UNIFORMLY-STABLE FINITE ELEMENT METHODS FOR DARCY-STOKES-BRINKMAN MODELS *

Transcription

1 Journal o Computational Matematics, Vol.26, No.3, 2008, UNIFORMLY-SABLE FINIE ELEMEN MEHODS FOR DARCY-SOKES-BRINKMAN MODELS * Xiaoping Xie Yangtze Center o Matematics, and Scool o Matematics, Sicuan University, Cengdu , Cina xpxiec@gmail.com Jincao Xu and Guangri Xue Center or Computational Matematics and Applications, Matematics Department, e Pennsylvania State University, PA 16802, USA xu@mat.psu.edu, xue@mat.psu.edu Dedicated to Proessor Junzi Cui on te occasion o is 70t birtday Abstract In tis paper, we consider 2D and 3D Darcy-Stokes interace problems. ese equations are related to Brinkman model tat treats bot Darcy s law and Stokes equations in a single orm o PDE but wit strongly discontinuous viscosity coeicient and zerotorder term coeicient. We present tree dierent metods to construct uniormly stable inite element approximations. e irst two metods are based on te original weak ormulations o Darcy-Stokes-Brinkman equations. In te irst metod we consider te existing Stokes elements. We sow tat a stable Stokes element is also uniormly stable wit respect to te coeicients and te jumps o Darcy-Stokes-Brinkman equations i and only i te discretely divergence-ree velocity implies almost everywere divergence-ree one. In te second metod we construct uniormly stable elements by modiying some well-known H(div)-conorming elements. We give some new 2D and 3D elements in a uniied way. In te last metod we modiy te original weak ormulation o Darcy-Stokes- Brinkman equations wit a stabilization term. We sow tat all traditional stable Stokes elements are uniormly stable wit respect to te coeicients and teir jumps under tis new ormulation. Matematics subject classiication: 65N12, 65N15, 65N22, 65N30. Key words: Darcy-Stokes equation, Brinkman, Finite element, Uniormly stable. 1. Introduction In tis paper, we consider te ollowing model equations on a bounded, connected, and polygonal domain Ω R d (d = 2, 3) (Fig. 1.1 is an example o two dimensional domain). A velocity u and a pressure p satisy wit piecewise-constant viscosity coeicient (ν(x) u) + α(x)u + p = in Ω, u = g in Ω, u = 0 on Ω, * Received Marc 15, 2008 / accepted Marc 29, 2008 / (1.1) ν(x) = ν i > 0, x Ω i, (1.2)

2 438 X.P. XIE, J.C. XU AND G. XUE Fig Domain. and piecewise-constant zerot-order term coeicient α(x) = α i 0, x Ω i. (1.3) e sub-domains Ω i are assumed to be bounded connected polygonal domains suc tat Ω i Ω j = or i j and Ω = m i=1 Ω i. By Γ ij, we denote te interace between two adjacent sub-domains Ω i and Ω j, namely, Γ ij = Ω i Ω j. For oter notations: σ(u, p) = ν(x) u pi is a stress tensor; n is te unit normal vector to Γ i,j ; [u] Γij = u Ωi Γ ij u Ωj Γ ij ; [σ(u, p)n] Γij = σ(u, p)n Ωi Γ ij σ(u, p)n Ωj Γ ij. For te interace boundary conditions, we ave [σ(u, p)n] Γij = 0, and [u] Γij = 0. In addition, te source term g is assumed to satisy te solvability condition: gdx = 0. (1.4) Ω Wen α i is big and ν i is small in some sub-domains, te equation is close to Darcy equation; in some sub-domain were ν i is big and α i is small togeter wit g = 0, te equation is close to te Stokes equation. is Darcy-Stokes equation is called Brinkman equation [1], wic models porous media low coupled wit viscous luid low in a single orm o equation. Among many applications to te Darcy-Stokes-Brinkman equations, our motivation comes rom computational uel cell dynamics [2 4]. A uel cell is a clean cemical energy conversion device wic as potential to replace te traditional combustion engine. In te uel cell, tere are porous gas diusion layers and gas cannels. e two-pase mixture low in te porous media is modeled by Darcy s law and low in te gas cannel is modeled by Navier-Stokes equations [5 10]. Reviews or tis area can be ound in [11,12]. It is so-called single-domain approac tat models multi-domain problems using single set o equations wit igly discontinuous coeicients ν(x) and α(x). In tis approac, te internal interace conditions are straigtorward (te velocity and normal component o stress tensor are continuous), compared to oter types o multi-domain Darcy-Stokes models tat couple troug tree interace conditions [13 21]. e goal o tis paper is to explore inite element metods wic beave uniormly wit respect to te igly discontinuous coeicients, ν(x) and α(x), and teir jumps. We present tree dierent metods. In te irst two metods, we consider te original weak ormulation. As discussed above, our model problems can be reduced to two extreme cases. One is standard Stokes equation.

3 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 439 e oter one is Darcy s law (essentially a mixed orm o an elliptic problem). For te Stokes equation, tere are many stable elements available. But not all elements lead to uniormly stable approximations or te Darcy-Stokes-Brinkman problem (1.1). For te mixed orm elliptic problem or Darcy s law, tere are also stable H(div)-conorming elements available. However, none o tem usually work or te original problem (1.1). In te irst metod, we sow tat any stable Stokes element (i.e., satisies (H1) in Section 3) leads to a uniormly stable approximation or te Darcy-Stokes-Brinkman problem i and only i te assumption (H2) (in Section 3) olds. Rougly speaking, tis assumption says discretely divergence-ree velocity implies te almost everywere divergence-ree one. e element satisying (H2) is also stable or te limiting case, Darcy s law, o te equation. On te oter and, in te second metod, we consider te construction o uniormly stable elements based on some well-known H(div)-conorming elements. Under reasonable assumptions we ind tat H(div) stable elements are also uniormly stable or te Darcy-Stokes-Brinkman problem. Considering te approximation property, we still need to add someting to te H(div) inite element space to approximate H 1 space. Based on te analysis, we construct some new uniormly stable elements or te Darcy-Stokes-Brinkman equations. In te last metod, we consider a modiied equivalent ormulation by adding a proper stabilization term. Brezzi, Fortin and Marini [22] presented a stabilization tecnique tat allows te use o continuous inite element spaces. eir tecnique involves a modiication o te usual mixed equations. We employ tis tecnique to modiy te Darcy-Stoke-Brinkman models. Under tis modiication, traditional stable Stokes elements are indeed uniormly stable wit respect to te coeicients and teir jumps. ere are also oter stabilized approaces (see, e.g., Franca and Huges [23], Burman and Hansbo [21], and te reerences terein). is paper is organized as ollows. In Section 2, we describe te continuous and discrete weak ormulations and discuss ow to coose te appropriate norms. In Section 3, we investigate special stable stokes elements wic lead to uniormly stable inite element approximations to our model problem (1.1). In Section 4, we start rom standard H(div)-conorming elements to construct uniormly stable elements. In Section 5, we discuss te metod to modiy original weak ormulation by adding a proper stabilization term. Finally in Section 6, we give concluding remarks. Let us introduce some notations. In tis paper, H k (Ω) denotes te Sobolev space o scalar unctions on Ω wose derivatives up to order k are square integrable, wit te norm k. e notation k denotes te semi-norm derived rom te partial derivatives o order equal to k. Furtermore, k, and k, denote respectively te norm k and te semi-norm k restricted to te domain. e notation L 2 0 (Ω) denotes te space o L2 unctions wit zero mean values. e space H k 0 (Ω) denotes te closure in Hk (Ω) o te set o ininitely dierentiable unctions wit compact supports in Ω. For te corresponding d-dimensional vector spaces, we put superscript d on te scalar notation, suc as, H k (Ω) d and H k 0 (Ω)d. We also denote H(div) := H(div, Ω) := {v L 2 (Ω) divv L 2 (Ω)}, H 0 (div) := {v H(div) v n = 0, on Ω}. Here n is te unit normal vector on Ω. For simplicity, ollowing Xu [24], we use X ( ) Y to denote tat tere exists a constant C suc tat X ( ) CY. Here, te constant C is independent o te mes size, te viscosity coeicient ν, and te zerot-order term coeicient α.

4 440 X.P. XIE, J.C. XU AND G. XUE 2.1. Continuous problem 2. Model Descriptions We introduce te variational ormulation o te problem (1.1). Deine te velocity and pressure spaces respectively as V := H 1 0 (Ω)d and W := L 2 0 (Ω). Let V and W be te dual spaces o V and W respectively. en, te variational ormulation reads as ollows: given V and g W, ind {u, p} V W suc tat { a(u,v) (p,divv) =<,v > v V, (divu, q) =< g, q > q W. (2.1) Here a(u,v) = (ν(x) u, v) + (α(x)u,v), (, ) denotes te L 2 inner product o a pair o unctions on Ω, and <, > denotes te duality pairing o te spaces. In te limiting case o ν(x) 0, te problem (2.1) reduces to a mixed orm o an elliptic equation. en, te space H 1 0(Ω) d is no longer a proper unction space or u. Instead, te solution space is replaced by H 0 (div). For tis consideration, we introduce te ollowing parameter-dependent norms: u 2 := a(u,u) + M(divu,divu), u V, (2.2) and Here p = M 1/2 p 0, p W. (2.3) M = max(ν, α, 1). (2.4) Under tese norms, we sall sow below te uniorm stability conditions are straigtorward. Firs o all, by deinition we ave a(u,v) u v, u,v V, (2.5) a(v,v) = v 2, v Z, (2.6) were Z = {v V : divv = 0}. (2.7) Note tat divv 0 M 1/2 v, te continuity condition ollows immediately: (divv, q) v q, v V, q W. (2.8) Next, it is well-known tat te ollowing in-sup condition olds [25], sup v V Since v M 1/2 v 1, we ave te uniorm in-sup condition sup v V (divv, q) v 1 q 0, q W. (2.9) (divv, q) v q, q W. (2.10)

5 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 441 By te Brezzi teory or saddle-point problems [26, 27], te problem (2.1) as a unique solution and te ollowing estimate olds uniormly wit respect to ν and α: Here te norms on V and W are deined by <,v > V := sup v V v u + p V + g W. (2.11) and g W := sup q W < g, q >. (2.12) q Remark 2.1. Let us now take a closer look at te norms deined in (2.12) wen, g L 2. Obviously, we always ave g W = M 1/2 g 0. For V, i α α 0 > 0, we can easily see tat V α , since v 0 α v. I α 0, Olsanskii and Reusken [28] proved tat (νv,v) (ν v, v) i k = 2 and one o te ollowing assumptions is satisied: meas( Ω i Ω) > 0 or i = 1, 2, or meas( Ω 1 Ω) > 0 and ν 2 ν 1. As a result, V ν For te general case o k sub-domains, similar results also old i one o te ollowing assumptions is satisied: meas( Ω i Ω) > 0 or i = 1, 2,, k, or meas( Ω i Ω) > 0 or i S 1 and ν j ν jn or j n N j and j S 2. Here N j denotes te set tat consists o sub-domain indices or te neigbors o j saring te same d 1 dimensional simplex. e sets S 1, S 2 {1, 2,, n}, S 1 S 2 = {1, 2,, n}, and S 1 S 2 = Discrete problem Let be a sape-regular simplicial triangulation o te domain Ω, were te edges or aces o any element lie on te interaces. In te simplicial triangulation, te mes parameter o is given by = max {diameter o }, were denotes triangle in 2D and tetraedron in 3D. For 2D mes, let E() denotes te set o all edges in ; or 3D mes, let F() denotes te set o all aces in. Pk d (Ω) denotes te d-dimensional polynomial space on Ω. Wen d = 1, we drop te superscript. Let V ( or V ) and W W denote velocity and pressure inite dimensional spaces respectively. e discrete weak ormulation o te problem (2.1) reads as: Find {u, p } V W suc tat { a (u,v ) (p,divv ) =<,v > v V, (2.13) (divu, q ) =< g, q > q W. Here a (u,v ) is deined by a (u,v ) = Here (, ) denotes te L 2 inner product on. ((ν(x) u, v ) + (α(x)u,v ) ). (2.14) Remark 2.2. I te space V H(div), trougout tis paper, we view divv as div v, v V. e operator div denotes te piecewise divergence operator acting on element by element in simplicial triangulations. Similar to te continuous problem, we deine te discrete norm in V as ollows: v V, v 2 := a (v,v ) + M(divv,divv ). (2.15) e discrete pressure norm is te same as te continuous one since W W. Denote te discretely divergence-ree space Z as Z := {v V : (divv, q ) = 0, q W }. (2.16)

6 442 X.P. XIE, J.C. XU AND G. XUE 3. Application o Special Stable Stokes Elements In te ollowing, we sall sow te critical conditions wic lead to uniormly-stable inite element metods or te problem (1.1). First o all, one can expect tat te elements are stable in te standard H 1 (or discrete H 1 ) norm or te velocity and L 2 0 norm or te pressure. us, te irst assumption is tat te ollowing in-sup condition olds: (H1) (divv, q ) sup q 0 q W. v V v 1, Here discrete H 1 norm 1, is deined by v 2 1, := v 2 1,, v V. Wen V H 1 0 (Ω)d, te discrete norm recovers te standard H 1 norm. e second assumption is: (H2) Z = {v V, divv = 0}. is assumption means tat te discretely divergence-ree velocity implies almost everywere divergence-ree one. I te pressure space contains divergence o velocity one, it yields te assumption (H2). We state tis stronger assumption as (H2 ) divv W. Under te above two assumptions (H1) and (H2), we easily know te ollowing uniorm stability conditions old. a (v,v ) v 2, v Z, (3.1) (divv, q ) sup q, v V v q W, (3.2) a (u,v ) u v, u,v V, (3.3) (divv, q ) v q, v V, q W. (3.4) eorem 3.1. raditional stable Stokes elements (i.e., satisy te in-sup condition (H1)) are also uniormly stable or te model problem (1.1), i and only i te assumption (H2) olds. Proo. It is easy to see tat te assumption (H2) is suicient. For te necessity o te assumption (H2), we consider te in-sup condition a (u,v ) sup v, v Z. (3.5) u Z u Wen α(x) and ν(x) bot approac to zero, in order to ave te uniorm in-sup condition (3.5), we must ave divv = 0, v Z. For nonconorming inite element metods, multiplying v V to te irst equation o (1.1) and integrating by parts, we ave a (u,v ) (divv, p) =<,v > +E (u, p,v ), (3.6) were te consistency error term is deined by E (u, p,v ) = (ν(x) u pi)n v ds = σ(u, p)n v ds. (3.7) We are now in a position to state te ollowing quasi-optimal approximation property. For completeness, we give a proo by ollowing similar arguments in [25 27,29].

7 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 443 eorem 3.2. Assume tat (H1) and (H2) are satisied. en te problem (2.13) admits a unique solution {u, p } V W, suc tat u u in w Z (g) u w + sup v Z E (u, p,v ) v, (3.8) u u in v V u v + sup v Z E (u, p,v ) v, (3.9) p p in q W p q + in v V u v + sup v V E (u, p,v ) v. (3.10) Here, Z (g) := {v V (divv, q ) =< g, q >, q W }. (3.11) Proo. Applying Lemma I.4.1 in [25], te in-sup condition (3.2) implies Z (g) is not empty. Coose u 0 Z (g). By te conditions (3.1) and (3.3), tere exists a unique solution s Z, suc tat a (s,v ) =<,v > a (u 0,v ), v Z. Let u = s +u 0. Furtermore, it ollows rom Corollary I.4.1 in [25] tat tere exists a unique p in W suc tat te pair {u, p } is te only solution o Problem (2.13). For w Z (g), u w Z. By assumption (H2), div(u w ) = 0. us, it yields te ollowing identity: a (u w,u w ) = a (u w,u w ) + E (u,u w ). From te coercivity condition (3.1) and continuity condition (3.3), we get u w u w + E (u, p,u w ) u w. (3.12) aking inimum o w and using triangle inequality, we obtain (3.8). For v V, by Lemma I.4.1 in [25], in-sup condition (3.2) implies: tere exists a unique r Z suc tat (divr, q ) = (div(u v ), q ), q W, and r u v. Let w = v + r, ten w Z (g). Furtermore u w u v + r u v. (3.13) (3.12), (3.13), and triangle inequality imply (3.9). It remains to estimate p p 0. From (3.6) and (2.13), we derive tat Furter, we can get By in-sup condition (3.2), a (u u,v ) (divv, p p ) = E (u, p,v ), v V. (3.14) (divv, p q ) = a (u u,v ) E (u, p,v ) + (divv, p q ). (3.15) p q sup v V (divv, p q ) v. (3.16)

8 444 X.P. XIE, J.C. XU AND G. XUE From (3.15), continuity conditions (3.3) and (3.4), urter we can get p q u u + p q + E (u, p,v ) v. us (3.10) ollows immediately by using triangle inequality. Remark 3.1. Under a stronger assumption (H2 ), rom (2.1) and (2.13), we ave Q divu = divu. Here Q : W W is an L 2 -ortogonal projection. us we get divu divu 0 = (I Q )divu 0. (3.17) Examples (P d k -P k 1 type elements). In all tese metods, we approximate te velocity by te continuous piecewise polynomials o order k and te pressure by te discontinuous piecewise polynomials o order k 1. e metods are all conorming in te sense tat V V and W W. 1. Scott and Vogelius [30] proposed 2D amily o P 2 k P k 1 type triangular elements or any k 4, on singular-vertex ree mes. An internal vertex in 2D is said to be singular i edges meeting at te point all into two straigt lines. 2. Arnold and Qin [31] proposed a 2D inite element o P 2 2 P 1 type on macro square meses were eac big square is subdivided into our triangles by connecting te square s vertices to te point midway between te center o te square and its bottom edge. 3. Qin [32] proposed 2D inite elements o P 2 k P k 1 type, or k = 2 and k = 3, on macro triangular meses were eac big triangle is subdivided into tree triangles by connecting te barycenter wit tree vertices. 4. Zang [33] proposed 3D inite elements o P 3 k P k 1 type, or k 3, on macro tetraedron meses were eac big tetraedron is subdivided into our subtetraedra by connecting te barycenter wit our vertices. For tese inite element spaces, te assumption (H2 ) is trivially satisied by te deinition o V and W. 4. Application o Modiied H(div)-conorming Elements e second metod is to consider te construction o uniormly stable elements or te Darcy-Stokes-Brinkman equations on te basis o H(div)-conorming elements. We irst consider te ollowing mixed ormulation o an elliptic problem, wic can be viewed as one limiting case o te problem (2.13). Find u V 0 and p W 0, suc tat { (u,v ) (p,divv ) =<,v > v V 0, (divu, q ) =< g, q > q W 0, (4.1) Here, V 0 V 0 and W 0 L2 0 are two inite element spaces, and V 0 := H 0 (div), wit te norm v 2 H(div) := (divv,divv) + (v,v), v V 0.

9 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 445 e stability conditions or tis problem are (v,v ) v 2, v H(div) Z 0 := {v V 0 : (divv, q ) = 0, q W 0 }, (4.2) (divv, q ) sup q 0, q W v V v, 0 (4.3) H(div) (u,v ) u v H(div), u H(div),v V 0, (4.4) (divv, q ) v H(div) q 0, v V 0, q W 0. (4.5) In te lemma below, we sall sow tat any stable Stokes elements satisying (H1) and (H2) are also stable or te reduced problem (4.1). Lemma 4.1. Suppose tat te inite element spaces V V 0 and W L 2 0 satisy te assumptions (H1) and (H2). en te stability conditions (4.2)-(4.5) old or {V, W }. Proo. e condition (4.2) is trivial by te assumption (H2). Under te assumption (H1), (4.3) ollows rom te act tat v v H(div) 1,, v V. From te Caucy-Scwartz inequality, te conditions (4.4) and (4.5) also ollow immediately. In consequence, we ave eorem 4.1. Under te conditions in Lemma 4.1, te problem (4.1) admits a unique solution {u, p } V W, suc tat u u H(div) in u v v V H(div), (4.6) p p 0 in p q 0 + in u v. q W v V H(div) (4.7) In addition, i te assumption (H2 ) olds, we ave div(u u ) 0 = (I Q )divu 0. (4.8) Now, we want to construct modiied H(div) elements to approximate H 1 space on te basis o H(div) stable elements. In practice or common coices o V 0 and W 0, te stability conditions (4.2) and (4.3) amount to (S1) { ere exists Π : V 0 V 0 suc tat v V 0, divπ v = Q divv and Π v H(div) v H(div), and (S2) divv 0 = W 0. In act it is easy to see tat, under te assumption (S1), te assumption (S2) is equivalent to a weaker one: divv 0 W 0. (4.9) For example, te Raviart-omas [34, 35] or Brezzi-Douglas-Marini elements [36, 37] satisy (S1) and (S2). In tese elements, Π is te canonical interpolation operator deined element by element. In addition, Π satisies H 1 bound property, namely v V 0, Π v 1, v 1,. (4.10) In act, tis condition is crucial to te uniorm stability or te original problem (2.13).

10 446 X.P. XIE, J.C. XU AND G. XUE Lemma 4.2. Suppose tat te H 1 bound condition (4.10) and te H(div) stability conditions, (S1) and (S2), old. en te assumptions (H1) and (H2) old or V 0 and W 0. Proo. By Fortin s Lemma, (S1), (S2), and (4.10) imply te assumption (H1). e assumption (H2) is a direct consequence o (S2). However, in general, te spaces V 0 and W 0 do not usually work or te original problem, since tese spaces lead to te nonconorming approximation and a unction in V 0 as no continuity o tangential component. In order to ave desired accuracy or te consistency error estimate, at least we need to impose some weak continuity o tangential component o te velocity approximation. Now our task is to construct new element spaces V and W to approximate te original problem wile preserving te structure o V 0 and W 0, namely, by satisying te conditions (S1) and (S2). A natural coice or pressure space is W = W 0. is implies tat, in order to ave (S2) or V and W, our new space may take te orm: V = V 0 + curls. (4.11) Here S is some piecewise polynomial space on. Recall tat te curl operator on a scalar unction q in 2D is deined by ( curlq = q ) q,, x 2 x 1 and on a vector unction q in 3D is deined by ( q2 curlq = q 3, x 3 x 2 q 3 x 1 q 1 x 3, q 1 q ) 2, x 2 x 1 were q = (q 1, q 2, q 3 ). Now we look at te degrees o reedom o v V. Let Π : V V be te canonical interpolation operator deined element by element. We irst note te identity by Green s ormula, (divπ v Q divv)q dx = div(π v v)q = (Π v v) q dx + (Π v v) nq ds. In order to ave te commutativity property in (S1), we can take te degrees o reedom used in te standard Raviart-omas [34, 35] or Brezzi-Douglas-Marini elements [36, 37]. Furter we need additional degrees o reedom to ensure certain weak continuity o te velocity approximation. Examples. We deine te velocity inite element space on te element as ollowing: V := V 0 + curl(by ), (4.12) ere b is te bubble unction, namely b = Π d+1 i=1 λ i, and λ i, i = 1,, d + 1, is te barycentric coordinate o. For te space Y, coose te ollowing polynomial spaces: Y 1 := P 1 () 2D, Y = Y 2 := P1 3 () 3D, (4.13) Y 3 := P1 3 ()/span{(λ i 1 3 ) λ i} 4 i=1 3D.

11 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 447 By construction, it is easy to see tat q curl(by ) satisies divq = 0 and q n = 0. (4.14) For te space V 0, we coose te ollowing well-known H(div)-conorming inite element spaces. R 1 () := P1 2() + P 1 ()x 2D, V 0 BDM 1 () := P1 2 () 2D, = R 1 () := P1 3 () + P (4.15) 1 ()x 3D, BDM 1 () := P1 3() 3D, were P 1 () := P 1 ()/P 0 () (i.e., te omogeneous polynomial space o degree 1). R 1 () denotes bot te irst order 2D Raviart-omas [34] and 3D Nedelec [35] inite element spaces. BDM 1 () denotes bot te irst order 2D Brezzi-Douglas-Marini [36] and 3D Brezzi-Douglas- Duran-Fortin [37] inite element spaces. For dierent coices o V 0 and Y in (4.15) and (4.13), we give te corresponding degrees o reedom in te able 4.1. In tis table, R 0 () is te zerot-order Raviart-omas element space [34], i.e., R 0 () := P 2 0 () + P 0 x, n is te unit normal vector to an edge e E() or a ace F(), and t is te unit tangent vector along te edge e. For te element diagrams, see te Figs able 4.1: e Six Modiied H(div) Elements. Elements V 0 Y Degrees o Freedom (DOF) # o DOF First 2D R 1() Y 1 v nqds, q P1(e), 11 e Element 0 () v tds e Second 2D BDM 1() Y 1 v nqds, q P1(e), 9 e Element e First 3D R 1() Y 2 v nqds, q P1() 27 Element 0 () (v n) rds, r R0() Second 3D BDM 1() Y 2 v nqds, q P1() 24 Element (v n) rds, r R0() ird 3D R 1() Y 3 v nqds, q P1() 23 Element v qds, q P 3 0 () (v n) rds, r P 0 2 () Fourt 3D BDM 1() Y 3 v nqds, q P1() 20 Element (v n) rds, r P 0 2 () Among tese six elements, te second 2D element and te second 3D element ave been also proposed by Mardal-ai-Winter [38] and ai-winter [39] respectively or te problem (1.1) wit ν(x) ǫ 2 and α(x) 1.

12 448 X.P. XIE, J.C. XU AND G. XUE Fig DOF or te irst 2D modiied R element (let) and or te second 2D modiied BDM element (rigt). Fig DOF or te irst 3D modiied R element (let) and or te second 3D modiied BDM element (rigt) Unisolvence Denote te dimension o a polynomial space by dim( ). It is easy to see tat te ollowing relation olds: dim(v 0 + curl(by )) = dim(v 0 ) + dim(curl(by )) = dim(v 0 ) + dim(y ). From tis relation, we can see tat te dimension o polynomial space V is te same as te number o degrees o reedom or eac element. In addition, te ollowing lemma is useul or te unisolvence o te 3D elements. Lemma 4.3. ([39]) Assume tat v P 2 1 () is o te orm v = 3 c i (λ i 1 3 ) λ i, i=1 and satisies b v rdx = 0. Here r is te position vector: r = (x, y), and b is te cubic bubble unction associated wit ace. en c 1 + c 2 + c 3 = 0. Now, we give a uniied unisolvence proo or all o te six elements. Lemma 4.4. For all te six elements, v V is uniquely determined by te corresponding degrees o reedom. Proo. Assume tat all te degrees o reedom are zeros. Let v = v 0 + curl(bq), wit v 0 V 0 and q Y (in 2D, q is scalar). In 3D, curl(bq) n = curl (bq) = 0,

13 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 449 Fig DOF or te tird 3D modiied R element (let) and or te ourt 3D modiied BDM element (rigt). wit (bq) te tangential component o bq on. In 2D, curl(bq) n = (bq) t = 0. us, v n = v 0 n P 1 ( ). Furtermore, { e v0 nrds = 0, r P 1 (e) e E() in 2D, v0 nwds = 0, w P 1 () F() in 3D. (4.16) Also, by Stoke s teorem, Hence curl(bq) rdx = 0, r P d 0 (), d = 2, 3. v 0 rdx = v rdx = 0, r P d 0 (). (4.17) It is well known tat 2D Raviart-omas [34], 3D Nedelec [35], 2D Brezzi-Douglas-Marini [36], and 3D Brezzi-Douglas-Duran-Fortin [37] elements are all unisolvent. ereore, v 0 = 0. In wat ollows, we sall sow tat q = 0. For te 2D elements, q P 1 (e), 0 = v tds = curl(bq) tds = (bq) nds = e e e e b n qds. Since b/ n remains te same sign on eac edge e, we know tat q as zero point in te interior o e. us, q = 0. For te 3D elements, on eac ace, it is easy to calculate tat v n = curl(bq) n = b (n q) n. n Here b/ n is proportional to b on te ace. en we get b (n q) n rds = 0, r R 0 () or P0 2 (), F(). (4.18) e remaining proo is essentially rom Lemma 3 by ai and Winter [39]. Note r P 2 0 () R 0 (). en it is easy to get q t (x b ) = 0, F(). (4.19)

14 450 X.P. XIE, J.C. XU AND G. XUE Here q t := (n q) n is te tangential component o q, x b is te barycenter o te ace. For q P1 3 (), rom (4.19) we easily know tat q can be ormulated as q = 4 c i (λ i 1 3 ) λ i. (4.20) i=1 Here c 1, c 2,, c 4 are arbitrary constants. en, te tangential component o q on te ace 1 as te orm 4 q t = c i (λ i 1 3 )( λ i) t. (4.21) i=2 For te irst and second 3D elements, rom Lemma 4.3, we get c 2 + c 3 + c 4 = 0. By considering te oter tree aces we can also get c 1 + c 3 + c 4 = 0, c 1 + c 2 + c 4 = 0 and c 1 + c 2 + c 3 = 0. ese our relations imply tat c i = 0, i = 1, 2, 3, 4. For te tird and ourt elements, by te construction o te space Y 3, it ollows immediately c i = 0, i = 1, 2, 3, 4. ereore, q = Veriication o assumptions Let V (i), i = 1, 2,, 6, be te inite dimensional velocity spaces corresponding to te six velocity inite elements listed in te able 4.1 wit all degrees o reedom o v V (i) being zero on Ω. It is easy to see V (i) H(div, Ω), but H0 1 (Ω). ese coices o spaces lead to nonconorming inite element metods o te problem (2.13). For te pressure, let te inite dimensional space W (i), i = 1, 2,, 6, be as ollows: W (1) := {q W : q P 1 ()}, W (2) := {q W : q P 0 ()}, W (3) := {q W : q P 1 ()}, W (4) := {q W : q P 0 ()}, W (5) := {q W : q P 1 ()}, W (6) := {q W : q P 0 ()}. aking V = V (i) and W = W (i), te assumption (S2) is trivially satisied by te constructions. For eac inite element space V, te canonical interpolation operator Π : V V is deined by te corresponding degrees o reedom in V. us, q W (i) and v V, divπ vqdx = Π v qdx + Π v nqds, v qdx + v nqds = divvqdx = Q divvqdx. = ereore, by te assumption (S2), we get te commutativity property divπ v = Q divv. (4.22) Furtermore, since te operator Π preserves linear polynomials locally, we can prove tat tere old te interpolation error estimates. v Π v j, k+1 j v k+1,, 0 j k 1, (4.23)

15 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 451 and te H 1 bound property Π v 1, v 1,. (4.24) Notice tat tese elements are not invariant under te Piola transormation. Consequently, a dierent argument is required to prove te interpolation error estimate. e analysis can be done by scaling to a similar element o unit diameter using translation, rotation, and dilation and using te compactness argument [40,41]. us, te assumption (S1) and (4.10) old. Now we sall derive consistency error estimates or all te six elements. We consider te detailed discussion in 3D case (2D case is similar and easier). By te interace condition [σ(u, p)n] Γij = 0, we can rewrite te consistency error term (3.7) as E (u, p,v ) = σ(u, p)n [v ]ds. (4.25) F() On te ace, decompose te vector σ(u, p)n and v along te normal direction n and along te tangential direction to te ace, i.e., σ(u, p)n = (σ(u, p)n n)n + n (σ(u, p)n n), and v = (v n)n + n (v n). en we get E (u, p,v ) = (σ(u, p)n n) [v n]ds. (4.26) F() Let and + denote te two tetraedrons saring te same ace. Denote w + := w +, w := w, ere w can be eiter a scalar or a vector. In addition, denote σ + (u, p) := σ(u +, p + ) and σ (u, p) := σ(u, p ). For all te our 3D elements in Section 4, te ollowing uniorm consistency error estimate olds. Lemma 4.5. For u H0 1, v V (i), i = 3, 4, 5, 6, E (u, p,v ) ν 1/2 u 1, v. (4.27) Proo. By te continuity o te normal component o te stress tensor, on te ace we ave tat σ(u, p)n = σ + (u, p)n = σ (u, p)n. We irst estimate ( σ + (u, p)n n ) [v n]ds = (ν + u + ( n n) [v n]ds (p + I)n n ) [v n]ds = (ν + u + n n) [v n]ds = ν + ( u + n n λ)[v n µ]ds ν + in λ R u+ n n λ 0, in [v n µ] 0, 2 µ R 2 ν + u 2, + v 1, +. (4.28) Here te tird equality ollows rom te deinition o degrees o reedom, te irst inequality rom Caucy-Scwartz inequality, and te second inequality rom te standard scaling argument and Bramble-Hilbert Lemma. Similarly, we can get ( σ (u, p)n n ) [v n]ds ν u 2, v 1,. (4.29)

16 452 X.P. XIE, J.C. XU AND G. XUE From te above two estimates, (4.28) and (4.29), it ollows tat (σ(u, p)n n) [v n]ds ( ν + u 2 2, + + ν u 2 2, ) 1/2 ( ν + v 2 1, + + ν v 2 1, ) 1/2. Finally, applying Caucy-Scwartz inequality, we ave te consistency error estimate E (u, p,v ) ( ) ν + u 2 + ν u 2 2, + 2, F() ν 1/2 u 1, v. 1/2 F() ( ν + v 2 1, + + ν v 2 1, For te 2D case, on te edge e, decompose te vector σ(u, p)n and v along te normal direction n and along te tangential direction t, i.e., σ(u, p)n = (σ(u, p)n n)n + (σ(u, p)n t)t and v = (v n)n + (v t)t. en, we get E (u, p,v ) = (σ(u, p)n t)[v t]ds. (4.30) e E() e Similar to te 3D case, te ollowing uniorm consistency error estimate olds or te two 2D elements in Section 4. Lemma 4.6. For u H0 1, v V (i), i = 1, 2, ) E (u, p,v ) ν 1/2 u 1, v. (4.31) 1/2 5. Application o Stable Stokes Elements or a Modiied Formulation Brezzi, Frotin and Marini [22] studied te mixed orm o Poisson equation and modiied te variational ormulation suc tat te coercivity condition automatically eld on te discrete level. We can apply te same tecnique by considering te ollowing equivalent ormulation o (2.1). Find {u, p} V W suc tat { a(u,v) + M(divu,divv) (p,divv) =<,v > +M < g,divv > v V, (5.1) (divu, q) =< g, q > q W, were M is given by (2.4). Correspondingly, we ave te ollowing discrete weak ormulation. Find {u, p } V W suc tat v V and q W, { a (u,v ) + M(divu,divv ) (p,divv ) =<,v > +M < g,divv >, (5.2) (divu, q ) =< g, q >. Under tis modiied ormulation, any pair o stable Stokes elements tat satisy te in-sup condition (divv, q ) sup q 0 q W, (5.3) v V v 1, is uniormly stable under te norms given in (2.15) and (2.3).

17 Uniormly Stable Finite Element Metods or Darcy-Stokes-Brinkman Models 453 Furtermore, by te standard saddle point teory [25 27], we ave u u + p p in v V u v + in q W p q + sup v V E (u, p,v ) v. (5.4) Here te consistency error is deined by E (u, p,v ) = (σ(u, p)n v divu(v n)) ds. (5.5) It is easy to see tat under (H2 ) te new ormulation (5.2) is equivalent to te original one (2.13). 6. Concluding Remarks and Future Work We sow tat any traditional stable Stokes element is also uniormly stable or te Darcy- Stokes-Brinkman equations wit respect to te viscosity and zerot-order term coeicient and teir jumps i and only i te discretely divergence-ree velocity implies almost everywere divergence-ree one. We also discuss te construction o uniormly stable elements on te basis o H(div)-conorming elements. By keeping te structure o standard H(div)-conorming elements, we construct several new uniormly stable 2D and 3D elements in a uniied way. On te oter and, te original weak ormulation o Darcy-Stokes-Brinkman equation can be equivalently modiied in suc a way tat any traditional stable Stokes element is also uniormly stable. Among tese tree metods, te modiied H(div) elements ave te exact sequence property wic is an important tool to design and analyze preconditioner and multigrid metod or te resulting linear systems. In tis sense, we regard tese modiied H(div) elements as solver riendly ones, and we will discuss it in uture work. Acknowledgments. is work was supported in part by NSF DMS and by te Center or Computational Matematics and Applications o Penn State. Jincao Xu was also supported in part by NSFC and Alexander H. Humboldt Foundation. Xiaoping Xie was supported by te National Natural Science Foundation o Cina ( ), te National Basic Researc Program o Cina (2005CB321701), and te program or New Century Excellent alents in University (NCE ). Xiaoping Xie would like to acknowledge te support during is visit to te Center or Computational Matematics and Applications o Penn State. Reerences [1] D.A. Nield and A. Bejan, Convection in Porous Media, Springer-Verlag, [2] X. Li, Principles o Fuel Cells, aylor and Francis, [3] G. Hoogers, Fuel Cell ecnology Handbook, CRC Press, [4] J. Larminie and A. Dicks, Fuel Cell Systems Explained, Wiley, 2nd edition, [5] Z.H. Wang, C.Y. Wang and K.S. Cen, wo-pase low and transport in te air catode o proton excange membrane uel cells, J. Power Sources, 94 (2001), [6] P. Sun, G. Xue, C.Y. Wang and J. Xu, New numerical tecniques or two-pase transport model in te catode o a polymer electrolyte uel cell, submitted, (2007). [7] P. Sun, G. Xue, C.Y. Wang and J. Xu, A domain decomposition metod or two-pase transport model in te catode o a polymer electrolyte uel cell, submitted, (2007).

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