A FETI-DP method for Crouzeix-Raviart finite element discretizations 1

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1 COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vo. 1 (2001), No. 1, pp c 2001 Editoria Board of the Journa Computationa Methods in Appied Mathematics Joint Co. Ltd. A FETI-DP method for Crouzeix-Raviart finite eement discretizations 1 LESZEK MARCINKOWSKI Facuty of Mathematics, University of Warsaw Banacha 2, Warszawa, Poand E-mai: L.Marcinowsi@mimuw.edu.p TALAL RAHMAN Department of Computer Engineering, Bergen University Coege Nygårdsgaten 112, N-5020 Bergen, Norway E-mai: Taa.Rahman@hib.no Abstract This paper is concerned with the construction and anaysis of a parae preconditioner for a FETI-DP system of equations arising from the nonconforming Crouzeix-Raviart finite eement discretization of a mode eiptic probem of second order with discontinuous coefficients. We show that the condition number of the preconditioned probem is independent of the coefficient jumps, and grows ony as (1 + og(h/h) 2, where H and h are mesh parameters, in other words the preconditioner is quasi optima Mathematics Subject Cassification: 65N55, 65N30, 65F08. Keywords: FETI-DP method, Crouzeix-Raviart nonconforming finite eement method, domain decomposition, eiptic differentia equations of second order. 1. Introduction In many scientific appications, where partia differentia equations are used to mode, the Crouzeix-Raviart (CR) finite eement [18] appears to be among the most commony used nonconforming finite eement for the discretization, which incudes appications ie the Poisson probem, the Stoes or the Navier-Stoes probem (cf. [18, 38]), the Darcy-Stoes probem (cf. [15], the easticity probem (cf. [5, 22]), and probems on nonmatching grids (cf. [30, 34]). There is aso a cose reationship between mixed finite eements and the nonconforming finite eement for the second order eiptic probem which maes the CR finite eement interesting; cf. [1, 2]. The eement has aso been used in the framewor of finite voume eement method; cf. [16]. There are a number of effective sovers of the domain decomposition type for the CR finite eement which can be found in the iterature, see, e.g., [7, 17, 26, 31, 36, 37] for wor on two eve agorithms, [23, 33, 35] on mutieve agorithms, [4, 6] on mutigrid agorithms, and [32] on substructuring type agorithms. To our nowedge, however, there is not any 1 This wor was partiay supported by Poish Scientific Grant: N N

2 2 L. Marcinowsi and T. Rahman wor on FETI-DP type domain decomposition agorithms for the Crouzeix-Raviart (CR) finite eement. The FETI-DP methods form a cass of fast and efficient iterative sovers for the agebraic systems arising from the finite eement discretization of partia differentia equations of second and fourth order; see, e.g. [21, 25, 28, 29, 19, 20, 24, 10, 11, 14] and references therein. In this paper, we introduce a FETI-DP method for soving the system of equations arising from the nonconforming Crouzeix-Raviart finite eement discretization of a mode eiptic probem with discontinuous coefficients across subdomains interfaces. Modes with discontinuous coefficients pay an important roe in the scientific computing, as for instance, in the simuation of fuid fow in porous media where the permeabiity of the porous media may have arge jumps across subdomain interfaces. Large jumps in the coefficients may resut in bad convergence for the iterative methods. We propose an effective parae preconditioner for soving the FETI-DP probem. We show that the condition number of the preconditioned system is independent of the jumps of the coefficient, and grows as (1+og(H/h) 2, where H is the subdomain size and h is the mesh size. Consequenty, if the conjugate gradient method is used to sove the preconditioned probem, the method wi converge independenty of the jumps of the coefficient whie the number of iterations wi depend ony ogarithmicay on the ratio between the subdomain size and the mesh size. The remainder of the paper is organized as foows: We begin by describing the Crouzeix- Raviart discretization of a mode probem in Section 2. In Section 3 the FETI-DP probem is introduced and then a parae preconditioner for this probem is defined in Section 4. In Section 5 we discuss a few impementation issues. Finay, Sections 6-7 are devoted to estabishing bounds for the condition number of the preconditioned probem. Throughout the paper, the foowing notations are used: u v, x y and w z mean that there exist positive constants c and C independent of the subdomain- and mesh size parameters as we as of the coefficient, such that c u v C u, x c y and w C z, respectivey. 2. Discrete probem We consider a poygona domain Ω in the pane, and assume that we have a partition of Ω into disjoint set of poygona subdomains {Ω } =1,...,N such that N (empty) Ω = Ω, Ω Ω = an edge,, = 1,...,N =1 a vertex form a shape reguar trianguation of Ω in the sense of [9]. Our mode second order eiptic probem is as foows: Find u H 1 0(Ω) such that where, a(u,v) = f(v) v H 1 0(Ω) (1) a(u,v) = N =1 with ρ being positive constant coefficients. Ω ρ u v dx

3 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 3 We introduce T h = T h (Ω) = {τ} a trianguation of the domain Ω consisting of trianguar eements τ, where any common edge between two subdomains wi not cut through any eement of Ω, i.e. any τ T h is contained in ony one subdomain. Thus we get a trianguation of Ω, which is inherited from T h : T h (Ω ) := {τ : τ T h ; τ Ω }. Let h = max τ Th diam(τ) be the mesh size parameter. We assume that the trianguation T h is quasiuniform (cf. [3] or [13]). Let Γ = Ω Ω denote the open common (interface) Figure 1. The Crouzeix-Raviart (CR) finite eement. Degrees of freedom are the associated with the midpoints of the edges of the triange, aso caed the CR noda points or simpy the CR nodes. edge between the two neighboring subdomains Ω and Ω, and Γ = N =1 Ω \ Ω be the interface seeton. The degrees of freedom of the Crouzeix-Raviart finite eement are the function vaues associated with the midpoints of the edges of a triange and we ca such points the CR noda points or simpy the CR nodes. We et the sets of CR noda points that are contained in Ω, Ω, Ω, Ω, and Γ, be denoted by Ω CR h, ΩCR h, ΩCR,h, ΩCR,h, and Γ CR,h, respectivey. Our Crouzeix-Raviart finite eement space Ŵ h (Ω) consists of functions that are piecewise inear over the trianguation T h, continuous at the CR noda points of Ω CR h, and equa to zero at the CR noda points of Ω CR h ; cf. [3] or [13]. The Crouzeix-Raviart finite eement is a nonconforming finite eement since Ŵ h (Ω) H 1 (Ω). We equip our discrete space with the foowing broen H 1 -seminorm and -norm: u 2 H 1 h (Ω) := τ T h u 2 H 1 (τ), u 2 H 1 h (Ω) := u 2 L 2 (Ω) + u 2 H 1 h (Ω). where A discrete formuation of the probem (1) is then: Find u h Ŵ h (Ω) such that a h (u,v) := a h (u h,v) = f(v) v Ŵ h (Ω), (2) N a,h (u,v), a,h (u,v) := =1 τ T h (Ω ) τ ρ u v dx. (3) The probem (2) has a unique soution and optima error bounds can be estabished in the broen H 1 norm and seminorm; cf. [13]. 3. FETI-DP method In this section, we introduce our FETI-DP method for soving (2) using the framewor given in [39, Ch. 6].

4 4 L. Marcinowsi and T. Rahman First on each Ω, we introduce a oca CR finite eement subspace W h (Ω ) consisting of functions that are piecewise inear over the trianguation T h (Ω ), continuous at a CR nodes in Ω CR,h, and equa to zero at a CR nodes in ΩCR h Ω CR,h. In the same way as before, the oca subspace is equipped with the foowing oca broen H 1 -seminorm and -norm: u 2 := Hh 1(Ω ) τ T h (Ω ) u 2 H 1 (τ), and u 2 := Hh 1(Ω ) u 2 L 2 (Ω ) + u 2, respectivey. Let Hh 1(Ω ) W h (Ω) := Π N =1W h (Ω ) be the goba space defined on the domain Ω. Next on each Γ, the edge common to Ω and Ω, we introduce V h (Γ ) as the space of piecewise constant functions over T h (Γ ), which is a 1D trianguation of Γ inherited from the trianguation T h. Then, we et V h := Π Γ ΓV h (Γ ) be the auxiiary interface space, which, further, wi be used as the space of Lagrange mutipiers. The biinear form b(, ) : W h (Ω) V h R, associated with the two spaces W h (Ω) and V h, is then defined as b(u,ψ) := Γ Γ > Γ (u Γ u Γ )ψ ds, for u = (u j ) N j=1 W h (Ω) and ψ = (ψ ) Γ Γ V h with u W h (Ω ) and ψ V h (Γ ). Let (ψ m ) m Γ CR be an L2 -orthogona basis of V h (Γ,h ), where each basis function ψ m is piecewise constant over the trianguation T h (Γ ), being equa to 1/ e m on the eement e m T h (Γ ) whose midpoint is m and zero on the remaining eements of T h (Γ ). Let u W h (Ω ) and u W h (Ω ), for two subdomains Ω and Ω sharing an edge, and et e m T h (Γ ) be the 1D eement with the midpoint m. Then we have Γ (u Γ u Γ )ψ m ds = 1 e m e m (u em u em ) ds = u (m) u (m) (4) since u em and u em are two inear poynomias. Thus we see that u = u at Γ CR,h ony if (u Γ u Γ )ψ ds = 0 ψ V h (Γ ). Γ This brings us to the definition of Ŵ h (Ω), a constrained subspace of W h (Ω): Ŵ h (Ω) = {u W h (Ω) : b(u,ψ) = 0 ψ V h }. if and The vertices of subdomains, those ying inside of Ω, aso nown as crosspoints, pay an important roe. Let Γ Γ be an interface with a crosspoint c r being its one end. Then there is a CR node m Γ CR,h, that is the midpoint of e m T h (Γ ), such that c r e m. Let the set of a such CR nodes associated with the crosspoint c r be denoted by V CR (c r ) (cf. Figure 3), and et V CR = c r V VCR (c r ), where V is the set of a crosspoints. We define W h (Ω) as the subspace of W h (Ω), consisting of functions which are continuous at the CR nodes of V CR. Note that Ŵ h (Ω) W h (Ω) W h (Ω).

5 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 5 Figure 2. Iustrating CR nodes (both circes and bac dots) associated with the degrees of freedom of the CR finite eement space W h (Ω). Functions of W h (Ω) can be discontinuous at the CR nodes aong the subdomain interfaces. Four subdomains are shown in the figure. We wi see our soution in W h (Ω). The corresponding Lagrange mutipier space Ṽ h and the oca interface Lagrange mutipier spaces are defined as: We note that Ṽ h (Γ ) := {λ V h (Γ ) : λ(m) = 0, m Γ CR,h V CR }, (5) Ṽ h := {λ V h : λ(m) = 0, m V CR } = Π Γ ΓṼ h (Γ ). (6) Ŵ h (Ω) = {u W h (Ω) : b(u,ψ) = 0 ψ Ṽ h }. We next introduce a oca decomposition of any function u W h (Ω ) as u = P u + H u, where P u W h 0 (Ω ) is defined by where a,h (P u,v) = a,h (u,v) v W h 0 (Ω ), W h 0 (Ω ) = {u W h (Ω ) : u(m) = 0 m Ω CR,h } W h (Ω ), and H u W h (Ω ) is the discrete harmonic part of u, i.e. H u = u P u, which is defined as the soution of the foowing probem: Find H u such that { a,h (H u,v) = 0 v W h 0 (Ω ), H u (m) = u (m) m Ω CR,h. A function in W h (Ω ) is discrete harmonic if u = H u, and we introduce W = H W h (Ω ) (7)

6 6 L. Marcinowsi and T. Rahman as the oca space of discrete harmonic functions. Consequenty, Hu = (H u ) N =1 for u = (u ) N =1 W h (Ω). Let the goba spaces of discrete harmonic functions, corresponding to the three spaces W h (Ω), W h (Ω), and Ŵ h (Ω) be defined as: W := HW h (Ω) = Π N =1W W h (Ω), (8) W := H W h (Ω) = {u W : u is continuous at the CR nodes in V CR }, (9) Ŵ := HŴ h (Ω) = {w W : b(w,ψ) = 0 ψ Ṽ h }. (10) c r Figure 3. Iustrating V CR (c r ) that is the set of CR nodes (bac dots) associated with the crosspoint c r (bac square) of the four subdomains as shown in the figure. Note that we can decompose the soution of (2) as u h = (u h,) N =1 = u I + w h where u I = (P u h, )N =1, and P u h, for = 1,...,N can be computed soving N independent oca subprobems. The discrete harmonic part of u h, i.e. w h = (H u h, )N =1 Ŵ, is the unique soution of the foowing probem: Find wh such that a h (wh,v) = f(v) v Ŵ. (11) We reformuate the probem (11) as an optimization probem: Find wh W such that J(wh ) := 1 2 Sw h,w h f,w h min } Bwh = 0. (12) where S = diag(s () ) N =1 is a boc diagona matrix with S() being the standard Schur compement matrix with respect to the space W, B = diag(b () ) Γ is a boc diagona

7 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 7 Figure 4. Iustrating CR nodes associated with the degrees of freedom of the CR finite eement space W h (Ω). The functions of W h (Ω) are continuous at the CR nodes of V CR (bac dots). The figure is showing four subdomains. matrix with B () (for > ) containing ony zeros, ones and minus ones as matrix entries, and wh is the vector representation of the function w h W (denoted by the same symbo). For simpicity, we use the same notation to represent a function in a space and its vector representation with respect to the noda basis of that space. Probem (12) can be reformuated as a sadde point probem by introducing a Lagrange mutipier λ V h : Find (wh,λ) W V h such that Sw h + B T λ = f, Bw h = 0. (13) Here λ is the vector representation of the function λ V h (denoted by the same symbo). The probem has a soution with a unique first component, i.e. wh, which is aso the soution of (11). In the one eve FETI method, this probem woud be used; cf. [39, 6.3]. Note that the matrix S is a singuar matrix if there is a subdomain Ω without having any edge on Ω. This is one of the reasons why we repace the space W in the above probem with W, which is equivaent to assuming continuity at the CR nodes of V CR ; cf. Figure 4. As a consequence, we decompose W into two new subspaces, W and ŴΠ, such that W = ŴΠ W, (14) where W W is the (prima) subspace consisting of functions which vanish at the CR nodes of V CR, and ŴΠ Ŵ is the (dua) subspace consisting of functions which vanish at the CR nodes on Γ, except the ones in V CR. Accordingy, for any u W, we regroup its unnowns so that we can write it in its vector representation as (cf. (14)): u = (u Π,u ) T,

8 8 L. Marcinowsi and T. Rahman where u Π is the vector of (prima) unnowns associated with the CR nodes in V CR, and u is the vector of (dua) unnowns associated with the remaining CR nodes, i.e. the CR nodes on Γ except the ones in V CR ; cf. (14). Here we identify the vector u with the function u W, and the vector u Π with the function u Π ŴΠ. Simiary, we partition the vector w h as w h = (w Π,w )T. Let à be the stiffness matrix associated with W h (Ω), which is obtained by restricting the corresponding stiffness matrix associated with W h (Ω), i.e. A = diag(a 1,...,A N ), where A is the oca stiffness matrix with respect to a,h (, ) in the standard CR noda basis. Note that, whie A is boc diagona, the matrix à is not. We partition the matrix à into bocs by regrouping its rows and coumns as foows: à = A II A IΠ A I A ΠI A ΠΠ A Π A I A Π A where the subscript I corresponds to the unnowns in the interior of the subdomains, and Π and correspond to the prima and dua unnowns, respectivey. We define the matrix for S : W W as S := A ( ) ( A A I A II A IΠ Π A ΠI A ΠΠ, ) 1 ( AI A Π ). (15) We aso get a reduced right hand side vector f from the oca right hand sides. The new optimization probem is then to find w W such that J(w ) := 1 2 Sw,w f,w min B w = 0, where the submatrix B is created from the matrix B by removing the rows and coumns corresponding to the (prima) unnowns associated with the CR nodes in V CR. Introducing a Lagrange mutipier we get an equivaent sadde point probem: Find (w, λ h ) W Ṽ h such that } (16) Sw + B T λ h = f, B w = 0 (17) Lemma 7.3 yieds that S is nonsinguar, see Section 7 beow. Thus we can eiminate w and arrive at the FETI-DP probem: Find λ h such that F λ h = d, (18) with F := B S 1 B T, d := B S 1 f.

9 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 9 4. Preconditioner In this section, we propose a preconditioner for the system (18); cf. [39, 6.4]. Let the matrix for S : W W be the submatrix of S obtained by restricting the matrix S to the subspace W. We first note that S = diag (S () ), with S () being the restriction of the oca Schur compement matrix S() to the subspace of functions from W which vanish at the CR nodes in V CR Ω CR,h. We define D = diag(d () )N =1, where D () representing D() : Π Γ Ω Ṽ h (Γ ) Π Γ Ω Ṽ h (Γ ) (cf. (5)), is a diagona scaing matrix such that, for any CR noda point m Γ CR,h \ VCR, we have We define a scaed jump operator as: ( D () λ ) (m) = λ(m). + B D, := [D (1) B(1),...,D(N) B(N) )], (19) where B () is obtained by extracting the coumns of B associated with the space W. Note that B D, is a row scaed B matrix. The preconditioner is then defined as: M 1 NN = B D, S B T D,. (20) This preconditioner is symmetric and positive definite; see Lemma 7.3 beow. In Section 7, we prove bounds for the condition number of the preconditioned system; see Theorem Impementation In this section we briefy discuss some impementation issues. For the simpicity of presentation we present our method in terms of the Richardson iteration, whie in practice the CG method is used. The Richardson iteration is then given as: λ (n+1) = λ (n) τm 1 NN (Fλ(n) d) = λ (n) + τs (n) n 0 where s (n) = M 1 NN (d Fλ(n) ), λ (0) Ṽ h is arbitrary, and τ is a propery chosen parameter. In each Richardson iteration, the two most computationay intensive tass are: computing r (n) := d Fλ (n), computing s (n) = M 1 NN r(n). In an appication of F = B S 1 B T to a vector, most of the computationa effort goes into computing the action of S 1, which can be done by soving the system with à directy with a propery defined right hand side. In order to ower the cost of directy soving this system, one has to use a proper ordering of the unnowns; we refer to [39, 6.4.1] for detais. In an appication of M 1 NN to a vector, the most expensive part of the computation is invoved in the action of S (), which requires soving a oca Dirichet probem in the corresponding subdomain; cf. [39, 6.4.1].

10 10 L. Marcinowsi and T. Rahman 6. Technica toos In this section we state and proof a few technica emmas necessary for the main resuts of the paper. Let X h/2 (Ω ) be the conforming space of piecewise inear continuous functions on the trianguation T h/2 (Ω ) which is constructed by joining the midpoints of the edges of the eements of T h (Ω ), and et X h/2 0 (Ω ) be the subspace of X h/2 (Ω ) consisting of functions with zero traces on Ω. Let the sets of a vertices of trianges from T h (Ω ) which are in Ω, Ω, and E an open edge of Ω, be denoted by Ω,h, Ω,h and E,h, respectivey. For each open edge E Ω we introduce a oca equivaence map (isomorphism), M E : WE h(ω ) X h/2 (Ω ) (cf. [30]), where WE h(ω ) is a subspace of W h (Ω ) formed by a functions which are zero in Ω CR,h \ ECR,h. This mapping is a sighty modified version of a simiar mapping introduced in [37]; see aso [8]. Definition 6.1. For given u W h E (Ω ), we introduce M E u Xh/2 (Ω ) by defining the vaues of M E u at the noda points of the trianguation T h/2(ω ). The noda points are divided into four sets of noda points: For p Ω CR,h ΩCR,h et ME u(p) = u(p). If p Ω,h, then M E u(p) = 1/(N(p)) τ h j u τ h j (p). where the sum is taen over a trianges τ h j with the common vertex p and N(p) is the number of eements with p as an vertex. For q a midpoint on the boundary, i.e. q Ω,h \ E,h, et M E u(q) = 0. If q on E is a vertex, i.e. q E,h,then M E qqr u(q) = u(q q q r ) + q q u(q q q r r), where q,q r are the eft and right neighboring CR noda points of q and ab is the ength of the segment with a,b as its ends. Note that M E u is piecewise inear between the CR nodes of ECR h, and ME u is zero on Ω \E. The mapping M E u has the foowing properties (cf. [30, Lemma 3.4]): u H 1 h (Ω ) M E u H 1 (Ω ), u L 2 (Ω ) M E u L 2 (Ω ). (21) We aso need the foowing technica resut which is a specia case of Lemma 4.1 in [27]. Let X h/2 (E) be the space of traces of functions from X h/2 (Ω ) onto this edge and H = diam Ω. Lemma 6.1. Let E = [a,b] Ω be an edge of Ω with a,b as its ends. Then for a u X h/2 (Γ ) such that u(a) = u(b) = 0 we have u H 1/2 00 (E) and u 2 H 1/2 00 (E) u 2 H 1/2 (E) + (1 + og(h /h) u 2 L (E). The next emma gives us an extension property of the CR discrete harmonic functions.

11 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 11 Lemma 6.2. Let u W and u = 0 in Ω CR,h \ ECR for an edge E Ω. Then we have u H 1 h (Ω ) M E u H 1/2 00 (E). For the proof of this emma we refer to the proof of Lemma 5.1 in [31]; cf. aso [37] where the definition of equivaence mapping is sighty different. Let E Ω be an edge. First for any u W we introduce u E W WE h(ω ) a oca discrete harmonic function, which is zero at the CR nodes of in Ω CR,h \ ECR h, and u E (m) = u(m) for m Eh CR. In the foowing emma we state an important property of an edge restriction of any function. Lemma 6.3. Let E Ω be an edge. Then for any u W we have ( ) u E 2 Hh 1(Ω ) (1 + og(h /h)) 2 H 2 u 2 L 2 (Ω ) + u 2 Hh 1(Ω, (22) ) where u E is a discrete harmonic function taing the same vaues as those of u at the CR noda points of Eh CR, and is equa to zero at the remaining CR noda points on Ω and H = diam Ω. The proof of this emma foows the ines of the proof of Lemma 4.4 in [30], the definition of the discrete harmonic extension is sighty changed there. The next emma is a Soboev ie inequaity for the CR eement; cf., e.g., [37]. Lemma 6.4. For any u W h (Ω ), we have where H = diam Ω. u 2 L (Ω ) (1 + og(h /h))(h 2 u 2 L 2 (Ω ) + u 2 H 1 h (Ω ) ), Let Γ Γ be the common edge of subdomains Ω and Ω. Note that there are two points in Γ CR,h VCR, namey the two CR noda points which ie cosest to the ends of this edge, one next to each end. Thus we can introduce I Γ H u s for s =, and u s W s as a unique inear poynomia such that I Γ H u (p) = u (p) for p Γ CR,h VCR. We aso need the foowing technica resut: Let Γ Γ be an edge of Ω and et for any u W define u H,Γ W as a function such that u H,Γ is zero at the CR nodes in Ω CR,h \ ΓCR,h and uh,γ (m) = I Γ H Lemma 6.5. It hods that M Γ where H = diam Ω. u(m) for m ΓCR,h. uh,γ 2 H 1/2 (Γ ) (1 + og(h /h)) ( ) H 2 u 2 L 2 (Ω ) + u 2 Hh 1(Ω, ) Proof. Let T h/2 (Γ ) be the one-dimensiona trianguation inherited from T h/2 (Ω ). Let a s, s = 1, 2 be the ends of this edge, and φ as be a continuous piecewise inear function which equas one at a s and zero at a remaining noda points of T h/2 (Γ ), i.e. at a remaining midpoints and vertices of T h (Γ ). Then note that M Γ uh,γ yieds that M Γ Finay, using Lemma 6.4 we end the proof. = I Γ H u s=1,2 (IΓ H u(a s))ψ as. Thus a standard argument uh,γ H 1/2 (Γ ) max s=1,2 IΓ H u(a s) u L (Ω ).

12 12 L. Marcinowsi and T. Rahman 7. Condition number estimate Note that formay S and S denote two matrices. In the foowing we wi use the same notations for their operators, i.e., S : W W and S : W W. Consequenty, if v = Sw for any functions w W, then v (the vector representing the function v W) is obtained by mutipying w (the vector representing the function w W) by the matrix S. We note that any function w W W can be represented as a function in W, in its vector representation, as (0,w ) T. Consequenty, since S is the restriction of S to the space W, we can write S w,w = ( 0 S w ) ( 0, w ). We use the foowing norms for w = w Π + w W and w W : w 2 S := Sw,w, w 2 S := S w,w, w 2 e S := Sw,w. Next, we state the foowing agebraic fact: for any u W, it hods that Su,u = inf Su,u, (23) u where the minimum is taen over a u = u +u Π W with u Π ŴΠ; cf. [39, Lemma 6.22]. We note that writing u = u + u Π using functions, is equivaent to writing u = (u,u Π ) T using their respective vector representations. Using an agebraic argument (cf. [39, 6.4, (6.78)]), we get the foowing resut: Let λ Ṽ h, then Fλ,λ = λ,b sup w 2 = sup 0 w W w 2 e S 0 w W λ,b w 2. (24) w 2 S A crucia roe, in our anaysis, is payed by the foowing projection operator, P : W W, defined as P w := B T D, B w, (25) where w on the right hand side is the vector representing the function w W, the uniquey defined component in the decomposition of w, i.e. w = w Π + w, with w Π ŴΠ. The main properties of this operator are stated in the foowing emma. Lemma 7.1. Let P be defined in (25). Then, For any interface Γ being the common edge of Ω and Ω, u = (u 1,...,u N ) W and m Γ CR,h we have (P u) (m) = (u + (m) u (m)) (26) P preserves the jumps of the functions from W, i.e. for any u W it hods that B P u = B u

13 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 13 For any u Ŵ W we have For any u W we have P u = 0 u P u Ŵ. Proof. First we prove (26). Let z := B u Ṽ h and w := B T D, z = BT D, B u = P u. We see that for any midpoint m Γ CR,h \ VCR : where w (m) = (BD, z)(m) T = (B D T () z)(m) = Next we see that δ := + { 1 >, 1 >. (B z)(m) T = δ z(m), + z(m) := (B u)(m) = δ (u (m) u (m)). (27) Thus combining the ast two equations we get (26). Next (26) and (27) yied that (B P u)(m) = (B w)(m) = δ (w (m) w (m)) [ ] = δ (u + (m) u (m)) (u + (m) u (m)) = δ (u (m) u (m)) = z(m), what ends the proof of the second statement of the emma. The third statement of this emma foows directy from the first statement, i.e. (26), as functions in Ŵ are continuous at a CR noda points. The ast statement foows again from (26). Namey, we see that (u P u) (m) = u (m) Thus we get (u + u )(m) = (u P u) (m) = (u P u) (m), u + (m) + i.e. u P u is continuous at a CR nodes on the subdomain boundaries. u + (m). (28) Next we obtain: Lemma 7.2. For any µ Ṽ h there is a w W such that µ = B w (29) with and P w = w, w es P w S. (30)

14 14 L. Marcinowsi and T. Rahman Proof. For any µ Ṽ h and m Γ CR,h \ VCR, we define w, (m) = δ + µ(m). Then using the fact that δ = δ, which equas one or minus one, we see that The formua (28) yieds that B w (m) = δ (δ w, (m) δ w, (m)) = µ(m). (w P w ) (m) = (w P w ) (m) = 0, for any m Γ CR,h \VCR, and by the definition of P and W we have (P w) j (m) = w j (m) = 0 for any m V CR. Thus we can concude that w P w = 0 ŴΠ. Next using (23) we get what proves (30). From Lemma 7.2 we get Coroary 7.1. It hods that We have aso the foowing emma: Lemma 7.3. The matrices S and M 1 NN w es w S = P w S, Ṽ h = Im B = {B w : w W }. are symmetric and positive definite. Proof. For the proof we foow the ines of the proof of Lemma 6.33 in [39]. From the definition of S (cf. (15)), and M 1 NN (cf. (20)), it foows that the both matrices are symmetric. Next (23) and the fact that both S and S are nonnegative definite yied that S and M 1 NN are aso nonnegative definite. Thus it is enough to show that the both matrices are nonsinguar. Let w W be such that Sw = 0. If w = w + w Π W is the minimizing eement in (23), then w is a constant in Ω, but due to the boundary conditions and the continuity of w at the CR nodes of V CR it foows that w = 0, thus w = 0. Let v Ṽ h be such that M 1 NN v = 0. Then by Lemma 7.2 there is w W such that v = B w and P w = w and we have 0 = M 1 NN v,v = B D, S B T D, B w,b w = S P w,p w = S w,w that means that w is a constant. We can repeat the argument used above to get w = 0. This yieds that v = B w = 0. The next emma gives us a crucia norm estimate of the norm of the operator P.

15 A FETI-DP method for Crouzeix-Raviart finite eement discretizations 15 Lemma 7.4. Let P be defined as in (25). Then ( ( )) H P w S 1 + og w S h w W, where H = max =1,...,N H and H = diam Ω. Proof. Let P u = w = (w 1,...,w N ) W W. We note that P u 2 S = N w 2 S = =1 N ρ w 2 Hh 1(Ω ). =1 We now estimate one term of the right hand side: ρ w 2 H 1 h (Ω ). For any function z W and each edge Γ, which is a common edge of Ω and Ω, we can introduce z,γ W such that { z (m) m Γ CR,h z,γ (m) = 0 m Ω CR,h \ ΓCR,h and an anaogous function z,γ W. Note that w,γ and w = (P u) equas zero at a noda points in V CR Ω CR,h. We see that w = Γ Ω w,γ. Then a triange inequaity and Lemma 6.2 yied that w H 1 h (Ω ) w,γ H 1 h (Ω ) MΓ w,γ. 1/2 H 00 (Γ ) Γ Ω Γ Ω Let us now estimate M Γ w,γ H 1/2 00 (Γ ) one term from the sum reated to the edge Γ. We see that (u u )(m) = (u,γ u,γ )(m) m Γ CR,h (31) M Γ u,γ = M Γ u,γ on Γ, (32) The equaity (32) foows from the fact that traces of M Γ s u,γ onto Γ for s =, are defined by the vaues of u s,γ in Γ CR,h ; cf. Definition 6.1. Lemma 7.1 and (31) yied that MΓ w,γ 1/2 H = 00 (Γ ) ρ1/2 + M Γ (u,γ u,γ ) H 1/2 00 (Γ ). Because u W, we see that u and u tae the same vaues at Γ CR,h VCR and we have (cf. Lemma 6.5) We define u H,Γ s is zero at the noda points of Ω CR s,h \ ΓCR Lemma 6.5. Note that u H,Γ I Γ H u = I Γ H u,γ = I Γ H u = I Γ H u,γ. W s for s =,, as a discrete harmonic function in Ω s such that u H,Γ s,h, and uh,γ s (m) = I Γ H u s(m) for m Γ CR,h, cf. = u H,Γ at Γ CR,h, so M Γ uh,γ = M Γ u H,Γ on Γ. (33)

16 16 L. Marcinowsi and T. Rahman Next by Lemma 6.1, (32), and (33) we see that MΓ w,γ 1/2 H ρ1/2 00 (Γ ) ρ1/2 + s=, s ( M Γ (u,γ u,γ ) 2 H 1/2 (Γ ) + + (1 + og(h/h) M Γ (u,γ u,γ ) L (Γ ) ( M Γ s (u s,γ u H,Γ s ) 2 H 1/2 (Γ ) + ) 1/2 s=, +(1 + og(h/h) M Γ s ( s (u + c s ) s,γ 2 H 1 h (Ωs) + MΓ s ) (u s,γ u H,Γ s ) 2 1/2 L (Γ ) ((u + c s ) H,Γ s ) 2 H 1/2 (Γ ) +(1 + og(h/h) (u + c s ) s,γ 2 L (Γ )) 1/2, where c s for s {,} are any constants. The ast inequaity foows from a standard trace theorem, (21), and the fact that M Γ s (u u H,Γ s ) s,γ = M Γ s (u + c s ) s,γ M Γ s ((u + c s ) H,Γ s ) s,γ on Γ for any constant c s, s =,. Then the Soboev-ie inequaities (cf. Lemma 6.4, Lemma 6.5, and Lemma 6.3) give us: ρ M Γ w,γ 2 (1 + og(h/h) ( ) 2 ρ H 1/2 00 (Γ s H 2 u + c s 2 L ) 2 (Ω ) + u + c s 2 Hh 1(Ωs) s=, (1 + og(h/h) 2 s=, ρ s u 2 H 1 h (Ωs). The ast estimate is obtained by a Poincare inequaity for the Crouzeix-Raviart eement, (cf. [12] or [37]), combined with a scaing argument since c s, s =,, is an arbitrary constant. Finay, summing over a edges in Ω, and then over a subdomains ends the proof. The foowing theorem is the main resut of the paper: Theorem 7.1. It hods that M NN λ,λ Fλ,λ ( 1 + og ( )) 2 H M NNλ,λ λ h Ṽ h. Proof. In the proof of this theorem we use resuts from this section and the agebraic arguments from [39, 6.4]. Lower bound Lemma 7.2 yieds that for any µ Ṽ h we have w W such that µ = B w, P w = w, w es P w S. Then taing µ = M NN λ and using (eq:oper-f-repres) yied that Fλ,λ λ,b w 2 w 2 e S λ,b w 2 P w 2 S = λ,µ 2 B D, S B T D, = λ,b w 2 B T D, B w S = λ,µ 2 B T D, µ S µ,µ = λ,µ 2 M 1 NN µ,µ = λ, M NNλ.

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