BCCS TCHNICAL RPORT SRIS The Crouzeix-Raviart F on Nonmatching Grids with an Approximate Mortar Condition Taa Rahman, Petter Bjørstad and Xuejun Xu RPORT No. 17 March 006 UNIFOB the University of Bergen research company BRGN, NORWAY
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The Crouzeix-Raviart F on Nonmatching Grids with an Approximate Mortar Condition Taa Rahman Petter Bjørstad Xuejun Xu March 006 Abstract A new approximate mortar condition is proposed for the owest order Crouziex- Raviart finite eement on nonmatching grids, which uses ony the noda vaues on the interface for the cacuation of the mortar projection. This approach aows for improved and more fexibe agorithms compared to those for the standard mortar condition where noda vaues in the interior of a subdomain, those cosest to a mortar side of the subdomain, are aso required in the cacuation. 1 Introduction The genera concept of the mortar technique was originay introduced by Bernardi, Maday and Patera in [3], which provides a usefu too for couping different discretization schemes. In recent years, the approach has argey been appied to nonmatching grids for the design of agorithms which are very we suited for parae impementation, and which can easiy hande compicated geometries and heterogeneous materias. In order to ensure that the overa discretization makes sense, an optima couping between the meshes is required. In a standard mortar technique, this condition is reaized by appying the condition of weak continuity on the soution, caed the mortar condition, saying that the jump of the soution aong the interface between two meshes is orthogona to some suitabe test space. The mortar technique has been extensivey studied by many authors. A sadde point formuation for the mortar technique was studied in []. Later, an extension to three dimensions was introduced in [1]. Further anaysis and extensions of the mortar technique can be found in [8, 11, 17, 18, 19, 9, 5], and the references therein. BCCS, Bergen Center for Computationa Science, Thormøhensgt. 55, 5008 Bergen, Norway mai: taa@bccs.uib.no). Institute for Mathematics, University of Bergen, Johannes Brunsgate 1, 5008 Bergen, Norway, mai: petter@ii.uib.no). LSC, Institute of Computationa Mathematics, Chinese Academy of Sciences, P.O.Box 719, Beijing, 100080, Peope s Repubic of China mai: xxj@sec.cc.ac.cn). 1
In this paper, we are interested in the appication of the mortar technique on nonmatching grids, where in each subgrid, the nonconforming P1 or the owest order Crouzeix- Raviart CR) finite eement is used for the discretization. The first anaysis of the mortar technique for such an eement was given by Marcinkowski in [11]. In the event of appying the mortar condition, it is necessary to know the function on the interface. For the conforming P1 finite eement, it is enough to know the noda vaues aong the interface. However, for the nonconforming P1 finite eement the owest order Crouzeix-Raviart finite eement), where the degrees of freedom are associated with the edge midpoints, the function on the interface depends on the noda vaues corresponding to interface nodes and some subdomain interior nodes ying cosest to the interface, see [11] and Fig. 1 for iustration. Mortar Nonmortar Figure 1: The owest order Crouzeix-Raviart CR) finite eement eft) and two nonmatching grids right). CR basis functions associated with the nodes on the mortar side, denoted by dots in the interior) and semi-dots on the mortar side of the interface), have nonzero support on the nonmortar side, denoted by shaded trianges. A variant of the mortar technique has recenty been proposed in [13], where the standard mortar condition, cf. [11], has been modified so that the method wi use ony the noda vaues on the interface. This is ceary an advantage as compared to the standard method, especiay if the method is extended to 3D probems. However, the error estimate as given in [13], is optima with respect to the H -norm ony, and moreover, it is necessary for the proof to assume that the mesh sizes are of the same order. In this paper, we improve this situation by modifying the mortar condition a step further, aowing us to prove an error estimate which is optima with respect to the H -seminorm, and to not require any assumption on the mesh sizes. The reason for introducing an approximate mortar condition in this paper is to faciitate the design of simper and improved agorithms for the CR finite eement on nonmatching grids. The concept of using approximate mortar condition has started to receive more and more attention. Recenty, an additive Schwarz preconditioner for an approximate CR mortar finite eement has been used to precondition the standard CR mortar finite eement,
see [1]. Approximate constraint has been used aso in other context, see [4], for instance, where approximate constraint was indeed necessary for couping the waveet and finite eement. The discrete probem Let Ω R be a bounded, simpy connected poygona domain, partitioned into a coection of nonoverapping poygona subdomains, Ω i, i = 1,, N. We consider the foowing probem: Find u H 1 0 Ω) such that au, v) = fv), v H 1 0 Ω), 1) where au, v) = N i=1 Ω i u v dx and fv) = N i=1 Ω i fv dx. We consider the subdomains to be geometricay conforming. With each subdomain Ω i, we associate a quasi-uniform trianguation T h Ω i ) of mesh size h i. The resuting trianguation can be nonmatching across subdomain interfaces. Let X hi Ω i ) or simpy X h Ω i )) be the nonconforming P1 Crouzeix-Raviart) finite eement space defined on the trianguation T hi Ω i ) or simpy T h Ω i )) of Ω i, consisting of functions which are piecewise inear in each triange τ T h Ω i ), continuous at the interior edge midpoints of Ω CR ih, and vanishing at the edge midpoints of ΩCR ih Ω ying on the boundary Ω. We use a subscript for h ony when we need to differentiate the discretization of one domain from the other. Let X 0 h Ω i) X h Ω i ) be the space of functions whose vaues at Ω CR ih are equa to zero. Here, ΩCR, and ΩCR ih ih represent the sets of edge midpoints, i.e., the nonconforming P1 Crouzeix-Raviart) noda points, of Ω i and Ω i, respectivey. Note that Ω CR ih = Ω CR ih ΩCR ih and ΩCR ih ΩCR ih is an empty set. Without the superscript CR, the sets represent the corresponding sets of triange vertices, i.e., the P1 conforming noda points. Using X h Ω i ), we define the product space X h on the whoe domain as X h Ω) = X h Ω 1 ) X h Ω )... X h Ω N ). Let Γ ij be an open edge common to Ω i and Ω j, i.e., Γ ij = Ω i Ω j. Note that each interface Γ ij inherits two different discretization from its two sides, T hi γ ij ) and T hj γ ij ). We seect one side of Γ ij as the master side, caed the mortar, and the other side as the save 3
side, caed the nonmortar. We define the skeeton S = Ω i ) \ Ω of the decomposition as foows: S = γ m, with γ m γ n = if m n, m where each γ m denotes an open mortar edge. We write γ m as γ mi), if it is an edge of Ω i, i.e., γ mi) Ω i. Let δ m = δ mj) Ω j be the corresponding open nonmortar edge of Ω j that occupies the same geometrica space as γ mi), i.e., γ mi) = Γ ij = δ mj) See Figure for an iustration). Γ ij Ω j h j h i Ω i Figure : A mortar- γ mi) ) and A nonmortar- δ mj) ) side of a subdomain interface Γ ij ) with nonmatching meshes on both sides. Since the trianguations on Ω i and Ω j do not match on their common interface Γ ij, the functions in X h Ω) are discontinuous on the set γmi)h CR or δcr mj)h of edge midpoints on Γ ij. In other words, X hi γ mi) ) = X hi Ω i ) γmi) differ from X hj δ mj) ) = X hj Ω j ) δmj) at those points. A weak continuity condition, caed the mortar condition, is therefore imposed. Let u h X h, where u h = {u i } N i=1. A function u h X h satisfies the mortar condition on δ mj) = Γ ij = γ mi), if Q m J m u i = Q m u j, ) where J m is an interpoation operator to be defined beow, and Q m is the L -projection operator: Q m : L Γ ij ) M h j δ mj) ) defined as Q m u, ψ) L δ mj) ) = u, ψ) L δ mj) ), ψ M h j δ mj) ), 3) where M hj δ mj) ) L Γ ij ) is the test space of functions which are piecewise constant on the trianguation of δ mj), and, ) L δ mj) ) denotes the L inner product on L δ mj) ). In the origina mortar method cf. [11]), we remark that the operator J m is simpy the identity operator. 4
Let P 1 τ) be the space of inear functions on a triange τ T h Ω i ), uniquey determined by their vaues at the vertices. Denote by Z h Ω i ) = Π τ Th Ω i )P 1 τ) the space of piecewise inear functions defined on the trianguation T h Ω i ). Let T h Ω i ) be the triangu- ation associated with the Ω i, which is obtained as a resut of dividing the edges of T h Ω i ). Denote by Y h Ω i ), the conforming space of piecewise inear and continuous functions on the trianguation T h Ω i ). The functions of this space are defined by their vaues at the set Ω i h of a edge endpoints of T h whose vaues at x Ω i h Ω i ). Let Y 0 h Ω i ) Y h Ω i ) be the space of functions are equa to zero. It is easy to see that Ω i h = Ω CR ih Ω ih. Let denote an edge a triange edge) or an edge segment. The midpoint, and the eft and right end points of each edge or edge segment), is denoted by x m, x and x r, respectivey. The ength of is denoted by h. We now define the operator J m : X h γ m ) Z h γ m ). It is based on the definition of another interpoation operator I m : X h γ m ) Y h γ m ), see Definition 1. J m u x x m x r u x e m x e m x m x r m x e m x e m h e h e h h r h e h e Figure 3: Iustrating the interpoation operator J m. Here u X h γ m ) is a discontinuous) piecewise inear function shown using soid ines) in the ower figure. The dashed-dotted ines in both upper and ower figures) correspond to I m u, cf. Definition 1. The soid ines in the upper figure correspond to J m u, cf. Definition. Definition 1. For u X h γ m ), I m u Y h γ m ) is defined by the noda vaues as I m ux) = h r h +h r ux e m ) + ux), x γmh CR, ux m) + h h +h ux r m ), x γ mh, r ) h e h e +h ux e m ) ux e m ) x γ mh. e 5
Here, and r are the eft- and the right neighboring edges of x γ mh, respectivey. e represents a triange edge of T h γ m ), touching γ m, and e is the corresponding neighboring edge. The interpoation is done basicay by first joining the neighboring edge midpoints using straight ines, and then simpy extending the two end straight ines towards the end of the mortar γ m, see Fig. 3. The operator J m can now be defined using I m. Definition. For u X h γ m ), J m u Z h γ m ) is a piecewise inear function on the edges, {}, of γ m, defined by its vaues at the two end points, x γ mh and x r γ mh, of each such edge. If is an interior edge of γ m, then { { ux J m ux) = m ) + 1 Im ux ) I mux r ) } {, x = x, ux m) + 1 Im ux r ) I m ux )}, x = x r. It is easy to see that, if is a boundary edge of γ m, then J m ux) = I m ux) for x = x, x r. As seen from Fig.3, for each edge with edge midpoint x m, the straight ine J m u passes through ux m), in other words J m ux m) = I m ux m) = ux m). It is not difficut to see that the operator J m preserves a inear functions on the mortar. Let V h X h be the subspace of functions which satisfy the mortar condition for a δ m S. Since functions of V h are not continuous, we use the broken biinear form a h, ) defined by N a h u, v) = a i u, v) = u v dx. i=1 τ T h Ω i ) τ The discrete probem takes the foowing form: Find u h = {u i} N i=1 V h such that a h u h, v h ) = fv h ), v h V h. 4) 3 Anaysis Since, by construction, both u X h γ m ) and J m u Z h γ m ) are inear on any edge beonging to an interface γ m, and they have the same vaue at the edge midpoint x m, their 1 integra averages over the edge are the same, in other words, J mu dx = 1 u dx. This we wi see is essentia for our anaysis. Definition 3. For any interior edge γ m, where γ m is a subdomain interface, such that is not touching the boundary γ m, then we ca B the set of a trianges τ such that 6
τ, that is, a trianges touching, see Figure 4 midde). If is a boundary edge of γ m, see Figure 4 eft), then B is the set of a trianges touching excuding the ones touching ony at the endpoint γ m of. We ca B i j the set of trianges connecting the two triange edges i and j, see Figure 4 right). j i Figure 4: Shaded areas representing B eft and midde), and B i j right). Using the continuity of the function u at edge midpoints, and the discrete equivaence of the H 1 -seminorm, it is easy to see, for any two edges i and j, that the foowing hods, ux i m) ux j m ) ) c where the sum is taken over the trianges of the set B i j. τ B i j u H 1 τ, 5) Now, we foow the definition of I m, cf. Definition 1. If is an interior edge of γ m, then a simpe cacuation yieds I m ux r ) I m ux ) [ h = h + h r { ux r m ) ux m) } + h h + h { ux m ) ux m) }]. where and r are respectivey the eft- and the right neighboring edges of. If is the right boundary edge of γ m, then we have the specia case I m ux r ) I m ux ) = h h + h and, simiary, if is the eft boundary edge of γ m, then I m ux r ) I m ux ) = h h + h r Hence, by appying 5), we get I m ux r ) I m ux )) c 7 { ux m ) ux m) }, { ux r m ) ux m) }. τ B u H 1 τ), 6)
for each edge γ m, where the sum is taken over the eements of the set B. Now, we foow the definition of J m, cf. Definition. For any edge γ m, the foowing is true. J m ux r ) ux m) = J m ux ) ux m) ) = 1 [ Im ux r ) I m ux ) ]. If x is a point ying on the right of x m, then J m ux) ux m) = distx, x m) h [ Im ux r ) I m ux ) ], or ying on the eft of x m, then J m ux) ux m) = distx, x m) h [ Im ux r ) I m ux ) ], where distx, x m) is the distance between the points x and x m. If x is a point anywhere on the edge, then by appying 6), we get J m ux ) ux m)) c τ B u H 1 τ), 7) for edge γ m, where the sum is taken over the eements of the set B. The properties of J m are stated in the foowing emmas. Lemma 1. For u i X h Ω i ), and each γ m, we have u i J m u i L ) ch i u i H 1 τ), τ B and u i J m u i L γ mi) ) ch i u i H 1 h Ω i). Proof: Let u i ) denote the average of u i over the edge, that is u i ) = 1 u i dx. From the trace- and the Poincaré inequaity, and 7), we get u i J m u i L ) u i u i ) L ) + J mu i u i ) L ) ch i u i H 1 τ ) + ch i J m u i x) u i x m)) x=x r,x m,x ch i u i H 1 τ ) + ch i u i H 1 τ) τ B ch i u i H 1 τ). τ B 8
The second inequaity foows immediatey from the first one by taking the sum over the triange edges of γ mi). The next emma is essentia for our anaysis. The factor h 1/ j for our estimate for the consistency error. Lemma. For u i X h Ω i ), we have in the estimate, is vita J m u i Q m J m u i L γ mi) ) ch 1/ j u i H 1 h Ω i). 8) Proof: We partition γ m into a coection of nonoverapping subintervas or edge segments, o, which are intersections between the edges from the mortar side and the edges from the nonmortar side. For simpicity we assume that each o, shown as a thick ine segment in Fig.5, corresponds to a compete edge. A more genera case wi ony contribute to the constant. We note that the function J m u i Q m J m u i is piecewise inear over these subintervas. Now, J m u i Q m J m u i L γ mi) ) = J m u i Q m J m u i L o ) o γ mi) c J m u i x) Q m J m u i x)). o γ mi) h o x=x o,x o m,x o r o o r r γ mi) δ mj) γ mi) δ mj) Figure 5: Iustrating h i < h j on the eft and h j < h i on the right. Case-A: Let h i < h j. For the sake of understanding, we first consider the case where ony three triange edges on the mortar side γ mi) ) intersect with an edge on the nonmortar side δ mj) ), cf. Fig.5 eft). On o, the function Q m J m u i is a constant. Writing h for the sum h + h o + h r, this constant can be written as Q m J m u i = 1 { } J m u i dx + J m u i dx + J m u i dx h o r = 1 { h u i x h m) + h o u i x o m ) + h r u i x r m ) }. 9
J m u i, on the other hand, is inear on o. Its vaues at the three points x o, x o m and x o r are given by the Definition. Hence, at x = x o r, J m u i x o r ) Q m J m u i x o = u i x o m ) + 1 r ) = h h ui x o m ) u i x + 1 { I m u i x o r ) I m u i x o ) m) ) + h r h I m u i x o r ) I m u i x o ) } Q m J m u i ui x o m ) u i x r m ) ) ) Now we extend the above expression to a more genera case where h i is arbitrariy smaer than h j. Using the inequaity i a i) n i a i, where n h j/h i ), we wi have Jm u i x o r ) Q m J m u i x o r ) ) h j c h i meas p ) 0 + hp h I m u i x o r ) I m u i x o ). ) ui x o m ) u i x p m ) ) ) }, where the sum is taken over the triange edges, { p }, those beonging to the mortar side. We note that the number of such edges in the sum is equa to h j h i. Consequenty, Jm u i x o r ) Q m J m u i x o r ) ) ) h j hi h j c u i H h i h j h 1 τ) + u i H 1 τ) i τ B f τ B o c u i H 1 τ). τ B Here the first sum, resuting from the use of 5), is taken over the set B f of trianges aong the mortar side, where f and are the two end first and ast) triange edges having nonzero intersection with on the nonmortar side. The second sum, resuting from the use of 6), is taken over the set B o of trianges aong the mortar side, those touching the triange edge o on the mortar side. Simiar estimates, for x = x o and x = x o m, can be found. The number of times the above sum wi occur in the fina cacuation is h j h i which is the number of triange edges ying on. This contributes an h j in the fina cacuation, since h o h j h i h j. Case - B: Let h j < h i, cf. Fig.5 right). On o, Q m J m u i is a constant equa to J m u i x o m ). The difference J m u i x) Q m J m u i x)) is zero at x = x o m, and at x = x o 10
and x o r can be written as foows. J m u i x o ) Q m J m u i x o ) = J m u i x o r ) Q m J m u i x o r ) ) { 1 = Im u i x r ) I m u i x ) ) h } ho where the term inside the cury brackets represents the sope of J m u i, or equivaenty of I m u i, aong the mortar. Hence, for x = x o J m u i x) Q m J m u i x)) c and x o r, it foows from 6) that ) ho Im u i x r ) I m u i x ) ) h c h j h i ) τ B u i H 1 τ), where the sum is taken over the set B of trianges aong the mortar side, those touching the edge. The number of occurrence of the above sum in the fina cacuation is h i /h j. This resuts into an h j in the fina cacuation, since h o hj h i ) hi h j Hence, in both Case - A and Case - B, we arrive at, h j due to hj h i 1. J m u i Q m J m u i L γ mi)) ch j u i H 1 h Ω i). In the foowing, we briefy introduce some specia operators from [11, 16], which we wi need for our anaysis. Let Π m : L δ m j)) Y 0 h δ mj)), where Y 0 h δ mj)) is the set of continuous and piecewise inear functions on δ mj), defined uniquey by their vaues at x δ CR mj)h, and taking zero vaues at the boundary. Π mu on δ mj), is defined as foows, cf. [11], Π m ux) = Q m ux), x δ CR mj)h. From the definition of Q m, cf. 3), it foows that Q m Q m u)x) = Q m u)x) for x δ CR mj)h, giving Π mq m u))x) = Q m Q m u))x) = Q m ux) = Π m ux), δ CR mj)h. Hence, on δ mj), x Π m u = Π m Q m u). 9) The stabiity of Π m is stated in the foowing emma, see [11] for proof. Lemma 3. For u H 1 oo δ mj) ), Π m u 1 c u 1, H oo δ mj) ) H ooδ mj) ) 11
and for u L δ mj) ), Π m u L δ mj) ) c u L δ mj) ). In the foowing, we define two discrete harmonic extension operators, H CR h X h Ω j ) and H h : Y h : X h Ω j ) Ω j ) Y h Ω j ), corresponding to the nonconforming P1 functions and the conforming P1 functions, respectivey. For u X h Ω j ), Hh CRu X hω j ) is the soution of where H CR h is the soution of a h H CR h u, v) = 0, v X 0 hω j ), 10) ux) = ux), for x ΩCR jh. Simiary, for u Y h Ω j ), H CR h u X h Ω j ) a j H h where H h ux) = ux), for x Ω jh. u, v) = 0, v Y 0 h Ω j ), 11) We state another usefu operator M + j : Y Ω h j ) X h Ω j ), cf. [16], which is defined as foows: for u Y h Ω j ), M + j ux) = ux), x ΩCR jh. For any u Y h Ω j ), the foowing hods, cf. [16], M + j u H 1 h Ωj) c u H 1 h Ω j). 1) In the next emma we estimate the H 1 h -seminorm over a subdomain, by the H 1 oo -norm on a nonmortar side of the subdomain. Lemma 4. Let u X h Ω j ) be discrete harmonic function in Ω j, in the sense of 10), with u = 0 at x Ω CR jh \ δcr mj)h. Then u H 1 h Ω j ) c Π m u H 1 oo δ mj) ) 13) Proof: u H 1 h Ω j ) c M + j H h m Π m u H 1 h Ω j ) c H h m Π m u H 1 h Ω j ) c Π m u H 1 oo δ mj) ) 1
The first inequaity foows from the fact that u is itsef discrete harmonic, and therefore has the minima energy over a functions whose boundary vaues at x Ω CR jh, are the same as those of u. The second inequaity foows from 1), and the third inequaity foows from the property of a conforming discrete harmonic functions, cf. [7]. m is the zero extension operator from L δ mj) ) to L Ω j ). Theorem 1 rror). Let u H Ω) and u h V h be the exact soution of 1) and 4), respectivey, then N u u h H 1 h Ω) c h k u H Ω k ) k=1 The proof foows from the second Strang emma, cf [10], ) 1. 14) and Lemma 5-6. u u h H 1 h Ω) c inf v V h u v H 1 h Ω) + sup N w V h k=1 τ T h Ω k ) τ u η w w H 1 h Ω) ds, 15) Lemma 5 Consistency error). Let u H Ω) be the exact soution of 1). Then for any w = {w k } V h, N k=1 τ T h Ω k ) τ u η w w H 1 h Ω) N 1/ ds c h k u H Ω k )) 16) k=1 Proof: Note that N k=1 τ T h Ω k ) τ u η w k ds = τ T h Ω k ), τ\s) + Γ ij S Γ ij u [w] ds η u [w] ds η Using Lemma 8.3.7 and Lemma 8.3.9 of [10], for the interior and boundary edge, respectivey, one can estimate the first term as foows τ T h Ω k ), τ\s) In the foowing, we estimate the second term. N 1/ u [w] ds c h η k u H Ω k )) w H 1 h Ω) k=1 Let [w] = w i w j be the jump across Γ ij, where γ mi) = Γ ij = δ mj). Let α = {α } γmi) be the piecewise constant function on γ mi), where α = 1 dσ for each triange edge γ mi). Simiary, we define the piecewise constant function on u η 13
δ mj) as β = {β } δmj), where β = 1 u η dσ for each triange edge δ mj). We get immediatey the foowing two reations, and γ mi) αw i J m w i ) dσ = 0 βj m w i w j ) dσ = 0. δ mj) The first one foows from the fact that 1 J mw i dx = 1 w i dx, by construction, for each γ mi), and the second one is due to the mortar condition, giving, for each δ mj), that w j x m) = 1 J mw i dx. Now, using these reations, we get u ) [w] dσ Γ ij η u γ mi) η α w i J m w i ) dσ ) u + η β J m w i w j ) dσ δ mj) u η α L γ mi) ) w i J m w i L γ mi) ) + u η β L δ mj) ) J m w i w j L δ mj) ) We have u η α L γ mi) ) = u η α L ) γ mi) c u η γ mi) h ch i u H Ω i ), H 1 τ ) where the ast sum is taken over the trianges τ T h Ω i ) those having γ mi) as one of their edges. Simiary, Aso, it foows from Lemma 1 that u η β L δ mj) ) ch j u H Ω j). w i J m w i L γ mi) ) ch i w i H 1 h Ω i). Now, using the mortar condition Q m J m w i = Q m w j and Lemma, we get J m w i w j L δ mj) ) J m w i Q m J m w i L δ mj) ) + Q m w j w j L δ mj) ) ) c h j w i H 1hΩi) + h j w j H 1hΩj). 14
Finay, Γ ij u η [w] dσ c [ h i u H Ω i ) w i H 1 Ω i ) )] +h j u H Ω j) w i H 1 h Ω i) + w j H 1 h Ω j). Now, summing over the interfaces Γ ij S, and using the Cauchy-Schwarz inequaity, i a ib i ) i a i ) 1 i b i ) 1, we get the proof. Lemma 6 Approximation error). For any function u H0 1 Ω), with u Ωk H Ω k ), we have N 1/ inf u v H 1 v V h Ω) c h k u H Ω k h )) 17) k=1 Proof: Let I h be the conforming P1-interpoant of u on T h Ω k ), for k = 1,... N, and define ṽ = I h u = {ṽ k } k=1,...,n X h. We note that ṽ does not satisfy the mortar condition, and so ṽ / V h. Let v = ṽ + w, where w = {w k } k=1,...,n X h, and w k = δ mk) Ω k w mk) X h Ω k ). The function w mk) X h Ω k ) is defined as foows. Q m J m ṽ ṽ k )x), x δ CR w mk) x) = 0, x Ω CR kh Discrete harmonic cf. 10), x Ω CR It is easy to see that v V h. mk)h, \ δcr mk)h, kh. u v Hh 1Ω k) u ṽ k H 1 Ω k ) + w k Hh 1Ω k) ch k u H Ω k ) + δ mk) Ω k w mk) H 1 h Ω k) Here we have used u I h u H 1 Ω k ) ch k u H Ω k ) from the Brambe-Hibert emma. Now using Lemma 4 and 9), we have w mk) H 1 h Ω k) Π m J m ṽ ṽ k ) 1 H ooδ mk) ) Π m J m ṽ ṽ ) 1 + Π m ṽ ṽ k ) 1 H ooδ mk) ) H ooδ mk) ) From [11], we can bound the second term as ) Π m ṽ ṽ k ) 1 c h H ooδ mk) ) u H Ω ) + h k u H Ω k ). 15
For the second term we use a trace inequaity to have Π m J m ṽ ṽ ) 1 H ooδ mk) ) ch 1 Π m J m ṽ ṽ ) L δ mk) ) ch 1 J m ṽ ṽ L γ m) ) = ch 1 J m ṽ ṽ L ) γ m) ch 1 γ m) h τ B ṽ H 1 τ). For each γ m), define Q u inside B, where Q u is the averaged Tayor poynomia of order of u as defined in Chapter 4 of [10]. Note that J m I h u I h u) = J m I h u Q u) I h u Q u) on, and hence using the Brambe-hibert emma, we have, cf. [10], Hence, J m ṽ ṽ L ) ch τ B I h u Q u) H 1 τ) ch ch 3 u Q u H 1 τ) τ B u H τ). τ B Π m J m ṽ ṽ ) 1 ch u H H ooδ mk) ) Ω ). 4 An additive Schwarz method Recenty, an additive Schwarz method for the CR mortar finite eement has been proposed in [14], which uses the standard mortar condition. A P1 mortar finite eement version of the method can be found in [6]. In this section, we present a simiar method for our discrete probem 4), formed as a natura extension of the previous method from the standard mortar case to the new approximate mortar variant. This is done by a simpe adjustment in the definition of the subspaces invoved in the decomposition of the discrete space, so that the subspaces adapt naturay to the new situation. In the standard mortar case, we reca from the definition of the subspaces that part of the subdomain interior nodes, that is, the set of edge midpoints those ying cosest to a mortar side were treated as if they were on the mortar side, in other words, the mortar side becomes thicker. Note that due to the new mortar condition we do not require this arrangement. 16
So, foowing the genera framework for additive Schwarz methods cf. [15]), we can decompose V h as V h = V S + V 0 + N i=1 V i. For i = 1,..., N, V i is the restriction of V h to Ω i, with functions vanishing at subdomain boundary edge midpoints Ω CR ih as on the remaining subdomains. S CR h as we V S is the space of functions given by their vaues on the skeeton edge midpoints = γ m γ CR mh, V S = {v V h : vx) = 0, x Ω CR h \ Sh CR }. We assume that there are no corner trianges, that is, trianges having more than one edge on a subdomain boundary. It is then easy to see that the corresponding stiffness matrix is a bock diagona matrix with each bock being associated to one mortar side ony. The coarse space V 0 is a specia space having a dimension equa to the number of subdomains. It is defined using the function χ i X h Ω i ) associated with the subdomain Ω i. χ i is defined by its noda vaues as: χ i x) = 1/ j ρ jx) at x Ω CR ih, where the sum is taken over the subdomains Ω j to which x beongs, and ρ j = 1, j. Note that the ρ j s may represent physica parameters with jumps across interfaces, see [14]. V 0 is given as the span of its basis functions, Φ i, i = 1,..., N, i.e., V 0 = span{φ i : i = 1,..., N}, where Φ i associated with Ω i, is defined as foows. 1, x Ω CR ih, ρ i χ i x), x γmi)h CR, ρ i Q m J m χ j )x), x δmi)h CR Φ i x) =, δ mi) = γ mj), ρ i Q m J m χ i )x), x δmj)h CR, δ mj) = γ mi), ρ i χ j x), x γmj)h CR, γ mj) = δ mi), 0, x Ω CR ih Ω, 18) and Φ i x) = 0 at a other x in Ω CR h. We use exact biinear forms for a our subprobems. The projection ike operators T i : V h V i are defined in the standard way, i.e., for i {S, 0,..., N} and u V h, T i u V i is the soution of a h T i u, v) = a h u, v), v V i. Let T = T S +T 0 +T 1 +...+T N. The probem 4) is now repaced by the preconditioned system T u h = g, 19) where g = T S u h + N i=0 T i u h. Let c and C represent constants independent of the mesh sizes h = inf i h i and H = max i H i, then the foowing theorem hods. Theorem. For a u V h, c h H a hu, u) a h T u, u) Ca h u, u). 0) 17
The theorem can be shown in the same way as the proof in [14], which uses the genera theory for Schwarz methods, cf. [15]. It foows from the theorem that the condition number of the operator T grows as H h. 5 Impementation issues The best way to find out how to impement the method, is to ook into the matrix representation of the method. When it comes to impementation, the mortar method of this paper differs from the one in [13] ony in the mortar condition. We consider therefore ony the mortar condition, and present its matrix representation here. For the rest we refer to the matrix formuation section of [13]. For each mortar γ m, et { k : k T h γ m )} k=1,...,n and { o : o T h δ m )} o=1,...,p be the sets of n and p triange edges aong the mortar and the corresponding nonmortar side, respectivey. Let J m be the matrix representation of the interpoation operator J m : X h γ m ) Z h γ m ), in the sense that if u γm is the vector containing the vaues of the function u X h at the nodes of γ CR mh, that is the set of edge midpoints of T hγ m ), then J m u γm γ LR wi return the vaues of the function J m u at the edge eft and right endpoints mh = {x : x = x k, x k r, k T h γ m )}. We note that γ LR mh contains edge endpoints which geometricay occupy the same as those of γ mh. In the set γ LR mh, each edge endpoint in the interior of γ m occurs twice, once as the eft endpoint of an edge and the other time as the right endpoint of the neighboring edge or vice versa the supperscript LR stands for Left and Right ). Let I m be the matrix representation of the interpoation operator I m : X h γ m ) Y h γ m ), in the foowing sense. Let u γm be the same vector as above, then I m u γm wi be the vector containing the vaues of the function I m u at the nodes of γ m h, that is the set of edge endpoints of T h γ m ). Note that γ m h is aso the set of edge endpoints and the edge midpoints of T h γ m ). Let h k be the ength of the edge k, then 18
I m = h 1+h h 1+h h1 h 1+h 1 h h 1 +h h 1 h 1 +h 1 h 3 h +h 3 h n h n +h n 1 1 h n h n 1+h n hn h n 1 +h n h n 1 h n 1+h n 1 h n 1+h n h n 1 +h n. We foow Definition, and introduce an intermediate matrix K m so that J m = K m I m. It is a bock matrix consisting of n rectanguar bocks of size 3, each corresponding to an edge k T h γ m ) and being equa to the rectanguar matrix [ 1 1 1 1 1 1 The coumns of this rectanguar matrix correspond to the eft endpoint x k, the midpoint x k m, and the right endpoint x k r of the edge k, respectivey, and the rows correspond to the endpoints x k m and x k r of the edge k, respectivey. Consequenty, the coumns of K m correspond to the set γ m h ]., whie the rows correspond to the set γ LR mh. We remark that the extra work invoved in our agorithm, compared to the one presented in [13], is associated with the appication of K m. Let ϕ k and ϕ be the standard CR basis functions associated with the edge midpoint x k m γ CR mh k T h γ m )) and x m δ CR mh T h δ m )), respectivey. Let ψ o be the basis function of the M h δ m ), associated with the edge midpoint x o m o T h δ m )), defined as ψ o = 1 in o and zero otherwise. Finay, et ξ q be the basis function of P 1 q ), associated with one of the edge endpoints x q eft) and x q r right). Define the master matrix as M γm = { J m ϕ k, ψ o ) L δ m )}, and the corresponding save matrix as S δm = { ϕ, ψ o ) L δ m )}, where x k m be the vector defined as above, and et u δm vaues of the function u X h at the edge midpoints δ CR mh, then γ CR mh and x m, x o m δ CR mh. Now, et u γ m be the corresponding vector containing the S δm u δm = M γm u γm = N γm K m I m u γm 19
is the matrix discrete) representation of the mortar condition, where the supporting master matrix N γm = { } ξ q, ψ o ) L δ m ) with x q, x q r γ LR mh and xo m δmh CR. We note that S δ m is a diagona matrix containing the engths of the triange edges o T h δ m ) aong the nonmortar side, as entries. 6 Numerica resuts We present our numerica resuts here. We consider our mode probem on a unit square domain with the forcing function f = π sinπx) sinπy) and a homogeneous Dirichet boundary condition. The exact soution u equas to sinπx) sinπy). For our experiments, in genera, we decompose the domain into a d d = d number of square subdomains subregions), and then uniformy trianguate each subdomain. In order to get nonmatching grids across a interfaces, each pair of subdomains sharing an interface, are trianguated using a fixed m and n number of right ange trianges, where m is different from n. Note that the number d is inversey proportiona to the subdomain size H, whereas the numbers m and n are proportiona to H h, the ratio between the subdomain size and the mesh size h. A sides of a subdomain are chosen to be either mortar or nonmortar. We appy both the standard mortar [11] and the proposed approximate mortar technique the CR eement for soving the boundary vaue probem, resuting into two different agebraic systems, which are then soved using the Conjugate Gradients CG) method. Standard CR Mortar Proposed CR Mortar {m, n} # dofs error L error H 1 h error L error H1 {07, 05} 1009.03e-3 5.815e- 1.98e-3 5.858e- {14, 10} 4088 5.019e-4.905e- 4.970e-4.916e- {8, 0} 16456 1.49e-4 1.454e- 1.43e-4 1.456e- {56, 40} 6603 3.116e-5 7.7e-3 3.109e-5 7.78e-3 Tabe 1: L -norm error L ) and Hh 1-seminorm error H1) of the error, using the standard and the proposed mortar techniques for the CR eement on d d = 3 3 nonmatching grids. In our first experiment, we investigate the approximation property of our new mortar technique, and compare it with that of the standard mortar technique. The L -norm and the H 1 -seminorm of the error for different mesh sizes, are shown in Tabe 1. As we can see from the tabe that the errors for both mortar techniques are very cose, and they vary as h in the L -norm and as h in the H 1 -seminorm, supporting the theory. 0
In our next experiment, we appy the additive Schwarz methods, the one proposed in this paper and the one from [14], as preconditioners to the CG method for their corresponding agebraic systems. The condition number estimates and the number of iterations required to reduce the discrete L -norm of the residua by a factor of 10 6 are shown in the Tabe. As seen from the tabe, for both mortar techniques, the condition number estimates as we as the iteration counts remain bounded for fixed H h ratio, thereby supporting the theory. Again, the condition number estimates for both mortar techniques are very cose, however, the proposed mortar technique seems to require sighty fewer iteration for the same toerance. Standard CR Mortar Proposed CR Mortar d d # dofs κ # iterations κ # iterations 03 03 1009 31.31 7 33.70 4 06 06 3984 35.47 41 35.85 39 1 1 16104 36.35 46 36.07 43 4 4 6475 36.58 48 36.14 43 Tabe : Condition number estimates and iteration counts using the standard and the proposed mortar techniques for the CR eement with H h fixed with m = 07 and n = 05. To concude, both approaches, the standard mortar and the proposed approximate mortar, exhibit a very simiar numerica behavior. The approximate mortar approach has, however, the advantage that the mortar condition uses ony the noda vaues on the interface, thereby making a its agorithms comparativey simper to design and impement, and ower in the compexity. References [1] F. B. BLGACM AND Y. MADAY, The mortar eement method for three dimensiona finite eements, RAIRO Modé. Math. Na. Numér, 31):89 30, 1997. [] F. BN BLGACM, The mortar eement method with Lagrange mutipiers. Université Pau Sabatier, Tououse, France, 1994. [3] C. BRNARDI, Y. MADAY, AND A. T. PATRA, A new non conforming approach to domain decomposition: The mortar eement method, in H. Brezis and J.-L. Lions, eds., Coége de France Seminar, Pitman, 1994. This paper appeared as a technica report about five years earier. 1
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