform equivalence v V v H 1 (Ω i ) on v V i. Now, noting tat r (p, v) = 0 for v V i, and employing (4.11) and (4.2) to control q L 2 (Ω i ) and q L 2 (Ω\Ω i ), respectively, proves (4.21). One easy verifies tat a is continuous 5. Well posedness and error estimates a (u, v) C a u V v V u, v V, wit some C a > 0 independent of and te position of Γ. e continuity and coercivity of te a (u, v) form (Lemma 1) and te inf-sup stability of te b (v, q) form (Corollary 1) readily imply te stability for te bilinear form of te finite element metod (3.2) wit respect to te product norm, A (u, p ; v, q) (5.1) C s u, p sup {v,q} V Q v, q {u, p } V Q,

12 12 J. GUZMÁN AND M.A. OLSHANSKII wit some C s > 0 independent of and te position of Γ and A (u, p; v, q) := a (u, v) + b (v, p) b (u, q) + J (p, q), ( ) 1 v, q := v 2 V + q 2 L 2 (Ω) + J 2 (q, q). e proof of (5.1) extends standard arguments, cf., e.g., [21], for J 0. or completeness we sketc te proof ere. or given {u, p } V Q, tanks to (4.21), one can find z V suc tat z V = p L 2 (Ω) and c b p 2 L 2 (Ω) b (z, p ) + J 1 2 (p, p ) p L 2 (Ω) = A (u, p ; z, 0) + a (u, z) + J 1 2 (p, p ) p L 2 (Ω) A (u, p ; z, 0) + C2 a c b u 2 V + c b 4 p 2 L 2 (Ω) + 1 c b J (p, p ) + c b 4 p 2 L 2 (Ω). Combining tis inequality wit we get J (p, p ) + c 0 u 2 V A (u, p ; u, p ), c u, p 2 A (u, p ; u αz, p ), for a suitable α > 0 and a constant c > 0 depending only on c b, C a, and c 0. Inequality (5.1) follows by noting u, p 1 1+α v, q, wit v = u αz, q = p. One verifies tat A is continuous (5.2) A (u, p; v, q) C c u, p v, q {u, p}, {v, q} V Q, wit some C c > 0 independent of and te position of Γ. Note also tat A is symmetric. erefore, by te Babuška-Brezzi teorem te problem (3.2) is well-posed and its solution satisfies te stability bound u, p Cs 1 f V. urter in tis section we assume tat te solution to te Stokes problem is sufficiently smoot, i.e., u H s+1 (Ω) and p H kp+1 (Ω). Since we are assuming tat Γ is smoot, tere exists extensions of u and p wic we also denote by u, p suc tat u H s+1 (S) and p H kp+1 (S). We let I u be te Scott-Zang interpolant of u onto [ W ku ] d. We also let I p be te Scott- Zang interpolant of p in te case Q = Q c and te L2 projection onto discontinuous piecewise polynomials of degree k p if Q = Q d. Applying trace inequality (4.5), standard approximation properties of I, and extension results one obtains te approximation property in te product norm: (5.3) u I u, p I p C min{ku,kp+1} ( u H ku+1 (Ω) + p H kp+1 (Ω) ) + ku s+1 l ku 1 u H l (Ω). l=k u+1 Denote by e u = u u and e p = p p te finite element error functions. Since te metod (3.2) is consistent for u and p sufficiently smoot as stated above, te Galerkin ortogonality

13 olds, IN-SUP SABILIY O UNIED SOKES ELEMENS 13 (5.4) A (e u, e p ; v, q ) = 0, for all v V and q Q. e optimal order error estimate in te energy norm is given in te next teorem. eorem 1. or sufficiently smoot u, p solving (2.1) and u, p solving (3.2), te error estimate olds, u u H 1 (Ω) + p p L 2 (Ω) u u, p p C min{ku,kp+1} ( u H ku+1 (Ω) + p H kp+1 (Ω) ) + s+1 ku l=k u+1 l ku 1 u H l (Ω), wit a constant C independent of and te position of Γ wit respect to te triangulation. Proof. e results follows from te inf-sup stability (5.1), continuity (5.2), Galerkin ortogonality (5.4), and approximation property (5.3) by standard arguments, see, for example, section 2.3 in [21]. Using te Aubin-Nitsce duality argument one sows te optimal order error estimate for te velocity in L 2 (Ω)-norm. Consider te dual adjoint problem. Let w [ H0 1(Ω)] d and r L 2 0 (Ω) be te solution to te problem w r = e u in Ω, (5.5) div w = 0 in Ω, w = 0 on Ω. We assume tat Ω is suc tat (5.5) is H 2 -regular, i.e. for e u [ L 2 (Ω) ] d it olds w [ H 2 (Ω) ] d and r H 1 (Ω) and w H 2 (Ω) + r H 1 (Ω) C(Ω) e u L 2 (Ω). By te standard arguments (section 2.3 in [21]) te results in (5.1), (5.2), (5.4), (5.3), and te above regularity assumption lead to te following teorem. eorem 2. or sufficiently smoot u, p solving (2.1) and u, p solving (3.2), te error estimate olds, u u L 2 (Ω) C u u, p p, wit a constant C independent of and te position of Γ wit respect to te triangulation. 6. Example of spaces satisfying Assumption 3 [ 6.1. Generalized aylor-hood elements. Consider Q = Q c := W k and V = W k+1 k 1. In tis case, te proof of estimate (4.6) from Assumption 3 is given in section 8 of [9] for d = 2 (two-dimensional case). In tree-dimensional case and k = 1, te result can be found in Lemma 4.23 in [21]. Below we extend te proof for all k 1 in 3D. We require eac i to ave at least tree edges in te interior of Ω i. Note tat te proof in [9] for d = 2 does not ] d,

14 14 J. GUZMÁN AND M.A. OLSHANSKII need a similar assumption. or any edge from te set of internal edges of i, E Ei, we denote a unit tangent vector be t E (any of two, but fixed), x E is te midpoint of E, and ω(e) is a set of tetraedra saring E. Also we denote by φ E W 2(Ωi ) a piecewise quadratic function suc tat φ E (x E ) = 1 and φ E (x) = 0, were x is any vertex or a midpoint of any oter edge from E i. or p Q we set v(x) = E E i 2 Eφ E (x) [t E p(x)]t E. Since te pressure tangential derivative t E p is continuous across faces tat contain E, it is easy to see tat v V i. We compute div v p dx = v p dx = 2 E φ E t E p 2 dx c 2 E t E p 2 dx. Ω i Ω i E E i ω(e) E E i ω(e) e constant c > 0 in te last inequality depends only on te polynomial degree k and sape regularity condition (3.1). rom te condition (3.1) we also infer E for ω(e). is gives after rearranging terms te estimate div v p dx c t E p 2 dx c p 2 dx = c p 2 H,i. 1 Ω i i E Ω i 2 or te last inequality we used te assumption tat at least tree edges of te tetraedra are internal and we apply te sape regularity condition one more time. Due to te finite element inverse inequalities and te obvious estimate φ E + φ E c on, wit E, we ave v 2 H 1 (Ω i ) c v 2 dx ( φ E 2 p 2 + φ E 2 2 p 2 ) dx is sows (4.6). Ω i i E Ω i 4 i i 2 2 ( p p 2 ) dx c p 2 H 1,i Bercovier-Pironneau element. is is a ceap version of te lowest order aylor-hood element. In 2D te element in defined in [6], te 3D version can be found, e.g., in [21]. o define te velocity space, one refines eac triangle of by connecting midpoints on te edges in 2D, wile in 3D one divides a tetraedron into six tetraedra by te same procedure. en te velocity space consists of piecewise linear continuous function wit respect to te refined triangulation, V = [ d, W/2] 1 and Q = Q c := W 1. or tis element, one sows (4.6) following te lines of te proof of eorem 8.1 in [9] for k = 1 in 2D or te arguments from te section 6.1 wit obvious modifications: or example, in te 3D case one substitutes edge-bubbles φ E by tere P1isoP2 counterparts P k+2 Pk disc (for d = 2) and P k+3 Pk disc (for d = 3) elements. We only consider te two dimensional case d = 2 as te case d = 3 is similar. We let Q = Q d be te space of

15 IN-SUP SABILIY O UNIED SOKES ELEMENS 15 [ piecewise polynomial functions of degree k and let V = W k+2 in tis case, take q Q d. We can coose v V i suc tat v wdx = 2 q w for all w P k 1 ( ), and for all i. Also for every interior edges E of Ωi rv n + dx = E r(q + n + + q n ) n + E E ] d. o sow Assumption 3 olds for all r P k (E), were E = + and +, i. Also, n± is te outward pointing unit normal of ±. o pin down v V i we make v vanis on all vertices and ave tangential components vanis on all edges. inally, we make v 0 on Ω i. Using elementwise integration by parts, we get div v q dx = v qdx + qv nds. Ω i rom te construction of v, we see tat div v q dx = Ω i i i 2 q 2 L 2 ( ) + E E i [q] 2 L 2 ( ) = q 2 H i It is not difficult to sow, using a scaling argument tat v H 1 (Ω i ) C q H i. rom tis we see tat Assumption 3 olds Bernardi-Raugel element. In a similar fasion we can sow tat te Bernardi-Raugel spaces satisfy Assumption 3. e space of Bernardi-Raugel elements consists of piecewise constant pressure and for te velocity one takes P 1 continuous functions enriced wit te normal components of te velocity as a degree of freedom at barycentre face nodes [8] Mini-Element. Wit k p = 1, Q = Q c and V = [W 1]2 + {v : v b c, were c [P 0 ( )] 2, for all e}. Here b is te cubic bubble. o prove (4.6) we consider an arbitrary q Q q. A simple argument gives 2 q 2 L 2 ( ) C 2 b (q q) 2 L 2 ( ). i i Integration by parts gives b q 2 L 2 ( ) = div(b q)qdx. If we define w V i following way w := 2 b q ten we ave 2 b q 2 L 2 ( ) = div w qdx. Hence, we get i i 2 q 2 L 2 ( ) C sup (div v, q) w v V i v H 1 H 1 (Ω i ) (Ω i ). Ω i in te

16 16 J. GUZMÁN AND M.A. OLSHANSKII Now, using Poincare s inequality w 2 H 1 (Ω i ) C w 2 L 2 ( ) C 4 b 2 L ( ) q 2 L 2 ( ). i Since 2 b 2 L ( ) C, we get w 2 H 1 (Ω i ) i 2 q 2 L 2 ( ). i e result now follows Generalized conforming Crouzeix-Raviart element. is element is defined by Q = Q d wit k p = k 1 for d = 2 or k 2 for d = 3, we define te velocity space to be V = [W k+1 ] d + {v : v b P k ( ), for all e}. were b is cubic bubble in two dimensions or quartic bubble in tree dimensions. is Pk+1 bubble Pk disc spaces was first introduced in [18]. e proof of (4.6) in tis case will be similar to tat of mini-elment. We leave te details to te reader. References [1] S. Amdouni, K. Mansouri, Y. Renard, M. Arfaoui, and M. Moaker. Numerical convergence and stability of mixed formulation wit X-EM cut-off. European Journal of Computational Mecanics/Revue Européenne de Mécanique Numérique, 21(3-6): , [2] J. W. Barrett and C. M. Elliott. A finite-element metod for solving elliptic equations wit neumann data on a curved boundary using unfitted meses. IMA Journal of Numerical Analysis, 4(3): , [3] J. W. Barrett and C. M. Elliott. inite element approximation of te diriclet problem using te boundary penalty metod. Numerisce Matematik, 49(4): , [4] J. W. Barrett and C. M. Elliott. itted and unfitted finite-element metods for elliptic equations wit smoot interfaces. IMA journal of numerical analysis, 7(3): , [5] R. Becker, E. Burman, and P. Hansbo. A Nitsce extended finite element metod for incompressible elasticity wit discontinuous modulus of elasticity. Computer Metods in Applied Mecanics and Engineering, 198(41): , [6] M. Bercovier and O. Pironneau. Error estimates for finite element metod solution of te stokes problem in te primitive variables. Numerisce Matematik, 33(2): , [7] C. Bernardi, M. Costabel, M. Dauge, and V. Girault. Continuity properties of te inf-sup constant for te divergence. SIAM Journal on Matematical Analysis, 48(2): , [8] C. Bernardi and G. Raugel. Analysis of some finite elements for te stokes problem. Matematics of Computation, pages 71 79, [9] D. Boffi,. Brezzi, and M. ortin. inite elements for te stokes problem. In D. Boffi and L. Gastaldi, editors, Mixed finite elements, compatibility conditions, and applications, pages Springer, Lecture Notes in Matematics. Springer Verlag. Vol [10] D. Boffi and L. Gastaldi. A finite element approac for te immersed boundary metod. Computers & structures, 81(8): , [11] M. Bogovskii. Solution of te first boundary value problem for te equation of continuity of an incompressible medium. In Dokl. Akad. Nauk SSSR, volume 248, pages , [12] E. Burman. Gost penalty. Comptes Rendus Matematique, 348(21): , [13] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. Cutfem: Discretizing geometry and partial differential equations. International Journal for Numerical Metods in Engineering, 104(7): , [14] E. Burman and P. Hansbo. ictitious domain finite element metods using cut elements: II. A stabilized Nitsce metod. Applied Numerical Matematics, 62(4): , 2012.

17 IN-SUP SABILIY O UNIED SOKES ELEMENS 17 [15] E. Burman and P. Hansbo. ictitious domain metods using cut elements: III. A stabilized Nitsce metod for Stokes problem. ESAIM: Matematical Modelling and Numerical Analysis, 48(03): , [16] L. Cattaneo, L. ormaggia, G.. Iori, A. Scotti, and P. Zunino. Stabilized extended finite elements for te approximation of saddle point problems wit unfitted interfaces. Calcolo, 52(2): , [17] A. Y. Cernysenko and M. A. Olsanskii. Non-degenerate eulerian finite element metod for solving pdes on surfaces. Russian Journal of Numerical Analysis and Matematical Modelling, 28(2): , [18] M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element metods for solving te stationary stokes equations i. Revue française d automatique, informatique, recerce opérationnelle. Matématique, 7(3):33 75, [19] K. Deckelnick, C. M. Elliott, and. Ranner. Unfitted finite element metods using bulk meses for surface partial differential equations. SIAM Journal on Numerical Analysis, 52(4): , [20] J. Dolbow and I. Harari. An efficient finite element metod for embedded interface problems. International journal for numerical metods in engineering, 78(2): , [21] A. Ern and J.-L. Guermond. eory and practice of finite elements, volume 159. Springer Science & Business Media, [22].-P. ries and. Belytscko. e extended/generalized finite element metod: an overview of te metod and its applications. International Journal for Numerical Metods in Engineering, 84(3): , [23] G. P. Galdi. An introduction to te matematical teory of te Navier-Stokes equations: Steady-state problems. Springer Science & Business Media, [24] S. Groß, M. A. Olsanskii, and A. Reusken. A trace finite element metod for a class of coupled bulk-interface transport problems. ESAIM: Matematical Modelling and Numerical Analysis, 49: , [25] S. Groß and A. Reusken. An extended pressure finite element space for two-pase incompressible flows wit surface tension. Journal of Computational Pysics, 224(1):40 58, [26] A. Hansbo and P. Hansbo. An unfitted finite element metod, based on Nitsces metod, for elliptic interface problems. Computer metods in applied mecanics and engineering, 191(47): , [27] P. Hansbo, M. G. Larson, and S. Zaedi. A cut finite element metod for a Stokes interface problem. Applied Numerical Matematics, 85:90 114, [28] M. Kircart, S. Groß, and A. Reusken. Analysis of an XEM discretization for Stokes interface problems. SIAM J. Sci. Comput, 38(2):A1019 A1043, [29] G. Legrain, N. Moës, and A. Huerta. Stability of incompressible formulations enriced wit x-fem. Computer Metods in Applied Mecanics and Engineering, 197(21): , [30] Z. Li. e immersed interface metod using a finite element formulation. Applied Numerical Matematics, 27(3): , [31] Z. Li,. Lin, and X. Wu. New cartesian grid metods for interface problems using te finite element formulation. Numerisce Matematik, 96(1):61 98, [32] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. A stabilized Nitsce fictitious domain metod for te Stokes problem. Journal of Scientific Computing, 61(3): , [33] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. A stabilized Nitsce overlapping mes metod for te Stokes problem. Numerisce Matematik, 128(1):73 101, [34] M. A. Olsanskii, A. Reusken, and J. Grande. A finite element metod for elliptic equations on surfaces. SIAM journal on numerical analysis, 47(5): , [35] M. A. Olsanskii, A. Reusken, and X. Xu. An eulerian space-time finite element metod for diffusion problems on evolving surfaces. SIAM journal on numerical analysis, 52(3): , [36] A. Reusken. Analysis of trace finite element metods for surface partial differential equations. IMA Journal of Numerical Analysis, 35: , [37] H. Sauerland and.-p. ries. e extended finite element metod for two-pase and free-surface flows: a systematic study. Journal of Computational Pysics, 230(9): , [38] B. Scott and W. Wall. A new face-oriented stabilized xfem approac for 2d and 3d incompressible navier Stokes equations. Computer Metods in Applied Mecanics and Engineering, 276: , [39] R. Verfürt. Error estimates for a mixed finite element approximation of te Stokes equations. RAIRO- Analyse numérique, 18(2): , [40] Q. Wang and J. Cen. A new unfitted stabilized Nitsces finite element metod for Stokes interface problems. Computers & Matematics wit Applications, 70(5): , 2015.

18 18 J. GUZMÁN AND M.A. OLSHANSKII Appendix A. Proof of Lemma 2 A.1. Proof of Lemma 2 wen Q = Q c. Consider te continuous (Lagrange) interpolant v W 1(Ωi ) of q Q. rom approximation property of v we immediately ave q = v + w, w L 2 ( ) + w H 1 ( ) C 2 q H 2 ( ), i. Wit te elp of inverse inequality in Q we also get: (A.1) (A.2) w L 2 ( ) + w H 1 ( ) C q H 1 ( ), v H 1 ( ) q H 1 ( ) + w H 1 ( ) C q H 1 ( ). Consider E,1 v W 1 (Ωe ) Q te extension built in Lemma 4.2 from [28], wic satisfies E,1 v H 1,e C v H 1,i. anks to (A.2) we ave (A.3) E,1 v H 1,e C q H 1,i. We build te extension E,2 w Q by setting E,2 w = w on Ω i, and E,2w = 0 in all Lagrangian nodes in Ω e, wic are not on Ωi. en E q := E,1 v + E,2 w is our desired extension. o prove te necessary stability bound, we need only to estimate E,2 w L 2 (Ω Γ). Assume Γ and as one or more faces (edges in 2D) on Ωi. Denote by neig te union of all tetraedra from Ω i saring a face wit. e polynomial E w vanises in all Lagrange nodes of except tose on Ω i. Due to tis, te trace inequality and (A.1) we ave: (A.4) E,2 w 2 L 2 ( ) C 1 E,2w 2 L 2 ( Ω i ) = 1 w 2 L 2 ( Ω i ( ) ) C w 2 L 2 ( ) + 2 w 2 L 2 ( ) i : neig C q 2 L 2 ( neig ) By te same argument we ave (A.5) E,2 w 2 L 2 ( ) C w 2 L 2 ( Ω i ) C2 q 2 L 2 ( neig ). We estimated E,2 w L 2 ( ) and so E q L 2 ( ) for all tetraedra saring at least one face wit Ω i. Denote te union of all suc tetraedra from ΩΓ by ΩΓ 1. It may appen tat ΩΓ \ ΩΓ 1 is still not empty. en we consider te union Ω Γ 2 of all tetraedra saring at least one face wit Ω Γ 1 and repeat our arguments in (A.4) (A.5) wit te only difference tat (A.5) is used instead of (A.1) for tetraedra in Ω Γ 1. We proceed wit a finite N (independent of, but dependent of te min-angle condition) number of steps until N k=1 ΩΓ k = Ω Γ. A.2. Proof of Lemma 2 wen Q = Q d. Let q Q. Now for every we let q ext P kp (R d ) be te natural of extension of q onto te entire plan. or Γ, we define E q = q ext K, were K Ω i is given by assumption 1. Since K W ( ) dist(k, ) C, it follows tat E q L 2 ( ) = q ext L 2 ( ) C q L 2 (K ).

19 IN-SUP SABILIY O UNIED SOKES ELEMENS 19 Hence, we ave (A.6) 2 E q 2 L 2 ( ) C 2 q 2 L 2 ( ). Γ i Let Γ and let =, wit at least one of, belonging to Γ. irst we assume te tat bot tetraedra belong to Γ. In tis case tere exists K and K belonging to i suc tat E q = qk ext and E q = q ext K. Now we know tat tere exists a sequence of tetraedra K = K 1, K 2,..., K M = K all belonging to i were K i, K i+1 sare a common face wic we denote by i. By assumption 1, te number M is bounded and only depends on te sape regularity of te mes. irst using inverse estimates we get 1/2 [E q] L 2 ( ) = 1/2 qext K 1 qk ext M L 2 ( ) C qk ext 1 qk ext M L 2 ( ) Easy to see tat since K 1 and belong to te same patc W ( ) tat anks to te triangle inequality we get q ext K 1 K M L 2 ( ) C q ext K 1 K M L 2 (K 1 ) q ext K 1 K M L 2 (K 1 ) q K1 K 2 L 2 (K 1 ) + q ext K 2 K M L 2 (K 1 ). Using equivalence of norms in finite dimensional case we obtain We also ave So we get, q ext K 1 K 2 L 2 (K 1 ) 1/2 1 [q] L 2 ( 1 ) + K1 (q K1 K 2 ) L 2 (K 1 ). q ext K 2 K M L 2 (K 1 ) C q ext K 2 K M L 2 (K 2 ). q ext K 1 K M L 2 (K 1 ) 1/2 1 [q] L 2 ( 1 ) + K1 (q ext K 1 K 2 ) L 2 (K 1 ) + q ext K 2 K M L 2 (K 2 ). If we continue tis we will get Again, we see tat Hence, we get M 1 qk ext 1 qk ext M L 2 (K 1 ) M 1 j=1 j=1 Kj (q ext K j 1/2 M 1 j [q] L 2 ( j ) + j=1 K j+1 ) L 2 (K j ) Kj (q ext K j M Kj q L 2 (K j ) j=1 1/2 M 1 [E q] L 2 ( ) = 1/2 j [q] L 2 ( j ) + A similar argument works if Γ inequality above would old true. j=1 K j+1 ) L 2 (K j ) M Kj q L 2 (K j ). j=1 and i. In tat case we would let K = and same

20 20 J. GUZMÁN AND M.A. OLSHANSKII If we now sum over Γ we get [E q] 2 L 2 ( ) C 2 q 2 L 2 ( ) + C [q] 2 L 2 ( ). Γ i i e result now follows by combining tis inequality wit (A.6). Division of Applied Matematics, Brown University, Providence, RI 02912, USA address: jonny guzman@brown.edu Department of Matematics, University of Houston, Houston, X 77204, USA address: molsan@mat.u.edu