A FETI-DP type domain decomposition algorithm for three-dimensional incompressible Stokes equations
aa r X i v : . [ m a t h . NA ] A p r A FETI-DP type domain decompositionalgorithm for three-dimensional incompressibleStokes equations
Xuemin Tu ∗ Jing Li † August 18, 2018
Abstract
The FETI-DP algorithms, proposed by the authors in [
SIAM J. Nu-mer. Anal. , 51 (2013), pp. 1235–1253] and [
Internat. J. Numer. MethodsEngrg. , 94 (2013), pp. 128–149] for solving incompressible Stokes equa-tions, are extended to three-dimensional problems. A new analysis of thecondition number bound for using the Dirichlet preconditioner is given.An advantage of this new analysis is that the numerous coarse level veloc-ity components, required in the previous analysis to enforce the divergencefree subdomain boundary velocity conditions, are no longer needed. Thisgreatly reduces the size of the coarse level problem in the algorithm, espe-cially for three-dimensional problems. The coarse level velocity space canbe chosen as simple as for solving scalar elliptic problems correspondingto each velocity component. Both Dirichlet and lumped preconditionersare analyzed using a same framework in this new analysis. Their con-dition number bounds are proved to be independent of the number ofsubdomains for fixed subdomain problem size. Numerical experiments inboth two and three dimensions demonstrate the convergence rate of thealgorithms.
Keywords domain decomposition, incompressible Stokes, FETI-DP, BDDC,divergence free
AMS
Mixed finite elements are often used to solve incompressible Stokes and Navier-Stokes equations. Continuous pressures have been used in many mixed finite ele- ∗ Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS66045-7594, [email protected] , ∼ xtu/ . This author’s work was sup-ported in part by National Science Foundation contract DMS-1115759. † Department of Mathematical Sciences, Kent State University, Kent, OH 44242, [email protected] , ∼ li/ . OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES et. al. [21], Benhassine andBendali [1], and Kim and Lee [13]. But the convergence rate analysis of thoseapproaches cannot be applied to the continuous pressure case due to the indefi-niteness of the linear systems; such difficulty can often be removed convenientlywhen discontinuous pressures are used in the discretization.Recently, the authors [16, 26] proposed and analyzed a FETI-DP (Dual-Priaml Finite Element Tearing and Interconnecting method) type domain de-composition algorithm for solving the incompressible Stokes equation in twodimensions. Both discontinuous and continuous pressures can be used in themixed finite element discretization. In both cases, the indefinite system of linearequations can be reduced to a symmetric positive semi-definite system. There-fore, the preconditioned conjugate gradient method can be applied and a scal-able convergence rate of the algorithm has been proved.The lumped and Dirichlet preconditioners have been studied in [16] and [26],respectively. For the lumped preconditioner it was shown both experimentallyand analytically in [16], that the coarse level space can be chosen the same asfor solving scalar elliptic problems corresponding to each velocity componentto achieve a scalable convergence rate. Similar observations for the lumpedpreconditioner have also been pointed out earlier by Kim and Lee [11, 12, 10,with Park], even though their studies are only for using discontinuous pressures.For the Dirichlet preconditioner studied in [26], a distinctive feature is theapplication of subdomain discrete harmonic extensions in the preconditioner.In other existing FETI-DP and BDDC (Balancing Domain Decomposition byConstraints) algorithms, cf. [15, 17], subdomain discrete Stokes extensions havebeen used and the coarse level velocity space has to contain sufficient com-ponents to enforce divergence free subdomain boundary velocity conditions.Those complicated and numerous coarse level velocity components, especiallyfor three-dimensional problems as discussed in [17], are not needed for the im-plementation of the Dirichlet preconditioner in [26]. But they are still requiredin [26] just for the analysis, where subdomain Stokes extensions were used, toobtain a scalable condition number bound.In this paper, we provide a new analysis for the algorithms in [16, 26], whichcan analyze both lumped and Dirichlet preconditioners in a same framework. Itdoes not use any subdomain Stokes extensions and those additional coarse levelvelocity components to enforce divergence free subdomain boundary velocityconditions are no longer needed. For both lumped and Dirichlet precondition-ers, the coarse level space can be chosen as simple as for solving scalar ellipticproblems corresponding to each velocity component. This greatly simplifiesthe requirements on the coarse level space for the case of Dirichlet precondi-tioner, especially in three dimensions. This paper is presented in the contextof solving three-dimensional problems; the same approach can be applied to
OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES
We consider solving the following incompressible Stokes problem on a bounded,three-dimensional polyhedral domain Ω with a Dirichlet boundary condition,(1) − ∆ u ∗ + ∇ p ∗ = f , in Ω , −∇ · u ∗ = 0 , in Ω , u ∗ = u ∗ ∂ Ω , on ∂ Ω ,where the boundary velocity u ∗ ∂ Ω satisfies the compatibility condition R ∂ Ω u ∗ ∂ Ω · n = 0. For simplicity, we assume that u ∗ ∂ Ω = without losing any generality.The weak solution of (1) is given by: find u ∗ ∈ (cid:0) H (Ω) (cid:1) = { v ∈ ( H (Ω)) (cid:12)(cid:12) v = on ∂ Ω } and p ∗ ∈ L (Ω), such that(2) ( a ( u ∗ , v ) + b ( v , p ∗ ) = ( f , v ) , ∀ v ∈ (cid:0) H (Ω) (cid:1) ,b ( u ∗ , q ) = 0 , ∀ q ∈ L (Ω) ,where a ( u ∗ , v ) = R Ω ∇ u ∗ · ∇ v , b ( u ∗ , q ) = − R Ω ( ∇ · u ∗ ) q, ( f , v ) = R Ω f · v . Wenote that the solution of (2) is not unique, with the pressure p ∗ different up toan additive constant.A mixed finite element is used to solve (2). In this paper we apply a mixedfinite element with continuous pressures, e.g., the Taylor-Hood type mixed finiteelements. The same algorithm and analysis can be applied to mixed finiteelements with discontinuous pressures as well; see [26]. Denote the velocityfinite element space by W ⊂ (cid:0) H (Ω) (cid:1) , and the pressure finite element spaceby Q ⊂ L (Ω). The finite element solution ( u , p ) ∈ W L Q of (2) satisfies(3) (cid:20) A B T B (cid:21) (cid:20) u p (cid:21) = (cid:20) f (cid:21) , where A , B , and f represent respectively the restrictions of a ( · , · ), b ( · , · ) and( f , · ) to the finite-dimensional spaces W and Q . We use the same notation inthis paper to represent both a finite element function and the vector of its nodalvalues. OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES A is symmetricpositive definite. Ker ( B T ), the kernel of B T , contains all constant pressures in Q . Im ( B ), the range of B , is orthogonal to Ker ( B T ) and consists of all vectorsin Q with zero average. For a general right-hand side vector ( f , g ) in (3), theexistence of solution requires that g ∈ Im ( B ), i.e., g has zero average; for theright-hand side given in (3), g = 0 and the solution always exists. When thepressure is considered in the quotient space Q/Ker ( B T ), the solution is unique.In this paper, when q ∈ Q/Ker ( B T ), we always assume that q has zero average.Let h represent the characteristic diameter of the mixed elements. We as-sume that the mixed finite element space W × Q , is inf-sup stable in the sensethat there exists a positive constant β , independent of h , such that(4) sup w ∈ W h q, B w i h w , A w i ≥ β h q, Zq i , ∀ q ∈ Q/Ker ( B T ) , cf. [3, Chapter III, § h· , ·i representsthe inner (or semi-inner) product of two vectors. The matrix Z representsthe mass matrix defined on the pressure finite element space Q , i.e., for any q ∈ Q , k q k L = h q, Zq i . It is easy to see, cf. [27, Lemma B.31], that Z isspectrally equivalent to h I for three-dimensional problems, i.e., there existpositive constants c and C , such that(5) ch I ≤ Z ≤ Ch I, where I represents the identity matrix. Here, as in other places of this paper, c and C represent generic positive constants which are independent of h and thesubdomain diameter H (described in the following section). The domain Ω is decomposed into N non-overlapping polyhedral subdomainsΩ i , i = 1 , , ..., N . Each subdomain is the union of a bounded number ofelements, with the diameter of the subdomain in the order of H . We use Γ torepresent the subdomain interface which contains all the subdomain boundarynodes shared by neighboring subdomains; we assume that the subdomain mesheshave matching nodes across Γ. Γ is composed of subdomain faces, which areregarded as open subsets of Γ shared by two subdomains, subdomain edges,which are regarded as open subsets of Γ shared by more than two subdomains,and of the subdomain vertices, which are end points of edges.The velocity and pressure finite element spaces W and Q are decomposedinto W = W I M W Γ , Q = Q I M Q Γ , where W I and Q I are direct sums of independent subdomain interior velocity OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES W ( i ) I , and interior pressure spaces Q ( i ) I , respectively, i.e., W I = N M i =1 W ( i ) I , Q I = N M i =1 Q ( i ) I . W Γ and Q Γ are subdomain interface velocity and pressure spaces, respectively.All functions in W Γ and Q Γ are continuous across Γ; their degrees of freedomare shared by neighboring subdomains.To formulate the domain decomposition algorithm, we introduce a partiallysub-assembled subdomain interface velocity space f W Γ , f W Γ = W ∆ M W Π = N M i =1 W ( i )∆ ! M W Π . W Π is the continuous, coarse level, primal velocity space which is typicallyspanned by subdomain vertex nodal basis functions, and/or by interface edge/face-cutoff functions with constant nodal values on each edge/face, or with values ofpositive weights on these edges/faces. The primal, coarse level velocity degreesof freedom are shared by neighboring subdomains. The complimentary space W ∆ is the direct sum of independent subdomain dual interface velocity spaces W ( i )∆ , which correspond to the remaining subdomain interface velocity degreesof freedom and are spanned by basis functions which vanish at the primal de-grees of freedom. Thus, an element in f W Γ typically has a continuous primalvelocity component and a discontinuous dual velocity component.It is well known that, for domain decomposition algorithms, the coarse space W Π should be sufficiently rich to achieve a scalable convergence rate. On theother hand, a large coarse level problem will certainly degrade the parallel per-formance of the algorithm. Therefore it is important to keep the size of thecoarse level problem as small as possible. When the Dirichlet preconditionerwas used in the FETI-DP algorithm for solving incompressible Stokes equa-tions [15] and similarly in the BDDC algorithm [17], subdomain discrete Stokesextensions were used and W Π has to contain sufficient subdomain interfacecomponents such that functions in W ∆ have zero flux across the subdomainboundaries. Such requirements lead to a large coarse level velocity space, espe-cially for three-dimensional problems, cf. [17].In [26], a FETI-DP type algorithm is proposed for solving two-dimensionalincompressible Stokes problems. A distinctive feature of the Dirichlet precondi-tioner used in that algorithm is the application of subdomain discrete harmonicextensions, instead of subdomain discrete Stokes extensions. As a result, thedivergence free subdomain boundary velocity conditions are not needed in thatalgorithm. However, the analysis, given in [26] for the Dirichlet preconditioner,still uses subdomain Stokes extensions and requires the same type coarse levelvelocity space as discussed in [17] to establish a scalable condition number boundestimate. In this paper, a new analysis is offered and it is sufficient for W Π tobe spanned just by the subdomain vertex nodal basis functions and subdomain OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES w ∆ in W ∆ are in general not continuous across Γ. To enforcetheir continuity, we define a Boolean matrix B ∆ of the form B ∆ = h B (1)∆ B (2)∆ · · · B ( N )∆ i , constructed from { , , − } . On each row of B ∆ , there are only two nonzeroentries, 1 and −
1, corresponding to one velocity degree of freedom shared by twoneighboring subdomains, such that for any w ∆ in W ∆ , each row of B ∆ w ∆ = 0implies that these two degrees of freedom from the two neighboring subdomainsbe the same. We note that, in three dimensions, a velocity degree of freedomon a subdomain edge is shared by more than two subdomains, e.g., by foursubdomains. In this case, a minimum of three continuity constraints can beapplied to enforce the continuity of this velocity degree of freedom among thefour subdomains, which corresponds to the use of non-redundant Lagrange mul-tipliers. In this paper, the fully redundant Lagrange multipliers are used, whichmeans, e.g., for a subdomain edge velocity degree of freedom shared by foursubdomains, six Lagrange multipliers are used to enforce all the six possiblecontinuity constraints among them, cf. [27, Section 6.3.1].We denote the range of B ∆ applied on W ∆ by Λ, the vector space of the La-grange multipliers. Solving the original fully assembled linear system (3) is thenequivalent to: find ( u I , p I , u ∆ , u Π , p Γ , λ ) ∈ W I L Q I L W ∆ L W Π L Q Γ L Λ,such that(6) A II B TII A I ∆ A I Π B T Γ I B II B I ∆ B I Π A ∆ I B TI ∆ A ∆∆ A ∆Π B T Γ∆ B T ∆ A Π I B TI Π A Π∆ A ΠΠ B T ΓΠ B Γ I B Γ∆ B ΓΠ B ∆ u I p I u ∆ u Π p Γ λ = f I f ∆ f Π ,where the sub-blocks in the coefficient matrix represent the restrictions of A and B in (3) to appropriate subspaces. The leading three-by-three block can bemade block diagonal with each diagonal block corresponding to one subdomain.The coefficient matrix in (6) is singular. The trivial null space vectors arethose with λ in the null space of B T ∆ and other components zero. Such singular-ity, due to the rank deficiency of B ∆ , needs not to be worried, since the Lagrangemultiplier vector λ will be confined in Λ, the range of B ∆ . The only meaningfulbasis vector in the null space of (6) corresponds to the one-dimensional nullspace of the original incompressible Stokes system (3), and is specified in thefollowing lemmas.We first need to introduce a positive scaling factor δ † ( x ) for each node x onΓ. Let N x be the number of subdomains sharing x , and we define δ † ( x ) = 1 / N x .Given such scaling factors at the subdomain interface nodes, we can define a OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES B ∆ ,D . We note that each row of B ∆ has only two nonzeroentries, 1 and −
1, connecting two neighboring subdomains sharing a node x onΓ. Multiplying each entry by the scaling factor δ † ( x ) gives us B ∆ ,D . Namely B ∆ ,D = h D ∆ B (1)∆ D ∆ B (2)∆ · · · D ∆ B ( N )∆ i , where D ∆ is a diagonal matrix and contains δ † ( x ) on its diagonal. We also seefrom the definition of B ∆ ,D that the scalings on all the Lagrange multipliersrelated to the same subdomain interface node are the same, from which we havethe following lemma. Lemma 1
The null of B T ∆ is the same as the null of B T ∆ ,D ; the range of B ∆ isthe same as the range of B ∆ ,D . The following lemma can be found at [27, Page 175].
Lemma 2
For any λ ∈ Λ , B ∆ B T ∆ ,D λ = B ∆ ,D B T ∆ λ = λ. Lemma 3
Let p I ∈ Q I , p Γ ∈ Q Γ represent vectors with value on each entry.Then (7) [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21) = B T ∆ λ, where (8) λ = B ∆ ,D [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21) ∈ Λ . Proof:
The left side of (7) contains face integrals of the normal component ofthe dual subdomain interface velocity finite element basis functions across thesubdomain interface. For a face velocity degree of freedom, which is shared bytwo neighboring subdomains, the face integrals of their normal components onthe two neighboring subdomains are negative of each other, since their normaldirections are opposite. This pair of opposite values can then be representedby the product of B T ∆ and a Lagrange multiplier with value equal to the faceintegral of the corresponding basis function.Now we consider a subdomain edge velocity degree of freedom, which isshared by more than two subdomains, e.g., by four subdomains Ω i , Ω j , Ω k , andΩ l . A two-dimensional illustration of such an edge node is shown in Figure 1,where the edge shared by the four subdomains points outward directly. Denotethe four faces having this edge in common by F ij , F jk , F kl , F li , where, e.g., F ij represents the face shared by Ω i and Ω j , while Ω i and Ω k have no common face.Denote the integration of the normal component of this velocity basis functionon these four faces by I ij , I jk , I kl , I li , with a chosen normal direction for eachface, e.g., upward on F ij and F kl , to the right on F jk and F li . Then the entriesof the left side vector in (7) corresponding to this edge velocity degree of freedomon the four subdomains Ω i , Ω j , Ω k , and Ω l , are I ij + I li , − I ij + I jk , − I jk − I kl , and OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES F ij F jk F li I li I ij I jk I kl F kl Ω i Ω j Ω k Ω l Figure 1: Illustration on a subdomain edge interface degree of freedom. I kl − I li , respectively. Here two neighboring subdomains sharing a common facehave opposite face integral values on that face because their normal directionsare opposite of each other. Take I ij , I jk , I kl , I li as the four Lagrange multipliervalues as illustrated in Figure 1. Then the four subdomain face integral values I ij + I li , − I ij + I jk , − I jk − I kl , and I kl − I li , can be represented as the productof corresponding B T ∆ with a Lagrange multiplier vector containing these fourLagrange multiplier values and zero elsewhere.The above has just shown that the left side of (7) can be represented by theproduct of B T ∆ with a Lagrange multiplier vector λ . If λ is not in Λ, i.e., notin the range of B ∆ , it can always be written as the sum of its components in Λand in the null of B T ∆ . Then we just take its component in Λ as λ , which doesnot change the product B T ∆ λ . By multiplying B ∆ ,D to both sides of (7) andusing Lemma 2, we have (8). (cid:3) Lemma 4
The basis vector in the null space of (6), corresponding to the one-dimensional null space of the original incompressible Stokes system (3), is (9) (cid:18) , p I , , , p Γ , − B ∆ ,D [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21) (cid:19) . Proof:
Since the null space of (3) consists of all constant pressures, substi-tuting the vector (9) into (6) gives zero blocks on the right-hand side, except atthe third block where(10) f ∆ = [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21) − B T ∆ B ∆ ,D [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21) , which also equals zero from (7) and (8) in Lemma 3. (cid:3) OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES The system (6) can be reduced to a Schur complement problem for the variables( p Γ , λ ). Since the leading four-by-four block of the coefficient matrix in (6) isinvertible, the variables ( u I , p I , u ∆ , u Π ) can be eliminated and we obtain(11) G " p Γ λ = g, where(12) G = B C e A − B TC , g = B C e A − f I f ∆ f Π , with(13) e A = A II B TII A I ∆ A I Π B II B I ∆ B I Π A ∆ I B TI ∆ A ∆∆ A ∆Π A Π I B TI Π A Π∆ A ΠΠ and B C = " B Γ I B Γ∆ B ΓΠ B ∆ . We can see that − G is the Schur complement of the coefficient matrix of (6)with respect to the last two row blocks, i.e., " I − B C e A − I A B TC B C I − e A − B TC I = " e A − G . From the Sylvester law of inertia, namely, the number of positive, negative, andzero eigenvalues of a symmetric matrix is invariant under a change of coordi-nates, we can see that the number of zero eigenvalues of G is the same as thenumber of zero eigenvalues (with multiplicity counted) of the original coeffi-cient matrix of (6), and all other eigenvalues of G are positive. Therefore G issymmetric positive semi-definite. The basis vectors of the null space of G alsoinherit those from the null space of (6), and the only interesting basis vector is(14) (cid:18) p Γ , − B ∆ ,D [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21) (cid:19) , which is derived from Lemma 4. The other null space vectors of G are allvectors with λ in the null of B T ∆ and p Γ = 0. The range of G contain all vectorsorthogonal to those null vectors. Denote X = Q Γ L Λ, where, as defined earlier,
OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES
10Λ is the range of B ∆ . Then the range of G , denoted by R G , is the subspace of X orthogonal to (14), i.e.,(15) R G = (" g p Γ g λ ∈ X (cid:12)(cid:12)(cid:12) g Tp Γ p Γ − g Tλ (cid:18) B ∆ ,D [ B TI ∆ B T Γ∆ ] (cid:20) p I p Γ (cid:21)(cid:19) = 0 ) . The restriction of G to its range R G is positive definite. The fact thatthe solution of (6) always exists for any given ( f I , f ∆ , f Π ) on the right-handside implies that the solution of (11) exits for any g defined by (12). Therefore g ∈ R G . When the conjugate gradient method (CG) is applied to solve (11) withzero initial guess, all the iterates are in the Krylov subspace generated by G and g , which is also a subspace of R G , and where the CG cannot break down. Afterobtaining ( p Γ , λ ) from solving (11), the other components ( u I , p I , u ∆ , u Π ) in(6) are obtained by back substitution.In the rest of this section, we discuss the implementation of multiplying G by a vector. The main operation is the product of e A − with a vector, cf. (12).We denote A rr = A II B TII A I ∆ B II B I ∆ A ∆ I B TI ∆ A ∆∆ , A Π r = A Tr Π = (cid:2) A Π I B TI Π A Π∆ (cid:3) , f r = f I f ∆ , and define the Schur complement S Π = A ΠΠ − A Π r A − rr A r Π , which is symmetric positive definite from the Sylvester law of inertia. S Π definesthe coarse level problem in the algorithm. The product A II B TII A I ∆ A I Π B II B I ∆ B I Π A ∆ I B TI ∆ A ∆∆ A ∆Π A Π I B TI Π A Π∆ A ΠΠ − f I f ∆ f Π can then be represented by " A − rr f r + " − A − rr A r Π I Π S − (cid:0) f Π − A Π r A − rr f r (cid:1) , which requires solving the coarse level problem once and independent subdomainStokes problems with Neumann type boundary conditions twice. Denote(16) f W = W I M f W Γ = W I M W ∆ M W Π . OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES w in f W , denote its restriction to subdomain Ω i by w ( i ) . A subdomain-wise H -seminorm can be defined for functions in f W by | w | H = N X i =1 | w ( i ) | H (Ω i ) . We also define e V = W I M Q I M W ∆ M W Π , and its subspace(17) e V = n v = ( w I , p I , w ∆ , w Π ) ∈ e V (cid:12)(cid:12) B II w I + B I ∆ w ∆ + B I Π w Π = 0 o . For any v = ( w I , p I , w ∆ , w Π ) ∈ e V , let w = ( w I , w ∆ , w Π ) ∈ f W . Then h v, v i e A = w I w ∆ w Π T A II A I ∆ A I Π A ∆ I A ∆∆ A ∆Π A Π I A Π∆ A ΠΠ w I w ∆ w Π = N X i =1 w ( i ) I w ( i )∆ w ( i )Π T A ( i ) II A ( i ) I ∆ A ( i ) I Π A ( i )∆ I A ( i )∆∆ A ( i )∆Π A ( i )Π I A ( i )Π∆ A ( i )ΠΠ w ( i ) I w ( i )∆ w ( i )Π = N X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( i ) I w ( i )∆ w ( i )Π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (Ω i ) (18)= | w | H , where the superscript ( i ) is used to represent the restrictions of correspondingvectors and matrices to subdomain Ω i . We can see from (18) that for any v ∈ e V ,the value h v, v i e A is independent of its pressure component p I . h· , ·i e A defines asemi-inner product on e V ; h v, v i e A = 0 if and only if the velocity component of v is constant on Ω and is in fact zero due to the zero boundary condition on ∂ Ω,while its pressure component can be arbitrary.Denote(19) e B = " B II B I ∆ B I Π B Γ I B Γ∆ B ΓΠ , cf. (6). The following lemma on the stability of e B can be found at [16, Lemma5.1]. Lemma 5
For any w ∈ f W and q ∈ Q , D e B w , q E ≤ | w | H k q k L . The following lemma will also be used and can be found at [8, Lemma 2.3].
Lemma 6
Let ( u , p ) ∈ W L Q satisfy (20) " A B T B u p = " f g , OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES where A and B are as in (3), f ∈ W , and g ∈ Im ( B ) ⊂ Q . Let β be the inf-supconstant specified in (4). Then k u k A ≤ k f k A − + 1 β k g k Z − , where Z is the mass matrix defined in Section 2. We first define certain jump operators across the subdomain interface Γ, whichwill be used for the analysis of the preconditioners.Denote the restriction operator from e V onto W ∆ by e R ∆ , i.e., for any v =( w I , p I , w ∆ , w Π ) ∈ e V , e R ∆ v = w ∆ . Define P D,L : e V → e V , by P D,L = e R T ∆ B T ∆ ,D B ∆ e R ∆ . Following this definition, given any v = ( w I , p I , w ∆ , w Π ) ∈ e V , the dualvelocity component of P D,L v , on any subdomain interface node x in subdomainΩ i , is given by, cf. [27, Equation (6.70)], (cid:16) e R ∆ ( P D,L v ) (cid:17) ( i ) ( x ) = X j ∈N x δ † ( x ) (cid:16) w ( i )∆ ( x ) − w ( j )∆ ( x ) (cid:17) , which represents the so-called jump of the dual velocity component w ∆ acrossthe subdomain interface Γ. All other components of P D,L v equal zero. We alsohave h P D,L v, P
D,L v i e A = ( e R T ∆ B T ∆ ,D B ∆ e R ∆ v ) T e A ( e R T ∆ B T ∆ ,D B ∆ e R ∆ v )= (cid:10) B T ∆ ,D B ∆ w ∆ , B T ∆ ,D B ∆ w ∆ (cid:11) A ∆∆ . (21)Together with (18), we have the following lemma, which can be found at [18,Section 6.1]. Lemma 7
There exists a constant C and a function Φ L ( H/h ) , such that forall v ∈ e V , h P D,L v, P
D,L v i e A ≤ C Φ L ( H/h ) h v, v i e A . Here, Φ L ( H/h ) = (
H/h )(1 +log (
H/h )) , when the coarse level space is spanned by the subdomain vertex nodalbasis functions and subdomain edge-cutoff functions corresponding to each ve-locity component. When applying P D,L to a vector, the jump of the dual subdomain interfacevelocities is extended by zero to the interior of subdomains. To improve thestability of the jump operator, the jump can be extended to the interior ofsubdomains by subdomain discrete harmonic extension. We define a Schurcomplement operator H ( i )∆ : W ( i )∆ → W ( i )∆ by, for any u ( i )∆ ∈ W ( i )∆ ,(22) " A ( i ) II A ( i ) I ∆ A ( i )∆ I A ( i )∆∆ u ( i ) I u ( i )∆ = " H ( i )∆ u ( i )∆ . OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES H ( i )∆ by a vector u ( i )∆ , a subdomain elliptic problem on Ω i with givenboundary velocity u ( i )∆ and u ( i )Π = needs to be solved. We let H ∆ : W ∆ → W ∆ to represent the direct sum of H ( i )∆ , i = 1 , . . . , N .Using H ( i )∆ , we define the second jump operator P D,D : e V → e V , by: forany given v = ( w I , p I , w ∆ , w Π ) ∈ e V , the subdomain interior velocity partof P D,D v on each subdomain Ω i is taken as u ( i ) I in the solution of (22), withgiven subdomain boundary velocity u ( i )∆ = B ( i ) T ∆ ,D B ∆ w ∆ . Here B ( i ) T ∆ ,D representsrestriction of B T ∆ ,D on subdomain Ω i and is a map from Λ to W ( i )∆ . The othercomponents of P D,D v are kept zero. Therefore h P D,D v, P
D,D v i e A = N X i =1 " u ( i ) I u ( i )∆ T " A ( i ) II A ( i ) I ∆ A ( i )∆ I A ( i )∆∆ u ( i ) I u ( i )∆ = N X i =1 u ( i ) T ∆ H ( i )∆ u ( i )∆ = N X i =1 w T ∆ B T ∆ B ( i )∆ ,D H ( i )∆ B ( i ) T ∆ ,D B ∆ w ∆ = N X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" B ( i ) T ∆ ,D B ∆ w ∆ H / ( ∂ Ω i ) ≤ C Φ D ( H/h ) N X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" w ( i )∆ w ( i )Π H / ( ∂ Ω i ) (23) ≤ C Φ D ( H/h ) N X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( i ) I w ( i )∆ w ( i )Π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H (Ω i ) = C Φ D ( H/h ) | w | H (Ω i ) . The first inequality in (23) is a well established result, cf., [27, Lemma 6.36].Since for any v ∈ e V , h v, v i e A = | w | H (Ω i ) , cf. (18), we have the following lemma. Lemma 8
There exists a constant C and a function Φ D ( H/h ) , such that forall v ∈ e V , h P D,D v, P
D,D v i e A ≤ C Φ D ( H/h ) h v, v i e A . Here Φ D ( H/h ) = (1 +log (
H/h )) , when the coarse level space is spanned by the subdomain vertexnodal basis functions and subdomain edge-cutoff functions corresponding to eachvelocity component. To introduce the preconditioners, we write G , defined in (12) and (13), in atwo-by-two block structure. Denote the first row of B C by e B Γ = [ B Γ I B Γ∆ B ΓΠ ] , and note that e R ∆ is the restriction operator from e V onto W ∆ . Then G can bewritten as(24) G = " G p Γ p Γ G p Γ λ G λp Γ G λλ , where G p Γ p Γ = e B Γ e A − e B T Γ , G p Γ λ = e B Γ e A − e R T ∆ B T ∆ ,G λp Γ = B ∆ e R ∆ e A − e B T Γ , G λλ = B ∆ e R ∆ e A − e R T ∆ B T ∆ . OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES G p Γ p Γ of G can be shown spectrally equivalentto h I p Γ , where I p Γ is the identity matrix of the same dimension as G p Γ p Γ ; see[16, 26]. Therefore, in the following block diagonal preconditioners, the inverseof G p Γ p Γ is approximated by αh − I p Γ . Here α is a given constant. We willshow in the next section that α has only a minor effect on the condition numberbound of the preconditioned operator and its value is typically taken as 1, cf.Remark 2. We introduce α in the preconditioner just for the convenience in thenumerical experiments to demonstrate the convergence rates of the proposedalgorithm.The inverse of the second diagonal block B ∆ e R ∆ e A − e R T ∆ B T ∆ , can be approx-imated by the lumped block(25) M − λ,L = B ∆ ,D e R ∆ e A e R T ∆ B T ∆ ,D . This leads to the following lumped preconditioner for solving (11)(26) M − L = " αh − I p Γ M − λ,L . Applying subdomain discrete harmonic extensions in the preconditioning step,we have the following Dirichlet preconditioner(27) M − D = " αh − I p Γ M − λ,D , where(28) M − λ,D = B ∆ ,D H ∆ B T ∆ ,D . We can see from Lemma 1 that both M − λ,L and M − λ,D are symmetric positivedefinite when restricted on Λ. Therefore both the lumped and the Dirichletpreconditioners M − L and M − D are symmetric positive definite in the range of G . In the following, we use the same framework to establish the condition numberbounds for both lumped and Dirichlet preconditioned operators M − L G and M − D G . Let M − , M − λ , P D , and Φ to represent both M − L , M − λ,L , P D,L , Φ L ,for the lumped preconditioner case, and M − D , M − λ,D , P D,D , Φ D , for the Dirichletpreconditioner case, respectively, when they apply in the proofs.When the conjugate gradient method is applied to solving the preconditionedsystem(29) M − Gx = M − g, OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES M − G and the vector M − g , which is a subspace of the range of M − G . We denote the range of M − G by R M − G and note that both precondi-tioners are symmetric positive definite in the range of G . We have the followinglemma, cf. [26, Lemma 6]. Lemma 9
The conjugate gradient method applied to solving (29) with zero ini-tial guess cannot break down.Proof:
We just need to show that for any 0 = x ∈ R M − G , h x, Gx i 6 = 0, i.e.,to show Gx = 0. Let 0 = x = M − Gy , for a certain y ∈ X and y = 0. Then Gx = GM − Gy , which cannot be zero since Gy = 0 and y T GM − Gy = 0. (cid:3) The following lemma will be used to provide the upper eigenvalue bound ofthe preconditioned operator. It is similar to [16, Lemma 6.4] and [26, Lemmas8 and 11].
Lemma 10
There exists a constant C , such that for all v ∈ e V , (cid:10) M − B C v, B C v (cid:11) ≤ C ( α + Φ( H/h )) D e
Av, v E , where Φ( H/h ) is defined in Lemmas 7 and 8, respectively.Proof: Given v = ( w I , q I , w ∆ , w Π ) ∈ e V , let g p Γ = B Γ I w I + B Γ∆ w ∆ + B ΓΠ w Π . From (13), (25)–(28), (21), and (23), we have (cid:10) M − B C v, B C v (cid:11) = αh − h g p Γ , g p Γ i + (cid:16) B ∆ e R ∆ v (cid:17) T M − λ B ∆ e R ∆ v = αh − h g p Γ , g p Γ i + h P D v, P D v i e A ≤ αh − h g p Γ , g p Γ i + C Φ( H/h ) h v, v i e A , (30)where we used Lemmas 7 and 8 for the last inequality. It is sufficient to boundthe first term of the right-hand side in the above inequality.Since v ∈ e V , we have B II w I + B I ∆ w ∆ + B I Π w Π = 0, cf. (17). Then h g p Γ , g p Γ i = (cid:20) B II w I + B I ∆ w ∆ + B I Π w Π B Γ I w I + B Γ∆ w ∆ + B ΓΠ w Π (cid:21) T (cid:20) B II w I + B I ∆ w ∆ + B I Π w Π B Γ I w I + B Γ∆ w ∆ + B ΓΠ w Π (cid:21) = D e B w , e B w E , where e B is defined in (19) and w = ( w I , w ∆ , w Π ) ∈ f W . From (5) and thestability of e B , cf. Lemma 5, we have h − h g p Γ , g p Γ i = h − D e B w , e B w E ≤ C D e B w , e B w E Z − = C max q ∈ Q D e B w , q E h q, q i Z (31) ≤ C max q ∈ Q | w | H k q k L k q k L = C | w | H = C h v, v i e A , OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES (cid:3)
The following lemma will be used to provide the lower eigenvalue bound ofthe preconditioned operator. In [26, Lemmas 9 and 12], the lower eigenvaluebounds for the lumped and Dirichlet preconditioners were analyzed differently.In the analysis of the Dirichlet preconditioner, subdomain discrete Stokes ex-tensions were used. Such extensions require enforcing the same type divergencefree subdomain boundary velocity conditions as discussed in [17], even thoughthey are not necessary for implementing the algorithm in [26]. The new proofgiven in the next lemma works for both lumped and Dirichlet preconditioners.It does not use the subdomain Stokes extensions and those additional subdo-main divergence free boundary conditions are no longer needed. For both typeof preconditioners, the coarse level velocity space can be chosen as simple as forsolving scalar elliptic problems corresponding to each velocity component.
Lemma 11
There exists a constant C , such that for any nonzero y = ( g p Γ , g λ ) ∈ R G , there exits v ∈ e V , which satisfies B C v = y , h v, v i e A = 0 , and D e
Av, v E ≤ C max (cid:8) , α (cid:9) (cid:16) β (cid:17) (cid:10) M − y, y (cid:11) .Proof: Given y = ( g p Γ , g λ ) ∈ R G , take u ( I )∆ = B T ∆ ,D g λ , u ( I )Π = , and p ( I ) = 0. On each subdomain Ω i , let u ( I,i ) I be zero for the lumped precon-ditioner, and be obtained for the Dirichlet preconditioner through the solu-tion of (22) with given subdomain boundary values u ( i )∆ = u ( I,i )∆ . Let v ( I,i ) = (cid:16) u ( I,i ) I , p ( I,i ) I , u ( I,i )∆ , u ( I,i )Π (cid:17) , the corresponding global vectors v ( I ) = (cid:16) u ( I ) I , p ( I ) I , u ( I )∆ , u ( I )Π (cid:17) ,and u ( I ) = (cid:16) u ( I ) I , u ( I )∆ , u ( I )Π (cid:17) . Then we have(32) B C v ( I ) = " B Γ I B Γ∆ B ΓΠ B ∆ u ( I ) I p ( I ) I u ( I )∆ u ( I )Π = " B Γ I u ( I ) I + B Γ∆ u ( I )∆ + B ΓΠ u ( I )Π g λ , where we have used Lemma 2. Also | u ( I ) | H = u ( I ) I u ( I )∆ u ( I )Π T A II A I ∆ A I Π A ∆ I A ∆∆ A ∆Π A Π I A Π∆ A ΠΠ u ( I ) I u ( I )∆ u ( I )Π (33) = ( | u ( I )∆ | A ∆∆ , for lumped preconditioner, | u ( I )∆ | H ∆ , for Dirichlet preconditioner . We consider a solution to the following fully assembled system of linear equa-tions of the form (3): find (cid:16) u ( II ) I , p ( II ) I , u ( II )Γ , p ( II )Γ (cid:17) ∈ W I L Q I L W Γ L Q Γ , OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES A II B TII A I Γ B T Γ I B II B I Γ A Γ I B TI Γ A ΓΓ B T ΓΓ B Γ I B ΓΓ u ( II ) I p ( II ) I u ( II )Γ p ( II )Γ = − B II u ( I ) I − B I ∆ u ( I )∆ − B I Π u ( I )Π g p Γ − B Γ I u ( I ) I − B Γ∆ u ( I )∆ − B ΓΠ u ( I )Π ,where we know that the particularly chosen right-hand side is essentially(35) − B II u ( I ) I − B I ∆ u ( I )∆ g p Γ − B Γ I u ( I ) I − B Γ∆ u ( I )∆ . Since ( g p Γ , g λ ) ∈ R G , we have, cf. (15),( − B I ∆ u ( I )∆ ) T p I +( g p Γ − B Γ∆ u ( I )∆ ) T p Γ = g Tp Γ p Γ − g Tλ B ∆ ,D (cid:0) B TI ∆ p I + B T Γ∆ p Γ (cid:1) = 0 . Meanwhile,( − B II u ( I ) I ) T p I + ( − B Γ I u ( I ) I ) T p Γ = − Z Ω (cid:16) ∇ · u ( I ) I (cid:17) . We have that the right-hand side vector (35) has zero average, which impliesexistence of the solution to (34).Denote u ( II ) = (cid:16) u ( II ) I , u ( II )Γ (cid:17) . Then from Lemma 6 and (5), we have | u ( II ) | H ≤ β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" − B II u ( I ) I − B I ∆ u ( I )∆ − B I Π u ( I )Π g p Γ − B Γ I u ( I ) I − B Γ∆ u ( I )∆ − B ΓΠ u ( I )Π Z − ≤ β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" B II u ( I ) I + B I ∆ u ( I )∆ + B I Π u ( I )Π B Γ I u ( I ) I + B Γ∆ u ( I )∆ + B ΓΠ u ( I )Π Z − + 1 β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" g p Γ Z − ≤ β | u ( I ) | H + Cβ h h g p Γ , g p Γ i , (36)where the bound on the first term is obtained in the same way as in (31).Split the continuous subdomain interface velocity u ( II )Γ into the dual part u ( II )∆ and the primal part u ( II )Π , and denote v ( II ) = (cid:16) u ( II ) I , p ( II ) I , u ( II )∆ , u ( II )Π (cid:17) . OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES v = v ( I ) + v ( II ) . Then we have from (34) that v ∈ e V , and B C v ( II ) = " B Γ I B Γ∆ B ΓΠ B ∆ u ( II ) I p ( II ) I u ( II )∆ u ( II )Π = " g p Γ − B Γ I u ( I ) I − B Γ∆ u ( I )∆ − B ΓΠ u ( I )Π . Together with (32), we have B C v = y . From (18) and (36), we have | v | e A = | u ( I ) + u ( II ) | H ≤ | u ( I ) | H + | u ( II ) | H = (cid:18) β (cid:19) | u ( I ) | H + Cβ h h g p Γ , g p Γ i = (cid:18) β (cid:19) | u ( I )∆ | A ∆∆ + Cβ h h g p Γ , g p Γ i , for lumped preconditioner, (cid:18) β (cid:19) | u ( I )∆ | H ∆ + Cβ h h g p Γ , g p Γ i , for Dirichlet preconditioner , where we used (33) in the last equality.On the other hand, we have from (25)–(28) (cid:10) M − y, y (cid:11) = αh h g p Γ , g p Γ i + g Tλ M − λ g λ = αh h g p Γ , g p Γ i + g Tλ B ∆ ,D A ∆∆ B T ∆ ,D g λ , for lumped preconditioner, αh h g p Γ , g p Γ i + g Tλ B ∆ ,D H ∆ B T ∆ ,D g λ , for Dirichlet preconditioner,= αh h g p Γ , g p Γ i + | u ( I )∆ | A ∆∆ , for lumped preconditioner, αh h g p Γ , g p Γ i + | u ( I )∆ | H ∆ , for Dirichlet preconditioner.It is not difficult to see that h v, v i e A = 0. Otherwise, all the velocity compo-nents of v would be zero, cf. (18), and then B C v would be zero, which conflictswith that B C v = y and y is nonzero. (cid:3) The proofs of the following two lemmas can be found at [16, Lemmas 6.6and 6.3].
Lemma 12
For any v = ( w I , p I , w ∆ , w Π ) ∈ e V , B C v ∈ R G . Lemma 13
For any x ∈ R M − G , h M x, x i = max y ∈ R G ,y =0 h y, x i h M − y, y i . OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES M − G is givenin the following theorem. Theorem 1
There exist positive constants c and C , such that for all x ∈ R M − G , min { , α } cβ (1 + β ) h M x, x i ≤ h
Gx, x i ≤ C ( α + Φ( H/h )) h M x, x i . Proof:
We only need to prove the above inequalities for any nonzero x ∈ R M − G . We know from Lemma 9 that0 = h Gx, x i = x T B C e A − B TC x = x T B C e A − e A e A − B TC x = D e A − B TC x, e A − B TC x E e A . Therefore e A − B TC x = 0. Also note that e A − B TC x ∈ e V and h· , ·i e A defines asemi-inner product on e V , cf (18), and then we have(37) h Gx, x i = max v ∈ e V , h v,v i e A =0 D v, e A − B TC x E e A h v, v i e A = max v ∈ e V , h v,v i e A =0 h B C v, x i D e
Av, v E . Lower bound:
From Lemma 11, we know that for any nonzero y ∈ R G , thereexits w ∈ e V , such that B C w = y , h w, w i e A = 0, D e
Aw, w E ≤ max (cid:8) , α (cid:9) C (1+ β ) β (cid:10) M − y, y (cid:11) .Then from (37), we have h Gx, x i ≥ h B C w, x i D e
Aw, w E ≥ c β max (cid:8) , α (cid:9) (1 + β ) h y, x i h M − y, y i . Since y is arbitrary, using Lemma 13, we have h Gx, x i ≥ c β max (cid:8) , α (cid:9) (1 + β ) max y ∈ R G ,y =0 h y, x i h M − y, y i = min { , α } cβ (1 + β ) h M x, x i . Upper bound:
From (37) and the fact that h Gx, x i 6 = 0, we have h Gx, x i = max v ∈ e V , h v,v i e A =0 h B C v, x i D e
Av, v E = max v ∈ e V , h v,v i e A =0 ,B C v =0 h B C v, x i D e
Av, v E , where the maximum only needs to be considered among v also satisfying B C v =0. Then using Lemmas 10, 12, and 13, we have h Gx, x i ≤ C ( α + Φ( H, h )) max v ∈ e V , h v,v i e A =0 ,B C v =0 h B C v, x i h M − B C v, B C v i≤ C ( α + Φ( H, h )) max y ∈ R G ,y =0 h y, x i h M − y, y i = C ( α + Φ( H, h )) h M x, x i . (cid:3) OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES Remark 2
We can see from Theorem 1 that, for α ≥
1, the condition numberbound of M − G is proportional to α + Φ( H, h ), and we should take smaller α to achieve faster convergence. When α ≤
1, the condition number boundis proportional to 1 + Φ( H,h ) α and we should take larger α . This explains whythe value of α in (26) and (27) is typically taken as 1. We introduce α in thepreconditioner just for the convenience to demonstrate the convergence rates ofthe proposed algorithm in the following section. We illustrate the convergence rate of the proposed algorithm by solving theincompressible Stokes problem (1) in both two and three dimensions, on Ω =[0 , and Ω = [0 , , respectively. Zero Dirichlet boundary condition is used.The right-hand side f is chosen such that the exact solution is u = " sin ( πx ) sin ( πy ) cos( πy ) − sin ( πx ) sin ( πy ) cos( πx ) , p = x − y , for two dimensions, and for three dimensions u = sin ( πx ) (sin(2 πy ) sin( πz ) − sin( πy ) sin(2 πz ))sin ( πy ) (sin(2 πz ) sin( πx ) − sin( πz ) sin(2 πx ))sin ( πz ) (sin(2 πx ) sin( πy ) − sin( πx ) sin(2 πy )) , p = xyz − . The Q - Q Taylor-Hood mixed finite element with continuous pressures isused; its inf-sup stability can be found at [2, 22]. In two dimensions, the velocityspace contains piecewise biquadratic functions and the pressure space containspiecewise bilinear functions; in three dimensions, piecewise triquadratic func-tions for the velocity and piecewise trilinear functions for the pressure.The preconditioned conjugate gradient method is used to solve (29); theiteration is stopped when the L − norm of the residual is reduced by a factor of10 − .The following tables list the minimum and maximum eigenvalues of theiteration matrix M − G , and the iteration counts for using both lumped andDirichlet preconditioners, respectively, for different cases. Here the extremeeigenvalues of M − G are estimated by using the tridiagonal Lanczos matrixgenerated in the iteration.Table 1 shows the performance for solving the two-dimensional problem. Thecoarse level velocity space in the algorithm is spanned by the subdomain vertexnodal basis functions corresponding to each velocity component. We take α = 1in both the lumped and the Dirichlet preconditioners (26) and (27). We can seefrom Table 1 that the minimum eigenvalue is independent of the mesh size forboth preconditioners. The maximum eigenvalue is independent of the numberof subdomains for fixed H/h ; for fixed number of subdomains, it depends on
H/h in the order of (
H/h )(1 + log (
H/h )) for the lumped preconditioner, andin the order of (1 + log (
H/h )) for the Dirichlet preconditioner. OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES , , α = 1 in(26) and (27). lumped Dirichlet H/h λ min λ max iteration λ min λ max iteration8 4 × × ×
16 0.3068 38.42 51 0.2556 5.28 2524 ×
24 0.3069 38.62 51 0.2397 5.33 2532 ×
32 0.3070 38.68 51 0.2304 5.36 25
H/h λ min λ max iteration λ min λ max iteration8 × α = 1;in Table 3, α = 1 / H/h ; for fixed number of subdomains, it depends on
H/h ,but not in the order of (
H/h )(1 + log (
H/h )) for the lumped preconditioner, nor(1 + log (
H/h )) for the Dirichlet preconditioner, as Φ( H/h ) does. Moreover,the convergence rate of the algorithm using the Dirichlet preconditioner is onlyslightly better than using the lumped preconditioner. The reason is that theupper eigenvalue bound in Theorem 1 depends on two terms α and Φ( H/h ),and in this case α = 1 dominates when H/h is small. Therefore, even thoughusing the Dirichlet preconditioner can reduce Φ(
H/h ) compared with using thelumped preconditioner, this improvement on the upper eigenvalue bound cannot show up in Table 2. What shows in Table 2 for λ max is essentially itsdependence on α . Only for larger H/h , e.g., for
H/h = 6 and
H/h = 8 inTable 2, the improvement on the upper eigenvalue bound by using the Dirichletpreconditioner becomes visible.To experiment the case when α is less dominant in the upper eigenvaluebound, we take α = 1 / OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES , , α = 1 in(26) and (27). lumped Dirichlet H/h λ min λ max iteration λ min λ max iteration4 3 × × × × × × × × H/h λ min λ max iteration λ min λ max iteration3 × × , , α = 1 / H/h λ min λ max iteration λ min λ max iteration4 3 × × × × × × × × H/h λ min λ max iteration λ min λ max iteration3 × × H/h ) for both preconditioners. They are independent of the number ofsubdomains for fixed
H/h ; for fixed number of subdomains, they depend on
H/h in the order of (
H/h )(1 + log (
H/h )) for the lumped preconditioner, andin the order of (1 + log (
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OMAIN DECOMPOSITION FOR INCOMPRESSIBLE STOKES