# A Preconditioner for FETI-DP Method of Stokes Problem with Mortar-TypeCR(h)-P0(h) Discretization

Authors: Chunmei Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

FETI-DP method is developed for incompressible Stokes problem
discretized by mortar-typeCR(h)−P0(h) mixed elements. A preconditioner
is proposed, and it is proved that the condition number of the preconditioned
FETI-DP operator is bounded by (1+log(H/h))2, where H and
h are mesh sizes. Finally, the numerical tests are presented to verify the
theoretical results.


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## Body

## 1. Introduction

FETI-DP method is a nonoverlapping domain decomposition method which has been implemented for large scale engineering applications. It has been developed for elliptic problems with heterogeneous coefficients, Stokes problems, and compressible elasticity problems (cf. [1–4]). Recently, extensions to irregular subdomains and inexact subdomain solvers have been done in [5–7]. Li [4] developed a FETI-DP method for the Stokes problem by adding primal continuity constraints to accelerate the convergence of the method. Li and Widlund [8] developed BDDC methods for incompressible Stokes equations which established a close connection between the BDDC and FETI-DP methods for the Stokes case. Kim et al. [9] developed a BDDC method discretized by mortar finite element methods for geometrically nonconforming partitions, and the BDDC preconditioner was shown to be closely related to the Neumann-Dirichlet version of the FETI-DP method.Nonconforming discretizations are important for multiphysics simulations, contact-impact problems, generations of meshes and partitions aligned with jumps in diffusion coefficients,hp-adaptive methods, and special discretizations in the neighborhood of singularities. In this paper, we use the inf-sup stable CR(h)-P0(h) mixed elements in each subdomain. In the event of applying the mortar condition, it is necessary to know functions on the interface. However, the degrees of freedom (d.o.f) of Crouzeix-Raviart element are associated with the edge midpoints and functions on the interface depend on the nodal values corresponding to interface nodes and some subdomain interior nodes lying closest to the interface. We adopt nonstandard mortar condition introduced in [10] which is only associated with nodal values on the interface.The inf-sup constant for mortar element space is crucial in the analysis of the approximation order. If the constant is independent of mesh size and subdomain size, then the optimal order of approximation follows independently of the number of subdomains and mesh size as in the case of elliptic problems.The fundamental idea of this paper is the same as [4], with additional technical complications caused by nonmatching grids across subdomain interfaces and the Crouzeix-Raviart element discretization for velocity space. We propose a preconditioner and analyze the condition number bounded by (1+log(H/h))2, where H and h are mesh sizes.The paper is organized as follows. In Section2, a domain decomposition procedure is described for solving incompressible Stokes problem. In Section 3, the matrix form of the mortar condition is introduced. In Section 4, we derive an FETI-DP operator for the Stokes problem and propose a preconditioner for the FETI-DP operator. Section 5 gives several technical lemmas. Section 6 is devoted to the proof of the condition number of the preconditioned FETI-DP operator. The numerical tests are presented to verify the theoretical results in Section 7.Throughout the paper, the following notations are used:u≍v,u⪯v, and u≽v mean that there exists positive constants c and C independent of the subdomain size and mesh size, such that
(1)cu≤v≤Cu,u≤Cv,u≥cv,respectively.


## 2. A Domain Decomposition Procedure for Stokes Problems

We consider the following two-dimensional incompressible Stokes problem:(2)-∆u+∇p=f,inΩ,-∇·u=0,inΩ,u=0,on∂Ω,
where Ω is a bounded polygonal domain in R2,u represents the velocity of fluid, p is the pressure, and f is the external force.We first decomposeΩ into a nonoverlapping subdomain partition {Ωi}i=1N. We consider the subdomains to be geometrically conforming. With each subdomain Ωi, we assume that the triangulation Th(Ωi) is quasiuniform. Let hi=maxτ∈Th(Ωi)diamτ be the mesh size parameter. Let Γij=∂Ωi∩∂Ωj denote the interface between the two neighboring subdomains Ωi and Ωj, and Γ=∪i=1N∂Ωi∖∂Ω be the interface skeleton. We denote the sets of the Crouzeix-Raviart element nodes and the P1 conforming element nodes that are contained in Ω,∂Ω,Ωi,∂Ωi, and Γij by ΩhCR, ∂ΩhCR, Ωi,hCR, ∂Ωi,hCR, Γij,hCR and Ωh, ∂Ωh, Ωi,h, ∂Ωi,h, Γij,h, respectively. The resulting triangulation can be nonmatching across the subdomain interface. We note that each interface Γij inherits two independent triangulations. One side of Γij associated with Ωi is defined as a mortar side denoted by γm(i), and the other side associated with Ωj is defined as a nonmortar side denoted by δm(j).For each subdomainΩi, we introduce the following spaces:
(3)HD1(Ωi)={v∈H1(Ωi):v=0on∂Ωi∩∂Ω},L02(Ωi)={p∈L2(Ωi):∫Ωi‍pdx=0},Π0={p0:p0|Ωiisconstantand∫Ω‍p0dx=0}.
Then, the variational form of the problem (2) is as follows: find (u,pI,p0)∈ ∏i=1N‍[HD1(Ωi)]2× ∏i=1N‍L02(Ωi)×Π0, such that
(4)∑i=1N‍(∇u,∇v)Ωi-∑i=1N‍(pI+p0,∇·v)Ωi=∑i=1N‍(f,v)Ωi,jkhjkhjkkyfujfhghhgfghfgh∀v∈∏i=1N[HD1(Ωi)]2,ghjkcighjhhj-∑i=1N‍(∇·u,qI)Ωi=0,∀qI∈∏i=1NL02(Ωi),cccccicccccc-∑i=1N‍(∇·u,q0)Ωi=0,∀q0∈Π0,xvxvxcccvvxvxcju|Ωi-u|Ωj=0,∀Γij⊂Γ.
Here, (·,·)Ωi denotes the inner product in [L2(Ωi)]n for n=1,2.We consider the inf-sup stableCR(hi)-P0(hi) elements in each subdomain Ωi. Let Wh(Ωi)={vi∈ [L2(Ωi)]2:vi|τ∈[P1(τ)]2,vi   is  continuous  at  the  midpoints  of  the edges  of τ, and vi(m)=0,m is  a  midpoint  on ∂Ω∩∂τ,∀τ∈Th(Ωi)},
(5)Qi={qi∈L02(Ωi):qi|τ∈P0(τ),∀τ∈Th(Ωi)},
where Pm(τ) is a set of polynomials of degree less than or equal to m in τ.The Lagrange multiplier space corresponding toΓij is defined as follows:
(6)M(δm(j))={λ∈[L2(Γij)]2:λ|τisaconstantvector,∀τ∈Th(δm(j))[L2(Γij)]2}.
Then we take the Lagrange multiplier space
(7)M=∏δm(j)⊂ΓM(δm(j)).
We introduce the nonstandard mortar condition (cf. [10]):
(8)Qm(Jm(vi))=Qm(vj),onΓij=γm(i)=δm(j),
where vk∈Wh(Ωk)(k=i,j),Qm:[L2(Γij)]2→M(δm(j)) is the L2 orthogonal projection defined as (Qmu,λ)[L2(δm(j))]2=(u,λ)[L2(δm(j))]2,∀λ∈M(δm(j)). The operator Jm is defined in the following. We can also write (8) as follows:
(9)∫δm(j)‍(Jmvi-vj)·λijds=0,∀λij∈M(δm(j)).We usee to denote a triangle edge. Let Zh(γm(i))=∏e∈Th(γm(i))‍P1(e) be the space of piecewise linear functions defined on the triangulation Th(γm(i)), and let Th/2(γm(i)) be the triangulation obtained as a result of dividing each edge of Th(γm(i)) into two equal segments. Let Yh/2(γm(i)) be the conforming space of piecewise linear continuous functions on the triangulation Th/2(γm(i)). The midpoint, left and right endpoints of each edge e∈Th(γm(i)) are denoted by xme, xle, and xre, respectively. The length of e is denoted by |e|.We now define the operatorJm:Wh(γm(i))=Wh(Ωi)|γm(i)→[Zh(γm(i))]2. It is based on the definition of another interpolation operator Im:Wh(γm(i))→[Yh/2(γm(i))]2.Definition 1.
Foru∈Wh(γm(i)),Imu∈[Yh/2(γm(i))]2 is defined by the nodal values as
(10)Imu(x)={u(x),x∈γm(i),hCR,|er||el|+|er|u(xmel)+|el||el|+|er|u(xmer),x∈γm(i),h,u(xmee)+|ee||ee|+|ee′|(u(xmee)-u(xme′e)),x∈∂γm(i),h.Here,el and er are the left- and the right-neighboring edges of x∈γm(i),h, respectively. ee represents a triangle edge of Th(γm(i)), touching ∂γm(i), and ee′ is the corresponding neighboring edge.Each component of the interpolation is done basically by first joining the neighboring edge midpoints using straight lines, and then simply extending the two end straight lines toward the end of the mortarγm(i) (cf. Figure 1). The operator Jm can now be defined using Im.Figure 1
Here each component ofu∈Wh(γm(i)) is a (discontinuous) piecewise linear function shown (using solid lines) in the lower figure. The dashed lines (in both upper and lower figures) correspond to each component of Imu. The solid lines in the upper figure correspond to each component of Jmu.Definition 2.
Foru∈Wh(γm(i)),Jmu∈[Zh(γm(i))]2 is a vector whose components are piecewise linear functions on the edges {e} of γm(i) defined by its values at the two endpoints xle,xre∈γ¯m(i),h of each edge e. If e is an interior edge of γm(i), then
(11)Jmu(x)={u(xme)+12{Imu(xle)-Imu(xre)},x=xle,u(xme)+12{Imu(xre)-Imu(xle)},x=xre.It is easy to see that, ife is a boundary edge of γm(i), then Jmu(x)=Imu(x) for x=xle,xre.We define the following spaces:(12)Wh(Ω)=∏i=1NWh(Ωi),W~h(Ω)={v∈Wh(Ω):∫Γij‍(vi-vj)ds=0,∀Γij⊂Γ},V={v∈W~h(Ω),vsatisfies(9)},QI=∏i=1NQi,P={q∈L02(Ω):q|Ωi∈Qi,∀i=1,…,N},
where V and P are the spaces for velocity and pressure, respectively.


## 3. Matrix form of Mortar Conditions

Forv=(v1t,…,vNt)t∈W~h(Ω),vi=(vIi,vΔi,vci)t, where the symbols I, Δ, and c represent the d.o.f corresponding to the interior nodes, nodes related to ∫Γij‍vds=0, and nodes related to the remaining parts of edges, respectively. We define the spaces WI, WΔ, and Wc which consists of vI, vΔ, and vc, respectively.LetBiij be a matrix with entries
(13)(Biij)lk=∫δm(j)‍ψl·ϕkds,∀l=1,…,L,∀k=1,…,K,
where {ψl}l=1L is basis for M(δm(j)) and {ϕk}k=1K is basis for JmWh(Ωi)|δm(j). Let Bjij be a matrix with entries
(14)(Bjij)lk=-∫δm(j)‍ψl·φkds,∀l=1,…,L,∀k=1,…,P,
where {ψl}l=1L is basis for M(δm(j)) and {φk}k=1P is basis for Wh(Ωj)|δm(j). Then we write (9) into a matrix form as
(15)Biijvi|δm(j)+Bjijvj|δm(j)=0.LetWi=Wh(Ωi)|∂Ωi,i=1,…,N. The standard discrete Stokes harmonic extension operator SHi: Wi→Wh(Ωi) is defined by the following: given a velocity ui∈Wi, find SHiui∈Wh(Ωi), pIi∈Qi and p0∈Π0, such that on each subdomain Ωi,
(16)∑i=1N‍(∇ui,∇vi)Ωi-∑i=1N‍(pIi+p0,∇·vi)Ωi=0,gfdgfgdgdgdf.gdfdfgdfbgfgg∀vi∈Wh(Ωi),ccccccccc-∑i=1N‍(∇·ui,qIi)Ωi=0,∀qIi∈Qi,ccccccccc-∑i=1N‍(∇·ui,q0)Ωi=0,∀q0∈Π0,cccccccccccccSHiui=ui,on∂Ωi.DefineEij:M(δm(j))→M to be an extension operator by zero and Rijl:Wl→Wl|δm(j) for l=i,j to be a restriction operator, and let Bi=∑δm(j)⊂Γ‍EijBiijRiji+∑δm(i)⊂Γ‍EijBiijRiji. Then (15) becomes
(17)Bw=0,
where
(18)B=(B1,…,BN),w=(w1t,…,wNt)t,wi=vi|∂Ωi,i=1,…,N.LetBi,c and Bi,Δ be matrices that consist of the columns of Bi corresponding to the d.o.f related to ∫Γij‍wds=0 and the d.o.f related to the remaining parts of the edges, respectively. Then, (17) can also be written into
(19)BΔwΔ+Bcwc=0,
where BΔ=(B1,Δ,…,BN,Δ) and Bc=∑i=1N‍Bi,cLci, where Lci:Wc→Wci is the restriction map that gives Lciwc=wci,∀i=1,…,N, for wc∈Wc. We will use any of these two expressions (17) and (19) for the sake of convenience.


## 4. FETI-DP Method

### 4.1. FETI-DP Operator

In this section, we formulate a FETI-DP operator with the mortar condition (19).We define the following spaces:(20)W={v∈∏i=1NWi:∫Γij‍(vi-vj)ds=0}.To solve the Stokes problem efficiently and correctly, we will consider the following primal constraints:(21)∫δm(j)‍(Jmvi-vj)ds=0,∀δm(j)⊂Γ.Note that (21) holds by replacing λij=(10) or (01) in (9), because constant multipliers belong to the Lagrange multiplier space M(δm(j)). These primal constraints were introduced by Li (cf. [4]). The primal constraints enlarge the size of the coarse problem, so that it may lead to a fast convergence of the FETI-DP iteration.We rewrite (21) as
(22)Rt(BΔwΔ+Bcwc)=0,
where the matrix R is a Boolean matrix, the number of columns of the matrix R equals to twice the number of Γij, and the number of rows of the matrix R equals to the d.o.f of the space M. For λ∈M, at each interior nodal point of Γij, λ|Γij has two components corresponding to horizontal and vertical parts of velocity function. For λ∈M,Rtλ=0  means that for all Γij, the sums of λ|Γij corresponding to the horizontal and vertical parts of velocity function are zero.LetN be the Lagrange multiplier space corresponding to the constraints (22), and for μ∈N,μ|Γij has two components that correspond to the constraints for horizontal and vertical velocity. By introducing Lagrange multipliers λ and μ to enforce the constraints (19) and (22), the following is induced from the Galerkin approximation to (4) as follows: find (uI,pI,uΔ,uc,p0,μ,λ)∈ WI×QI×WΔ×Wc× Π0×N×M, such that
(23)(AIIGIIAIΔAIcGI000GIIt0GΔItGcIt000AΔIGΔIAΔΔAΔcGΔ0BΔtRBΔtAcIGcIAcΔAccGc0BctRBctGI0t0GΔ0tGc0t00000RtBΔRtBc00000BΔBc000)(uIpIuΔucp0μλ)=(fI0fΔfc000).Here,(AIIAIΔAIcAΔIAΔΔAΔcAcIAcΔAcc) is a stiffness matrix induced from ∑i=1N‍(∇u,∇v)Ωi, (GIItGΔItGcIt) is a matrix induced from ∑i=1N‍(-∇·v,pI)Ωi, and (GI0tGΔ0tGc0t) is a matrix induced from ∑i=1N‍(-∇·v,p0)Ωi. Since p0|Ωi is constant and vI|∂Ωi=0 for vI∈WI, we have GI0=0 by the divergence theorem.Let(24)KΔΔ=(AIIGIIAIΔGIIt0GΔItAΔIGΔIAΔΔ),KΔc=(AIcGI00GcIt00AΔcGΔ0BΔtR),Kcc=(AccGc0BctRGc0t00RtBc00),B~Δ=(00BΔ),B~c=(Bc00),f~Δ=(fI0fΔ),f~c=(fc00),zΔ=(uIpIuΔ),zc=(ucp0μ),
where zc is a primal variable, then (23) can be written as
(25)(KΔΔKΔcB~ΔtKΔctKccB~ctB~ΔB~c0)(zΔzcλ)=(f~Δf~c0).In Section5, we will show that the matrix KΔΔ is invertible (cf. Lemma 9). After eliminating zΔ, we obtain the following equations for zc and λ:
(26)(-FccFclFcltFll)(zcλ)=(-dcdl),
where
(27)Fll=B~ΔKΔΔ-1B~Δt,Fcl=KΔctKΔΔ-1B~Δt-B~ct,Fcc=Kcc-KΔctKΔΔ-1KΔc,dl=B~ΔKΔΔ-1f~Δ,dc=f~c-KΔctKΔΔ-1f~Δ.
The matrix Fcc which is a coarse problem in the FETI-DP method is invertible (cf. Lemma 10). Hence by eliminating zc, we obtain the following equation for λ:(28)FDPλ=dl-FcltFcc-1dc,
where FDP=Fll+FcltFcc-1Fcl, and we call it the FETI-DP operator. Since we add the primal constraints to the coarse problem, λ is not uniquely determined in the space M. We define a subspace of M as follows:
(29)M~={λ∈M:Rtλ=0}.
The matrix FDP is symmetric and positive definite (s.p.d) on M~, and the solution λ of (28) is uniquely determined in M~.


### 4.2. Preconditioner

Forwi∈Wi, we define Siwi by
(30)(AIIiGIIiAIΔiAIci(GIIi)t0(GΔIi)t(GcIi)tAΔIiGΔIiAΔΔiAΔciAcIiGcIiAcΔiAcci)(uIipIiwΔiwci)=(00Si(wΔiwci)),
where the subscript i for a matrix denotes the submatrix corresponding to subdomain Ωi. Since the upper left 2×2 matrix
(31)(AIIiGIIi(GIIi)t0)
represents the local Stokes problem with a Dirichlet boundary condition, it is invertible, so that the Schur complementary Si is well defined. We then assemble local Schur complementary matrices and define
(32)S=diag(S1,…,SN)
and it can be seen easily that S is an s.p.d operator on W. Hence, we define
(33)∥w∥W=(∑i=1N‍〈Siwi,wi〉)1/2
as a norm for w∈W. Here, 〈·,·〉 denotes the L2-inner product of vectors.We define the following spaces:(34)Wij0={wij∈Wh(Ωi)|δm(j):∫δm(j)‍wijds=0},W0=∏i∏δm(j)⊂ΓWij0.For a functionwij∈Wij0, let w~ij be the zero extension of wij into Wi. For w∈W0 we define an extension w~∈W by
(35)w~=(w~1,…,w~N)withw~i=∑δm(j)⊂Γ‍w~ij,∀i=1,…,N,
and define a norm on W0 by
(36)∥w∥W0=∥w~∥W.
We introduce the following subspaces with the norms induced from the spaces W and W0:
(37)W~={w∈W:∫δm(j)‍(Jmwi-wj)ds=0,∀δm(j)⊂Γ},(38)W~Δ={w∈W~:∫∂Ωi‍wi·ni=0,i=1,…,N},(39)W~0={w∈W0:w~∈W~},
where ni denotes the outward unit normal vector on the subdomain boundary ∂Ωi.Recall the definition ofM~ in (29), and let 〈·,·〉m be a duality pairing between M~ and W~0 defined as
(40)〈λ,w〉m=∑i‍∑δm(j)⊂Γ‍∫δm(j)‍λij·wijds.
Then we define a dual norm for λ∈M~ by
(41)∥λ∥M~2=maxw∈W~0∖{0}〈λ,w〉m2∥w∥W02.
Now, we will find an operator F^DP which gives
(42)〈F^DPλ,λ〉=∥λ∥M~2
and propose F^DP-1  as a preconditioner for FDP in (28). Define Rij:W0→Wij0 as a restriction operator and Eiji:Wij0→Wi as an extension operator by zero. Then for w∈W~0,
(43)∥w∥W02=∥w~∥W2=∑i=1N‍〈Siw~i,w~i〉cccccc=∑i=1N‍〈Si(∑δm(j)⊂Γ‍EijiRijw),∑δm(j)⊂Γ‍EijiRijw〉.
Let S^=∑i=1N‍(∑δm(j)⊂Γ‍EijiRij)tSi(∑δm(j)⊂Γ‍EijiRij). Moreover, we have
(44)〈λ,w〉m=〈B^w,λ〉,
where B^=diagi=1,…,Ndiagδm(j)⊂ΓB^iij and B^iij is a matrix obtained from Biij after deleting the columns corresponding to the d.o.f. of ∫Γij‍wds=0. Note that B^iij is invertible. We define the following L2-orthogonal projections:
(45)PW~0:W0⟶W~0,PM~:M⟶M~.Forλ∈M~ and w∈W~0, we have
(46)〈λ,w〉m=〈B^pw,λ〉,∥w∥W02=〈S^pw,w〉,
where
(47)S^p=PW~0S^PW~0,B^p=PM~B^PW~0.Then it can be shown that the operatorsS^p:W~0→W~0,B^p:W~0→M~ are invertible and S^p is s.p.d on W~0. Hence, using (46), the maximum in (41) occurs when w∈W~0 satisfies S^pw=B^ptλ. Therefore, we have
(48)∥λ∥M~2=〈B^pS^p-1B^ptλ,λ〉.
Let
(49)F^DP-1=(B^pS^p-1B^pt)-1=(B^pt)-1S^p(B^p)-1
be the preconditioner for the FDP.Define theL2-orthogonal projections
(50)PW~0ij:W0|δm(j)⟶W~0|δm(j),PM~ij:M|δm(j)⟶M~|δm(j).Then the projection operatorsPW~0 and PM~ are composed of diagonal blocks of PW~0ij and PM~ij, respectively. Moreover, it can be shown easily that
(51)PM~ijB^iijPW~0ij:W~0|δm(j)⟶M~|δm(j)
is invertible. Hence, it follows that
(52)B^p-1=diagi=1,…,Ndiagδm(j)⊂ΓB^ij-1,
where B^ij=PM~ijB^iijPW~0ij, and we obtain
(53)F^DP-1=∑i=1N‍(∑δm(j)⊂Γ‍EijiB^ij-1Rij)tSi(∑δm(j)⊂Γ‍EijiB^ij-1Rij).


## 4.1. FETI-DP Operator

In this section, we formulate a FETI-DP operator with the mortar condition (19).We define the following spaces:(20)W={v∈∏i=1NWi:∫Γij‍(vi-vj)ds=0}.To solve the Stokes problem efficiently and correctly, we will consider the following primal constraints:(21)∫δm(j)‍(Jmvi-vj)ds=0,∀δm(j)⊂Γ.Note that (21) holds by replacing λij=(10) or (01) in (9), because constant multipliers belong to the Lagrange multiplier space M(δm(j)). These primal constraints were introduced by Li (cf. [4]). The primal constraints enlarge the size of the coarse problem, so that it may lead to a fast convergence of the FETI-DP iteration.We rewrite (21) as
(22)Rt(BΔwΔ+Bcwc)=0,
where the matrix R is a Boolean matrix, the number of columns of the matrix R equals to twice the number of Γij, and the number of rows of the matrix R equals to the d.o.f of the space M. For λ∈M, at each interior nodal point of Γij, λ|Γij has two components corresponding to horizontal and vertical parts of velocity function. For λ∈M,Rtλ=0  means that for all Γij, the sums of λ|Γij corresponding to the horizontal and vertical parts of velocity function are zero.LetN be the Lagrange multiplier space corresponding to the constraints (22), and for μ∈N,μ|Γij has two components that correspond to the constraints for horizontal and vertical velocity. By introducing Lagrange multipliers λ and μ to enforce the constraints (19) and (22), the following is induced from the Galerkin approximation to (4) as follows: find (uI,pI,uΔ,uc,p0,μ,λ)∈ WI×QI×WΔ×Wc× Π0×N×M, such that
(23)(AIIGIIAIΔAIcGI000GIIt0GΔItGcIt000AΔIGΔIAΔΔAΔcGΔ0BΔtRBΔtAcIGcIAcΔAccGc0BctRBctGI0t0GΔ0tGc0t00000RtBΔRtBc00000BΔBc000)(uIpIuΔucp0μλ)=(fI0fΔfc000).Here,(AIIAIΔAIcAΔIAΔΔAΔcAcIAcΔAcc) is a stiffness matrix induced from ∑i=1N‍(∇u,∇v)Ωi, (GIItGΔItGcIt) is a matrix induced from ∑i=1N‍(-∇·v,pI)Ωi, and (GI0tGΔ0tGc0t) is a matrix induced from ∑i=1N‍(-∇·v,p0)Ωi. Since p0|Ωi is constant and vI|∂Ωi=0 for vI∈WI, we have GI0=0 by the divergence theorem.Let(24)KΔΔ=(AIIGIIAIΔGIIt0GΔItAΔIGΔIAΔΔ),KΔc=(AIcGI00GcIt00AΔcGΔ0BΔtR),Kcc=(AccGc0BctRGc0t00RtBc00),B~Δ=(00BΔ),B~c=(Bc00),f~Δ=(fI0fΔ),f~c=(fc00),zΔ=(uIpIuΔ),zc=(ucp0μ),
where zc is a primal variable, then (23) can be written as
(25)(KΔΔKΔcB~ΔtKΔctKccB~ctB~ΔB~c0)(zΔzcλ)=(f~Δf~c0).In Section5, we will show that the matrix KΔΔ is invertible (cf. Lemma 9). After eliminating zΔ, we obtain the following equations for zc and λ:
(26)(-FccFclFcltFll)(zcλ)=(-dcdl),
where
(27)Fll=B~ΔKΔΔ-1B~Δt,Fcl=KΔctKΔΔ-1B~Δt-B~ct,Fcc=Kcc-KΔctKΔΔ-1KΔc,dl=B~ΔKΔΔ-1f~Δ,dc=f~c-KΔctKΔΔ-1f~Δ.
The matrix Fcc which is a coarse problem in the FETI-DP method is invertible (cf. Lemma 10). Hence by eliminating zc, we obtain the following equation for λ:(28)FDPλ=dl-FcltFcc-1dc,
where FDP=Fll+FcltFcc-1Fcl, and we call it the FETI-DP operator. Since we add the primal constraints to the coarse problem, λ is not uniquely determined in the space M. We define a subspace of M as follows:
(29)M~={λ∈M:Rtλ=0}.
The matrix FDP is symmetric and positive definite (s.p.d) on M~, and the solution λ of (28) is uniquely determined in M~.


## 4.2. Preconditioner

Forwi∈Wi, we define Siwi by
(30)(AIIiGIIiAIΔiAIci(GIIi)t0(GΔIi)t(GcIi)tAΔIiGΔIiAΔΔiAΔciAcIiGcIiAcΔiAcci)(uIipIiwΔiwci)=(00Si(wΔiwci)),
where the subscript i for a matrix denotes the submatrix corresponding to subdomain Ωi. Since the upper left 2×2 matrix
(31)(AIIiGIIi(GIIi)t0)
represents the local Stokes problem with a Dirichlet boundary condition, it is invertible, so that the Schur complementary Si is well defined. We then assemble local Schur complementary matrices and define
(32)S=diag(S1,…,SN)
and it can be seen easily that S is an s.p.d operator on W. Hence, we define
(33)∥w∥W=(∑i=1N‍〈Siwi,wi〉)1/2
as a norm for w∈W. Here, 〈·,·〉 denotes the L2-inner product of vectors.We define the following spaces:(34)Wij0={wij∈Wh(Ωi)|δm(j):∫δm(j)‍wijds=0},W0=∏i∏δm(j)⊂ΓWij0.For a functionwij∈Wij0, let w~ij be the zero extension of wij into Wi. For w∈W0 we define an extension w~∈W by
(35)w~=(w~1,…,w~N)withw~i=∑δm(j)⊂Γ‍w~ij,∀i=1,…,N,
and define a norm on W0 by
(36)∥w∥W0=∥w~∥W.
We introduce the following subspaces with the norms induced from the spaces W and W0:
(37)W~={w∈W:∫δm(j)‍(Jmwi-wj)ds=0,∀δm(j)⊂Γ},(38)W~Δ={w∈W~:∫∂Ωi‍wi·ni=0,i=1,…,N},(39)W~0={w∈W0:w~∈W~},
where ni denotes the outward unit normal vector on the subdomain boundary ∂Ωi.Recall the definition ofM~ in (29), and let 〈·,·〉m be a duality pairing between M~ and W~0 defined as
(40)〈λ,w〉m=∑i‍∑δm(j)⊂Γ‍∫δm(j)‍λij·wijds.
Then we define a dual norm for λ∈M~ by
(41)∥λ∥M~2=maxw∈W~0∖{0}〈λ,w〉m2∥w∥W02.
Now, we will find an operator F^DP which gives
(42)〈F^DPλ,λ〉=∥λ∥M~2
and propose F^DP-1  as a preconditioner for FDP in (28). Define Rij:W0→Wij0 as a restriction operator and Eiji:Wij0→Wi as an extension operator by zero. Then for w∈W~0,
(43)∥w∥W02=∥w~∥W2=∑i=1N‍〈Siw~i,w~i〉cccccc=∑i=1N‍〈Si(∑δm(j)⊂Γ‍EijiRijw),∑δm(j)⊂Γ‍EijiRijw〉.
Let S^=∑i=1N‍(∑δm(j)⊂Γ‍EijiRij)tSi(∑δm(j)⊂Γ‍EijiRij). Moreover, we have
(44)〈λ,w〉m=〈B^w,λ〉,
where B^=diagi=1,…,Ndiagδm(j)⊂ΓB^iij and B^iij is a matrix obtained from Biij after deleting the columns corresponding to the d.o.f. of ∫Γij‍wds=0. Note that B^iij is invertible. We define the following L2-orthogonal projections:
(45)PW~0:W0⟶W~0,PM~:M⟶M~.Forλ∈M~ and w∈W~0, we have
(46)〈λ,w〉m=〈B^pw,λ〉,∥w∥W02=〈S^pw,w〉,
where
(47)S^p=PW~0S^PW~0,B^p=PM~B^PW~0.Then it can be shown that the operatorsS^p:W~0→W~0,B^p:W~0→M~ are invertible and S^p is s.p.d on W~0. Hence, using (46), the maximum in (41) occurs when w∈W~0 satisfies S^pw=B^ptλ. Therefore, we have
(48)∥λ∥M~2=〈B^pS^p-1B^ptλ,λ〉.
Let
(49)F^DP-1=(B^pS^p-1B^pt)-1=(B^pt)-1S^p(B^p)-1
be the preconditioner for the FDP.Define theL2-orthogonal projections
(50)PW~0ij:W0|δm(j)⟶W~0|δm(j),PM~ij:M|δm(j)⟶M~|δm(j).Then the projection operatorsPW~0 and PM~ are composed of diagonal blocks of PW~0ij and PM~ij, respectively. Moreover, it can be shown easily that
(51)PM~ijB^iijPW~0ij:W~0|δm(j)⟶M~|δm(j)
is invertible. Hence, it follows that
(52)B^p-1=diagi=1,…,Ndiagδm(j)⊂ΓB^ij-1,
where B^ij=PM~ijB^iijPW~0ij, and we obtain
(53)F^DP-1=∑i=1N‍(∑δm(j)⊂Γ‍EijiB^ij-1Rij)tSi(∑δm(j)⊂Γ‍EijiB^ij-1Rij).


## 5. Technical Tools

In this section, we state and prove a few technical lemmas necessary for the proof of Theorem17 of Section 6.LetYh/2(Ωi) be the conforming space of piecewise linear continuous functions on the triangulation Th/2(Ωi) which is constructed by joining the midpoints of the edges of the elements of Th(Ωi), and let Yh/20(Ωi) be the subspace of Yh/2(Ωi) consisting of functions with zero traces on ∂Ωi.For each open edgeε⊂∂Ωi, we introduce a local equivalence map (isomorphism) Miε: Wεh(Ωi)→ [Yh/2(Ωi)]2 (cf. [11]), where Wεh(Ωi) is a subspace of Wh(Ωi) formed by all vectors whose component are zero in ∂Ωi,hCR∖εhCR. This mapping is a slightly modified version of a similar mapping introduced in [12].Definition 3.
For givenu∈Wεh(Ωi), we introduce Miεu∈[Yh/2(Ωi)]2 by defining the values of Miεu at the nodal points of the triangulation Th/2(Ωi).(i)
Forp∈Ωi,hCR∪∂Ωi,hCR, let Miεu(p)=u(p).(ii)
Forp∈Ωi,h, then Miεu(p)=(1/N(p))∑τjh‍u|τjh(p), where the sum is taken over all triangles τjh with the common vertex p, and N(p) is the number of elements with p as an vertex.(iii)
Forp∈∂Ωi,h∖εh, let Miεu(p)=0.(iv)
Forp∈εh, then Miεu(p)=(|ppr|/|plpr|)u(pl)+(|plp|/|plpr|)u(pr), where pl,pr are the left- and right-neighboring CR nodal points of p and |ab| is the length of the segment with a,b as its endpoints.Note that each component ofMiεu is piecewise linear between the CR nodes of εhCR, and Miεu=0 on ∂Ωi∖ε. The mapping Miεu has the following properties (cf. [11]):
(54)|u|Hh1(Ωi)≍|Miεu|H1(Ωi),∥u∥L2(Ωi)≍∥Miεu∥L2(Ωi),∥Miεu-u∥L2(Ωi)⪯hi|u|Hh1(Ωi),∥Miεu-u∥L2(ε)⪯hi1/2|u|Hh1(Ωi).Lemma 4 (see [13]).
Letu∈Wj and u=0 in ∂Ωj,hCR∖δm(j),hCR for a slave edge δm(j)⊂∂Ωj. Then we have
(55)|SHju|Hh1(Ωj)⪯∥Mjδm(j)SHju∥H001/2(δm(j)),
where SHju is the discrete harmonic extension of u.Lemma 5 (see [11]).
Letδm(j)⊂∂Ωj be a slave, then for any u∈Wh(Ωj), we have
(56)|uδm(j)|Hh1(Ωj)2⪯(1+log(Hjh))2×(Hj-2∥u∥L2(Ωj)2+|u|Hh1(Ωj)2),
where uδm(j) is a discrete harmonic function taking the same vector values as those of u at CR nodal points on δm(j),hCR and is equal to zero vector at the remaining CR nodal points on ∂Ωj and Hj=diamΩj.We also need a special mortar operator defined over trace spaces:Πδm(j):[L2(δm(j))]2→W0(δm(j)), where W0(δm(j)) is a set of vectors whose components are continuous and equal to zero at the endpoints of δm(j) and are piecewise linear over all segments that have their ends in δm(j),hCR. Each Πδm(j)u∈W0(δm(j)) is uniquely determined by its values in δm(j),hCR.Definition 6 (see [11]).
LetΠδm(j):[L2(δm(j))]2→W0(δm(j)) be defined by
(57)Πδm(j)u(x)=Qmu(x),∀x∈δm(j),hCR.TheL2 and H001/2 stability of Πδm(j) is stated in the next lemma.Lemma 7 (see [11]).
ForΠδm(j) defined previously, we have
(58)∥Πδm(j)u∥L2(δm(j))⪯∥u∥L2(δm(j))∀u∈[L2(δm(j))]2,∥Πδm(j)u∥H001/2(δm(j))⪯∥u∥H001/2(δm(j))∀u∈[H001/2(δm(j))]2.Lemma 8 (see [10]).
Letu∈Wh(Ωj) be a discrete harmonic vector in Ωj, with u=0 at x∈ ∂Ωj,hCR∖ δm(j),hCR. Then we have
(59)|u|Hh1(Ωj)⪯∥Πδm(j)u∥H001/2(δm(j)).Lemma 9 (see [14]).
The matrixKΔΔ is invertible on WI×QI×WΔ.Lemma 10 (see [14]).
Assume that the domainΩ has the triangulation to satisfy that
(60)-∑i‍(p0,∇·vΔ)Ωi+∑i,j‍λ·∫Γij‍(vΔi-vΔj)ds=0,vcccbvvvvvvvvvvvcbcbcbcbccbcbcbccb∀vΔ∈WΔ,-∑i‍(p0,∇·vc)Ωi+∑i,j‍λ·∫Γij‍(vci-vcj)ds=0,vcccbvvvvvvvvvvvcbcbcbcbccbcbcbccb∀vc∈Wc
for the solution (p0λ). Then the coarse problem matrix Fcc is invertible.Remark 11.
Most triangulations of the domainΩ satisfy the assumption of Lemma 10 because the number of velocity unknowns vΔ and vc is usually greater than the number of unknowns p0 and λ.Lemma 12 (see [14]).
We haveB(W~Δ)=B(W~)=M~.Lemma 13 (see [14]).
Forλ∈M~, we have
(61)〈FDPλ,λ〉=maxw∈W~Δ∖{0}〈Bw,λ〉2∥w∥W2.Lemma 14 (see [15]).
For anyλ∈M, we have
(62)〈F^DPλ,λ〉1/2=supw∈W~Δ∖{0}∑δm(j)⊂Γ‍∫δm(j)‍wλds∥w∥W.Proof.
The proof follows from the definitions ofW and W~Δ, (40), and a standard algebraic argument; see [16].Definition 15 (see [17]).
Givenu∈Wi, we define an operator Pi:Wi→W~ as follows:(i)
forp∈γm(i),hCR, let Piu(p)=u(p);(ii)
forp∈δm(j),hCR, let Piu(p)=QmJmu(p);(iii)
forp∈δm(i),hCR, for any slave side δm(i)⊂∂Ωi, or if the edge midpoint p is on ∂Ω or on any remaining master or slave side, let Piu(p)=0.We define a projection operatorP:W~→W~Δ by
(63)(Pu)|δm(j)=Qm(Jmui-uj)onδm(j).
where ui and uj are, respectively, the restriction of u∈W~ to the mortar side γm(i) and the slave side δm(j) of an interface Γij.Lemma 16.
For allu∈W~, we have
(64)∥Pu∥W⪯(1+log(Hh))∥u∥W,
where H=maxiHi and h=minihi.Proof.
LetPu=(P1u,…,PNu)∈W~, and we have ∥Pu∥W2=∑i=1N‍∥Piu∥Wi2. Next we will estimate the term ∥Piu∥Wi2.
Note thatPiu can be nonzero only over Ωi and the neighboring subdomain Ωj that share a common edge Γij with Ωi, such that the slave side δm(j) is associated with Ωj. Let Ni denote the set of indices of such subdomains. Thus
(65)∥Piu∥Wi2=|SHiPiu|Hh1(Ωi)2+∑j∈Ni‍|SHjPiu|Hh1(Ωj)2.
Then consideringΩi, we have Piu=u-∑δm(i)⊂∂Ωi‍uδm(i) with uδm(i) being a discrete harmonic function which equals to u at the CR nodes of δm(i),hCR and to zero at all remaining nodes of ∂Ωi,hCR. Thus, by Lemma 5,
(66)|SHiPiu|Hh1(Ωi)2⪯|SHiu|Hh1(Ωi)2+∑δm(i)⊂∂Ωi|uδm(i)|Hh1(Ωi)2xzcdccdccdciic⪯(1+log(Hihi))2(Hi-2∥SHiu∥L2(Ωi)2bbbbbbbbbbbbcbcbbbcdcsdcdcdccd+|SHiu|Hh1(Ωi)2).Next we consider those Ωj for which j∈Ni, consequently the slave side δm(j) of the edge Γij is an edge of Ωj, and the mortar side γm(i) is an edge of Ωi. Then by Lemma 4, we get
(67)|SHjPiu|Hh1(Ωj)⪯∥Mjδm(j)SHjPiu∥H001/2(δm(j)).
We also need to consider the function Πδm(j)Miγm(i)uγm(i). From Definitions 3 and 15, we have
(68)Mjδm(j)SHjPiu(p)=SHiPiu(p)=QmSHiPiu(p)=QmJmSHiu(p),∀p∈δm(j),hCR.
Thus we get
(69)∥Mjδm(j)SHjPiu∥H001/2(δm(j))≤∥∏δm(j)Miγm(i)uγm(i)Mjδm(j)SHjPiuzxzxzxcvbsssbsbdbffghngfg-Πδm(j)Miγm(i)uγm(i)∥H001/2(δm(j))sdsadasdsdasasdaasdsadsd+∥Πδm(j)Miγm(i)uγm(i)∥H001/2(δm(j)).
The pervious second term can be estimated as
(70)∥Πδm(j)Miγm(i)uγm(i)∥H001/2(δm(j))2⪯∥Miγm(i)uγm(i)∥H001/2(δm(j))2⪯|Miγm(i)uγm(i)|H1(Ωi)2⪯|uγm(i)|Hh1(Ωi)2⪯(1+log(Hihi))2(Hi-2∥SHiu∥L2(Ωi)2+|SHiu|Hh1(Ωi)2),
by Lemmas 5 and 7 and the trace theorem. The first term is bounded using an inverse inequality, (68) and Definition 6 as follows:
(71)xc∥Mjδm(j)SHjPiu-Πδm(j)Miγm(i)uγm(i)∥H001/2(δm(j))2⪯hj-1∥Mjδm(j)SHjPiu-Πδm(j)Miγm(i)uγm(i)∥L2(δm(j))2⪯∑p∈δm(j),hCR‍|Mjδm(j)SHjPiu(p)-Πδm(j)Miγm(i)uγm(i)(p)|2=∑p∈δm(j),hCR‍|Qm(JmSHiu-Miγm(i)uγm(i))(p)|2⪯hj-1∥Qm(JmSHiu-Miγm(i)uγm(i))∥L2(δm(j))2.Next, by using the fact that Qm is an orthogonal L2 projection, we get
(72)hh∥Qm(Miγm(i)uγm(i)-JmSHiu)∥L2(δm(j))⪯∥Miγm(i)uγm(i)-JmSHiu∥L2(δm(j)).
Finally, the fact that JmSHiu=Jmuγm(i) yields
(73)iiii∥Miγm(i)uγm(i)-JmSHiu∥L2(δm(j))≤∥Miγm(i)uγm(i)-uγm(i)∥L2(δm(j))ffffff+∥uγm(i)-Jmuγm(i)∥L2(δm(j)).
Applying the property of Miγm(i)u to the first term and Lemma 3.2 from [10] to the second term, Lemma 5 and the quasiuniform triangulation, we get
(74)hj-1∥Miγm(i)uγm(i)-JmSHiu∥L2(δm(j))2≤hihj|uγm(i)|Hh1(Ωi)⪯(1+log(Hihi))2(Hi-2∥SHiu∥L2(Ωi)2+|SHiu|Hh1(Ωi)2)⪯(1+log(Hihi))2|SHiu|Hh1(Ωi)2.
Finally, summing over all edges in ∂Ωi and then over all subdomain ends the proof.


## 6. Condition Number Estimate

In this section, we give the condition number estimate of the preconditioned operator in the following main theorem of this paper.Theorem 17.
For anyλ∈M, it holds that
(75)〈F^DPλ,λ〉⪯〈FDPλ,λ〉⪯(1+log(Hh))2〈F^DPλ,λ〉,
where H=maxiHi and h=minihi.Proof.
In the proof of this theorem we use results from this section and the algebraic arguments from [16, Section 6.4].
Lower Bound. From Lemma 13, the zero extension E(w0)∈W~Δ for w0∈W~0, and (41), we obtain
(76)〈FDPλ,λ〉=supw∈W~Δ∖{0}〈Bw,λ〉2∥w∥W2≥supw0∈W~0∖{0}〈BE(w0),λ〉2∥E(w0)∥W2=〈F^DPλ,λ〉.Upper Bound. For any w∈W~,
(77)〈Bw,λ〉=∑δm(j)⊂Γ‍∫δm(j)(Jmui-uj)·λijds=∑δm(j)⊂Γ‍∫δm(j)Qm(Jmui-uj)·λijds=∑δm(j)⊂Γ‍∫δm(j)‍(Pw)|δm(j)·λijds.Hence by Lemmas 13, 14, and 16, we conclude that
(78)〈FDPλ,λ〉=supw∈W~Δ∖{0}〈Bw,λ〉2∥w∥W2=supw∈W~Δ∖{0}(∑δm(j)⊂Γ‍∫δm(j)‍(Pw)|δm(j)·λijds)2∥w∥W2⪯〈F^DPλ,λ〉supw∈W~Δ∖{0}∥Pw∥W2∥w∥W2⪯(1+log(Hh))2〈F^DPλ,λ〉.


## 7. Numerical Tests

Consider the following Stokes problem:(79)-Δu+∇p=finΩ,-∇·u=0inΩ,u=0on∂Ω,
where Ω=[0,1]×[0,1]⊂R2 and f is given by the exact solution of the problem (79) as follows:
(80)u=(sin3(πx)sin2(πy)cos(πy)-sin2(πx)sin3(πy)cos(πx)),p=x2-y2.LetN denote the number of subdomains. We only consider the uniform partition of Ω.N=4×4 means that Ω is partitioned into 4×4 square subdomains. For all subdomains, we take the same number of nodes n, including endpoints, in horizontal and vertical edges with n=4k+1 for some positive integer k. We solve (79) on both matching and nonmatching grids. For matching grids, we make uniform triangulations in each subdomain with (n-1)/2+1 nodes on horizontal and vertical edges of subdomain. For nonmatching grids, we take (n-1)/2+1 random quasi-uniform nodes on each horizontal and vertical edges of subdomain and generate nonuniform structured triangulations.Now, we solve the FETI-DP operator with and without preconditioner varyingN and n on both matching and nonmatching triangulations. The Conjugate Gradient (CG) iteration is stopped when the relative residual is less than 10-6.In Tables1–3, the number of CG iterations and the corresponding condition numbers are shown with varying N and n. In Table 1, N=4×4 and n-1 increase by double. On both matching and nonmatching grids, the preconditioner performs well and the condition numbers seem to behave log2-growth as n increases. Especially on nonmatching grids, the preconditioner is much more efficient. In Tables 2 and 3, the CG iteration becomes stable as N increases with n=5 and n=9.Table 1
CG iterations (condition numbers) whenN=4×4.
n
Matching
Nonmatching
F
DP
F
^
DP
-
1
F
DP
F
DP
F
^
DP
-
1
F
DP
5
21
(
9.82
)
18
(
5.65
)
39
(
1.94
e
+
1
)
26
(
3.95
)
9
35
(
3.57
e
+
1
)
21
(
7.48
)
104
(
2.41
e
+
2
)
29
(
6.42
)
17
47
(
8.48
e
+
1
)
22
(
1.01
e
+
1
)
133
(
9.90
e
+
2
)
33
(
8.75
)
33
53
(
1.78
e
+
2
)
24
(
1.29
e
+
1
)
157
(
2.48
e
+
3
)
35
(
11.3
)
65
68
(
3.38
e
+
2
)
25
(
1.57
e
+
1
)
199
(
5.98
e
+
3
)
37
(
13.5
)Table 2
CG iterations (condition numbers) whenn=5.
N
Matching
Nonmatching
F
DP
F
^
DP
-
1
F
DP
F
DP
F
^
DP
-
1
F
DP
4
×
4
15
(
8.32
)
12
(
6.68
)
21
(
11.7
)
16
(
6.87
)
8
×
4
16
(
8.68
)
13
(
6.71
)
22
(
13.1
)
16
(
6.84
)
16
×
16
14
(
8.85
)
12
(
6.66
)
21
(
13.8
)
16
(
6.43
)
32
×
32
13
(
8.96
)
12
(
6.61
)
21
(
15.9
)
17
(
7.12
)Table 3
CG iterations (condition numbers) whenn=9.
N
Matching
Nonmatching
F
DP
F
^
DP
-
1
F
DP
F
DP
F
^
DP
-
1
F
DP
4
×
4
30
(
38.9
)
15
(
12.7
)
61
(
32.1
)
21
(
13.5
)
8
×
4
32
(
40.2
)
16
(
12.5
)
64
(
33.5
)
22
(
14.1
)
16
×
16
31
(
40.6
)
16
(
12.3
)
66
(
38.1
)
23
(
14.4
)
32
×
32
29
(
40.8
)
17
(
12.8
)
68
(
35.8
)
23
(
14.3
)Moreover, we have observed the convergence behaviors of the approximated solution. TheH1 and L2-errors for velocity and pressure are examined. uh and ph denote the approximated solutions for the velocity and pressure, and ∥u-uh∥1,* means the square root of ∑i=1N‍∥u-uh∥1,Ωi2. The errors are shown in Table 4 for various N and n with matching grids. Three cases are considered: when n-1 increases by double with N=4×4, when N increases by double in both edges of Ω with n=5 and n=9. For all cases, we can see that the H1-error for velocity ∥u-uh∥1,*,L2-error for pressure ∥p-ph∥0 reduce by half and L2-error for velocity ∥u-uh∥0 reduces by quarter. For the finite elements CR(h)-P0(h), these convergence behaviors are optimal.Table 4
H
1- and L2-errors on matching grids.
N
=
4
×
4
n
=
5
n
=
9
∥
u
-
u
h
∥
1
,
*
∥
u
-
u
h
∥
0
∥
p
-
p
h
∥
0
n
N
N
5
4
×
4
6.60
e
-
1
6.34
e
-
3
1.68
e
-
1
9
8
×
8
4
×
4
3.44
e
-
1
1.61
e
-
3
8.51
e
-
2
17
16
×
16
8
×
8
1.73
e
-
1
4.13
e
-
4
4.23
e
-
2
33
32
×
32
16
×
16
8.71
e
-
2
1.05
e
-
4
2.09
e
-
2
65
32
×
32
4.35
e
-
2
2.54
e
-
5
1.06
e
-
2For the case of nonmatching grids, the errors are shown in Tables5–7 with various N and n. In Table 5, we observe that the error ∥u-uh∥1,* and ∥p-ph∥0 reduce by half and the error ∥u-uh∥0 reduces by quarter as n-1 increases by double with N=4×4. When n=5 and n=9, as N increases, the errors also show the optimal convergence behaviors in Tables 6 and 7.Table 5
H
1- and L2-errors on nonmatching grids: N=4×4.
n
∥
u
-
u
h
∥
1
,
*
∥
u
-
u
h
∥
0
∥
p
-
p
h
∥
0
5
6.41
e
-
1
7.08
e
-
3
1.81
e
-
1
9
3.28
e
-
1
1.79
e
-
3
8.91
e
-
2
17
1.67
e
-
1
4.39
e
-
4
4.36
e
-
2
33
8.43
e
-
2
1.12
e
-
4
2.24
e
-
2
65
4.32
e
-
2
2.98
e
-
5
1.15
e
-
2Table 6
H
1- and L2-errors on nonmatching grids: n=5.
N
∥
u
-
u
h
∥
1
,
*
∥
u
-
u
h
∥
0
∥
p
-
p
h
∥
0
4
×
4
3.76
e
-
1
2.11
e
-
3
1.08
e
-
1
8
×
8
1.85
e
-
1
5.19
e
-
4
5.21
e
-
2
16
×
16
9.08
e
-
2
1.28
e
-
4
2.59
e
-
2
32
×
32
4.56
e
-
2
3.15
e
-
5
1.28
e
-
2Table 7
H
1- and L2-errors on nonmatching grids: n=9.
N
∥
u
-
u
h
∥
1
,
*
∥
u
-
u
h
∥
0
∥
p
-
p
h
∥
0
4
×
4
7.21
e
-
1
7.73
e
-
3
2.91
e
-
1
8
×
8
3.58
e
-
1
1.92
e
-
3
1.47
e
-
1
16
×
16
1.81
e
-
1
4.52
e
-
4
5.19
e
-
2
32
×
32
8.94
e
-
2
1.15
e
-
4
2.61
e
-
2If the inf-sup constant for the spaceV×P is independent of N and n, then the optimality of approximation can be shown. Let β* and β be the inf-sup constants for the space V×P and the CR(h)-P0(h) finite elements, respectively, and β0 be the inf-sup constant for the space V×Π0. Then the constant β* depends on β and β0 from the trick conceived by Boland and Nicolaides [18]. Hence, if the constant β0 is independent of n and N, then the same holds for β*. In [19], for V×Π0 which is obtained from the Hood-Taylor finite elements, it was shown that the constant β0 is independent of n, but not shown for N. Following the proofs in [19], we can obtain the same results for the space V×Π0 of the CR(h)-P0(h) finite elements. We have no proof that β0 is independent of N. Instead, we compute the constant β0 numerically as N increases. The results are given in Table 8 both for matching and nonmatching grids when n=5 and n=9. We observe that the constant β0 becomes stable as N increases. Table 9 gives the constant β0 as n increases with N=4×4. This confirms that the constant β0 is independent of n.Table 8
Inf-sup constantβ0 when n=5 and n=9.
N
n
=
5
n
=
9
Nonmatching
Matching
Nonmatching
Matching
4
×
4
0.7537
0.7543
0.7868
0.7870
8
×
8
0.7050
0.7052
0.7109
0.7111
16
×
16
0.6765
0.6768
0.6798
0.6801
32
×
32
0.6584
0.6586
0.6611
0.6613Table 9
Inf-sup constantβ0 when N=4×4.
n
Nonmatching
Matching
5
0.7537
0.7543
9
0.7050
0.7052
17
0.6765
0.6768
33
0.6584
0.6586
65
0.6586
0.6587



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*Source: 485628-2013-04-11.xml*