A convergence theory of multilevel additive Schwarz ... - CiteSeerX

Baltzer Journals

October 4, 1996

A convergence theory of multilevel additive Schwarz methods on unstructured meshes 1

Tony F. Chan and Jun Zou

2

1 Department

of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555 E-mail: [email protected] 2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong E-mail: [email protected]

We develop a convergence theory for two level and multilevel additive Schwarz domain decomposition methods for elliptic and parabolic problems on general unstructured meshes in two and three dimensions. The coarse and ne grids are assumed only to be shape regular, and the domains formed by the coarse and ne grids need not be identical. In this general setting, our convergence theory leads to completely local bounds for the condition numbers of two level additive Schwarz methods, which imply that these condition numbers are optimal, or independent of ne and coarse mesh sizes and subdomain sizes if the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. In particular, we will show that additive Schwarz algorithms are still very ecient for non-selfadjoint parabolic problems with only symmetric, positive de nite solvers both for local subproblems and for the global coarse problem. These conclusions for elliptic and parabolic problems improve our earlier results in [15, 12, 16]. Finally, the convergence theory is applied to multilevel additive Schwarz algorithms. Under some very weak assumptions on the ne mesh and coarser meshes, e.g. no requirements on the relation between neighboring coarse level meshes, we are able to derive a condition number bound of the order O(2 L2 ), where = max1lL (hl + hl?1 )=l , hl is the element size of the l-th level mesh, l the overlap of subdomains on l-th level mesh, and L the number of mesh levels. Subject classi cation: AMS(MOS) 65N30, 65F10. Keywords: convergence, multilevel additive methods, unstructured meshes. The work was partially supported by the NSF under contract ASC 92-01266, and ONR under contract ONR-N00014-92-J-1890. The second author was also partially supported by HKRGC grants no. CUHK 316/94E and the Direct Grant of CUHK.

T. Chan and J. Zou / Convergence of Unstructured Multilevel Methods

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1 Introduction In recent years, unstructured meshes have become very popular in scienti c computing. One of their advantages is the exibility in adapting eciently to complicated geometries and rapid changes in the solution eld, cf. Barth [3] and Mavriplis [28]. Domain decomposition methods (DDMs) have proved to be very ecient for solving elliptic and parabolic problems on structured meshes. A natural question is: what is the eciency of these methods for unstructured meshes ? Our recent studies show that the existing well-developed convergence theory for overlapping DDMs on structured meshes can be fully extended to unstructured meshes. By unstructured meshes we mean here that the nite element meshes are highly non-quasi-uniform, and not generated by using the usual techniques of recursive re nement of coarser meshes. So in general, no coarse mesh nested to a unstructured one exists for the use in domain decomposition algorithms. Therefore, practical coarse meshes for DDMs on a ne unstructured mesh are generally nonnested to that ne mesh and the coarse domains formed by the coarse meshes are nonmatching to the ne mesh domain. In this general setting, we have previously investigated the convergence of two level additive type DDMs for elliptic problems in Chan-Zou[15] and Chan-Smith-Zou[12] (see also Cai [8], which assumed the quasi-uniform coarse mesh and the matching of the coarse domain boundary to the ne domain boundary), for parabolic problems in Chan-Zou [16]. All these results are valid both in two and three dimensions. Moreover, the multilevel additive Schwarz algorithm (see Bramble-Pasciak-Xu [6] and Zhang [42]) on unstructured meshes was considered in our earlier technical report [14], but with some very restrictive assumptions on the meshes; for example, the ne mesh and all the coarse meshes were required to be quasi-uniform, each coarse element on every level had to contain suciently many ne elements, and coarser boundaries were required to be matching to the boundary of the original ne domain, etc. These restrictions will be removed in the present paper by using some new technical tools. In this paper, by extending the convergence theory developed in Xu [36] for structured meshes (which assumes all involved local and coarser spaces are subspaces of a nest space) to unstructured meshes, we provide a uni ed convergence theory for additive type preconditioning iterative methods on unstructured meshes. Here we allow all involved local, coarser spaces and nest space to be independent of each other by introducing certain ne-to-coarse and coarse-to- ne operators. Applying this new analysis framework to two- and multi-level additive DDMs, we can unify our earlier convergence results in [12, 14, 15, 16] and greatly simplify those convergence proofs. Moreover, combining with some new technical lemmas, we can remove several previous restrictions on meshes required in [14], and are able to improve the conclusions in [12, 15, 16] by giving completely local condition number bounds, which imply immediately a practically important convergence phenomenon of the concerned domain decomposition algorithms, that is, the algorithms are still optimal in the sense that the corresponding condition numbers are independent of all mesh parameters as long as the overlap amount of a subdomain with its neighbors varies proportionally to the subdomain size. For related multigrid methods, there are several existing theoretical works on nonnested spaces which are closely related to general unstructured meshes. For example, BramblePasciak-Xu [7] and Bramble [4] proposed a general multigrid convergence theory for symmetric multigrid methods with the same number of pre- and post-smoothings, and then applied the framework to obtain optimal convergence of multigrid algorithms with


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nonnested subspaces de ned on quasi-unform meshes in two dimension. Zhang[40, 41] and Scott-Zhang [33] proved optimal convergence of multigrid algorithms with nonnested subspaces induced by non-quasi-uniform meshes in two dimension ([40, 41]) and quasiuniform meshes in higher dimensions ([33]), where all grid domains on each level are required to have the same boundary. Douglas-Douglas [19] proposed a uni ed convergence theory for general multigrid or multilevel methods which use the recursive way to de ne coarse coecient matrices by ner coecient ones and are applicable to nonsymmetric, inde nite or singular linear systems of equations, and then Douglas-Douglas-Fyfe [20] extended and applied the theory in [19] to derive the convergence of nonnested multigrid methods for solving nite element equations and nite volume equations. Bank-Xu [1, 2] recently proposed an algorithm for coarsening unstructured meshes, which was then used for constructing hierarchical basis multigrid method for unstructured meshes. The content of the paper is arranged as follows: In Section 2, we formulate the general framework for the development of the convergence theory for the additive type DDMs, and a perturbation extension of the result of Section 2 will be presented in Section 2.3 for the application of additive Schwarz algorithms to non-selfadjoint parabolic problems. In Section 3 we introduce the ne and coarse nite element spaces, domain decompositions, the assumptions made for the ne and coarse nite element triangulations, and especially the L2-optimal approximation and H 1 -stability of the standard nite element interpolant and the Clement interpolant. Section 4 and Section 5 will be devoted to the application of the abstract convergence theory developed in Section 2 to self-adjoint elliptic problems and non-selfadjoint parabolic problems. Finally, in Section 6 we apply the abstract theory of Section 2 to the multilevel additive Schwarz methods. We will focus only on the convergence theory of algorithms in the present paper, no attention will be paid to the numerical implementation of the algorithms. For details of the matrix representation of the algorithms, we refer to our earlier work [11, 12, 14, 15]; for numerical experiments, we refer to the papers [11, 12, 9, 10] Throughout the paper, k km;r; and j jm;r; denote the norm and semi-norm of the usual Sobolev spaces W m;r ( ) for any integer m 0 and real number r 1, but the subscript r will be omitted if r = 2, and so W m;r ( ) = H m ( ). The sign k kA will be often used for the inner product (A; )-induced norm. The notation \ a < b " (\ a > b " resp.) for real functions a and b which depend on a set of parameters (e.g. coarse and ne mesh sizes, subdomain sizes or time steps if any) means that there are two positive constants c0 and c1 independent of all the parameteres such that c0 a < c1 b (c0 a > c1 b resp.).

2 Abstract framework of convergence analysis for additive preconditioners This section is devoted to the abstract framework of convergence analysis for two and multilevel additive preconditioners. 2.1 Additive preconditioners for symmetric positive de nite operators Let V , and V k , 0 k p be nite dimensional vector spaces with inner products (; ) and (; )k , resp. All spaces V k are not necessarily subspaces of V . The space V 0 is special,


4

usually referring to the coarse grid space. Given a symmetric positive de nite (SPD in short) operator A on V and f 2 V , we are interested in solving the equation Au = f on V , which arises from the discretization of elliptic or parabolic problems by using nite element methods. As A is ill-conditioned, our goal is to nd a good preconditioner M for A such that MA is better conditioned than A, and the action of M is inexpensive to calculate. Then one can use iterative methods, like Conjugate Gradient method, for MAu = Mf instead of Au = f . We will study in this paper preconditioners of the following additive type: p X

Ik Rk Qk ; (1) k=0 where the \interpolation" operators Ik : V k ! V are linear, and the \projection" operators Qk are the adjoints of Ik de ned by (Qk u; vk )k = (u; Ik vk ); 8 u 2 V; vk 2 V k ; (2) M=

and Rk : V k ! V k are given SPD operators, approximating the inverses of the restrictions of A on V k in some sense. It is easy to verify that M is an SPD operator on V . We remark that the preconditioner form (1) is a natural extension of the one introduced by Xu [36] with nested subspaces. The awareness of this general form (1) was due to Griebel-Oswald [24], see also [12], [34] and [29]. Following the theory of Xu [36] for structured meshes with all V k being subspaces of V , one can similarly bound the condition number of MA for the present unstructured cases in terms of three parameters K0 , !0 and 0 de ned as follows (with a dierent de nition for 0 compared to [36]):

(P1) (P2) (P3)

P For any u 2 V , there exist uk 2 V k (0 k p) such that u = pk=0 Ik uk and

p X

(Rk?1 uk ; uk )k K0 (Au; u):

k=0 k For any uk 2 V , k = 0; 1; ; p,

(AIk uk ; Ik uk ) !0 (Rk?1 uk ; uk )k : For any u 2 V and uk 2 V k (1 k p), p X

k=1

1

(Au; Ik uk ) 02 (Au; u) 12

p X k=1

1

(AIk uk ; Ik uk ) 2 :

Without loss of generality, we assume that K0 1, !0 1 and 0 1. From (1) we may write p X (3) MA = Ik Pk ; Pk = Rk Qk A: k=0

We have the following theorem:

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Theorem 1

Under the assumptions (P1) - (P3), (MA) !0 (0 + 1)K0 :

Proof

Let us rst estimate the minimum eigenvalue of MA, we obtain (Au; u) = (Au; =

p X k=0

p X

k=0

Ik uk ) =

p X k=0

(Qk Au; uk )k (by (P1) & (2))

(Rk?1 Pk u; uk )k ( p 1 X

p X k=0

(Rk?1 uk ; uk )k ) 21 (

p X k=0

(Rk?1 Pk u; Pk u)k ) 12 (Schwarz ineq)

K02 (Au; u) 2 ( (Qk Au; Pk u)k ) 12 = K02 (Au; u) 21 (MAu; Au) 21 ((P1) & (2)-(3)); 1

1

k=0

this implies (MAu; Au) K0?1 (Au; u) which gives min(MA) K0?1 : Next we estimate max (MA). We have for any uk 2 V k (0 k p) that p X k=0

(Au; Ik uk ) = (Au; I0 u0) +

p X

k=1

(Au; Ik uk ) p X

1

kukA kI0 u0kA + 02 kukA ( p X

(0 + 1) 21 kukA ( Thus, we derive that (MAu; Au) =

k=0

p X k=0

k=1

kIk uk k2A) 12 (Schwarz ineq & (P3))

kIk uk k2A) ) 12 (Schwarz ineq):

(Ik Pk u; Au) (0 + 1) kukA ( 1 2

p X

p X k=0

kIk Pk uk2A) 12 (by (4))

!0 (0 + 1) (Au; u) ( (Rk?1 Pk u; Pk u)k ) 12 (by (P2)) 1 2

1 2

1 2

k=0 p X

1 2

= !0 (0 + 1) 21 (Au; u) 21 ( 1 2

k=0

(Au; Ik Pk u)) 12 (by (3) & (2))

= !0 (0 + 1) (Au; u) (MAu; Au) 12 (by (3)); which indicates that max (MA) !0 (0 + 1): Combining the lower and upper bounds gives the desired bound on (MA). 1 2

1 2

(4)


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Remark 1

For the related work to the abstract framework of additive Schwarz methods, we refer to Xu [36], Griebel-Oswald [24], Dryja-Widlund [22], Le Tallec [26] and Smith-Bjrstad-Gropp [34]. (P1) and (P2) are natural extensions of assumptions from the theory for structured meshes by Xu [36], where all the spaces V k , 0 k p, are assumed to be subspaces of V . The kind of partition in (P1) was rst introduced by Nepomnyaschikh [30] and Lions [27] with exact local solvers, i.e. Rk?1 equal to inverses of restrictions of A on V k , and then was generalized by Xu [36] to allow with inexact local solvers. (P1) means that any function in V can be decomposed into a sum of functions in spaces V k and this partition is stable with the \energy" norm in some sense, this ensures the lower bound of smallest eigenvalue of MA. (P2) is equivalent to max (Rk Ak ) !0 where Ak = Qk AIk is the \restriction" of A on V k , it means that the approximation of Rk to the inverse of Ak cann't be \too bad". (P3) is a condition on the \local" properties of V k (1 k p), and requires that the overlapping of spaces V k be bounded independent of p in terms of the energy norm. Note that our (P3) is not the extension of the corresponding assumption used in [36]. It might be replaced by the extension of the so-called strengthened Cauchy-Schwarz inequality in [36] for nested subspaces with identity operators Ii (1 i p): (P3 ) Let "ij 2 (0; 1] be the smallest constants satisfying that

(AIi ui ; Ij uj ) "ij (AIi ui ; Ii ui ) 21 (AIj uj ; Ij uj ) 21 ;

8ui 2 V i ; uj 2 V j ; i; j = 1; ; p: It is straightforward to prove that (P1), (P2) and (P3 ) imply (P3). Thus (P3) is a weaker assumption than (P3 ). We prefer (P3) to (P3 ) as (P3) is more convenient to check than (P3 ), especially for subspaces de ned on unstructured meshes, and for multilevel additive type methods, see Section 6 for more details.

2.2 Multilevel additive preconditioners for SPD operators Though the formulation of preconditioners of this subsection may actually be included in (1) of the last section, we prefer to present a more detailed formulation of preconditioners and assumptions here for the convenience of the applications of these algorithms later on. Let V and V l (0 l L) be given nite dimensional spaces with inner products (; ) and (; )l (0 l L) respectively. All spaces V l are not necessarily subspaces of V . We assume further that for each l : 1 l L, the space V l can be decomposed into a sum of subspaces Vkl (1 k Nl ). Analogous to the last section, we are interested in the following type of preconditioners for a given SPD operator A de ned on the space V :

M=

Nl L X X

Ikl Rkl Qlk

(5)

l=0 k=1 where the \interpolation" operators Ikl : Vkl ! V are linear, and the \projection" operators Qlk are the adjoints of Ikl de ned by (Qlk u; vkl )l = (u; Ikl vkl ); 8 u 2 V; vkl 2 Vkl ; (6)


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and Rkl : Vkl ! Vkl are given SPD operators, approximating the inverses of the restrictions of A on Vkl in some sense. It is easy to verify that M is an SPD operator on V . Note that for l = 0, we adopt the notation N0 = 1; Ik0 = I 0 ; Q0k = Q0 ; Rk0 = R0 : As in the last section, the condition number of MA can be bounded in terms of three parameters K0 , ! and 0 de ned as follows:

(P10 )

For P any uP2 V , there exist ulk 2 Vkl (0 l L, 1 k Nl ) such that l u = Ll=0 Nk=1 Ikl ulk and Nl L X X

(P20 ) (P30 )

((Rkl )?1 ulk ; ulk )k K0 (Au; u):

l=0 k=1 l l For any uk 2 Vk , 0 l L, 1 k Nl , (AIkl ulk ; Ikl ulk ) !0 ((Rkl )?1 ulk ; ulk )l : For any u 2 V and ulk 2 Vkl , 0 l L, 1 k Nl , Nl Nl L X L X X 1 X 1 (AIkl ulk ; Ikl ulk ) 2 : (Au; Ikl ulk ) 02 (Au; u) 12 l=0 k=1 l=0 k=1

Analogous to Theorem 1, we have

Theorem 2

Under the assumptions (P10 ) - (P30 ), (MA) !0 (0 + 1)K0 : 2.3 Additive preconditioners for small perturbations of SPD operators The results of this section is applicable to the systems arising from the Galerkin discretization of general non-symmetric parabolic problems. Let V be a nite dimensional space with the scalar product (; ), and E a non-symmetric operator on V which is a small perturbation of the SPD operator A, that is, E = A + B , and we solve the equation Eu (A + B )u = f on V . Our goal is to nd a good preconditioner M for the non-symmetric operator E . Then we can use iterative methods, like GMRES or BiCGSTAB, to solve MEu = Mf instead of Eu = f . Let us consider the GMRES method. It is known (cf. [23]) that the convergence rate of GMRES depends on the following two parameters: (u; MEu)A ; = max kMEukA : (7) 1 = min 2 u6=0 kuk u6=0 (u; u)A A


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If 1 > 0, GRMES converges, and at the mth iteration the residual is bounded as (cf. [23]) 2 m=2

kMf ? MEumkA 1 ? 12

kMf ? MEu0kA : Suppose we are given nite dimensional spaces V k (1 k p) and the scalar products (; )k , where V k need not be subspaces of V . Let the linear operators Ik : V k ! V , their adjoints Qk and the SPD operators Rk : V k ! V k be de ned as in Section 2.1. We are 2

interested in the following additive type preconditioners

M=

p X

k=0

Ik Rk Qk ;

(8)

for operator E . Note that we still use an SPD preconditioner M even though E is nonsymmetric, this idea was earlier used by Xu-Cai [38] and Xu [37]. For later use, we write p X (9) ME = Ik Pk ; Pk = Rk Qk E: k=0

We introduce two assumptions for the perturbation operator B : (P4) For any u 2 V and uk 2 V k , 1 k p, p X

(P5)

k=1

1 2

(Bu; Ik uk ) 1 (Au; u)

1 2

p X k=1

1


There exists a constant 1 2 (0; 1) such that for any u; v 2 V , j(Bu; v)j 1 kukAkvkA :

Remark 2 (P4) is the analogue of (P3) and means B is \bounded" by A in some sense, while (P5) means that the perturbation B is small relatively to A and ensures the positive de niteness of E .

By the Cauchy-Schwarz inequality, (P4) and (P5) imply that for any u 2 V and

uk 2 V k (0 k p), p X

k=0

Theorem 3

(Bu; Ik uk ) (21 + 1 ) 21 (Au; u) 21

p X k=0

1


(10)

If in addition to (P1) - (P5), we assume further that 2 21 + 1 (12!? K1 ) ; (11) 0 0 then we have (u; ME )A (1 ? 1 )2 ; = max kMEukA 2! 12 (1 + + + 2 ) 12 : 1 = min 0 1 2 u6=0 kuk 0 0 1 u6=0 (u; u)A 4K0 A

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Proof

P We rst estimate 2 . For any u 2 V , let w = pk=1 Ik Pk u, the following holds from (P3):

kwk2A = (Aw;

p X

k=1

p X

Ik Pk u) 10=2 (Aw; w)1=2 ( (AIk Pk u; Ik Pk u))1=2 ; k=1

that gives

kwkA = k Using this, we can deduce that

kMEuk2A = k

p X k=0

p X k=1

p X

Ik Pk ukA 0 (

k=1

Ik Pk uk2A 2kI0 P0 uk2A + 2k

2kI0 P0 uk2A + 20 2!0 0

p X k=0

p X k=1

kIk Pk uk2A)1=2 :

p X k=1

(12)

Ik Pk uk2A ((9) & triangle ineq)

kIk Pk uk2A (by (12))

(Rk?1 Pk u; Pk u)k (by (P2));

(13)

but Rk?1Pk = Qk E by de nition of Pk , thus using E = A + B we obtain that p X

= =

(Rk?1 Pk u; Pk u)k

k=0 p X k=0 p X k=0

(Qk Eu; Pk u)k =

(Au; Ik Pk u) +

q

p X k=0

p X k=0

(Eu; Ik Pk u) (by (2))

(Bu; Ik Pk u)

2(1 + 0 + 1 + 21 )kukA

q

p X k=0

2(1 + 0 + 1 + 21 )!02 kukA 1

kIk Pk uk2A

p X k=0

1

2

(by (4) & (10))

(Rk?1 Pk u; Pk u)k

1 2

(by (P2));

which indicates that

p X

k=0

(Rk?1 Pk ; Pk u)k 2!0(1 + 0 + 1 + 21 )(Au; u):

This with (13) implies the bound for 2 . Next we bound 1 . By (P5), we see that (Au; u) (Eu; u) ? (Bu; u) (Eu; u) + 1 (Au; u):

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Then we deduce that (1 ? 1 )(Au; u) (Eu; u) = =

p X k=0

p X k=0

(Eu; Ik uk ) (by (P1))

(Qk Eu; uk )k =

1 2

K0 kukA

p X k=0

p X k=0

(Rk?1 Pk u; uk )k (by (2) & (9))

(Rk?1 Pk u; Pk u)k

from which we obtain (1 ? 1 )2 (Au; u) K0

p X k=0

1 2

(Schwarz ineq & (P1));

(Rk?1 Pk u; Pk u)k :

(14)

Finally we come to (MEu; u)A

= =

p X

(Ik Pk u; Au) =

k=0 p X k=0 p X

k=0

(Ik Pk u; Eu) ?

(Rk?1 Pk u; Pk u)k ?

p X k=0

p X k=0

(Ik Pk u; Bu) (by E = A + B )

(Bu; Ik Pk u) (Pk 's de nition)

(Rk?1 Pk u; Pk u)k ? !02 (21 + 1 ) 12 kukA

k=0 p X

12

p X

k=0

1

p X

(Rk?1 Pk u; u)k ? 21 !0 (21 + 1 )(Au; u)

k=0

(Rk?1 Pk u; Pk u)k

1 2

2 (1 4?K1 ) (Au; u) (by (11) & (14)): 0 where we have used (P2) & (10) for the rst inequality and the fact xy (x2 + y2 )=2, 8 x; y > 0, for the second inequality.

3 Technical lemmas In order to apply the general convergence framework of Section 2 to elliptic and parabolic problems, one needs to verify three assumptions given in Section 2. Here we develop some technical lemmas which will be used for such veri cations in the subsequent sections. We rst introduce the ne and coarse nite element spaces, domain decompositions and the assumptions required for the relations between the ne and coarse triangulations of the original domain . Then we demonstrate the local L2 -optimal approximation and H 1 stability of the standard ne nite element interpolant on the coarse spaces, and the local L2 -optimal approximation and H 1 -stability of the Clement's interpolant.


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3.1 Finite elements and domain decompositions Given an open domain Rd (d = 2; 3) on which certain elliptic and parabolic problems are de ned, we will solve these problems by nite element methods. Suppose we are given a family of triangulations fT h g on , consisting of simplices. We will not discuss the eects of approximating but always assume in this paper that the triangulations fT h g of are exact, i.e., we assume is polygonal or polyhedral, and

= h [ h 2T h h : Let h = diam h , h = max h 2T h h , h = min h 2T h h , = the radius of the largest ball inscribed in h . We say an element 2 T h is 0 -shape regular if

h = 0 ;

(15)

and T h is 0 -shape regular (or shape regular for short) if all its elements are 0 -shape regular. Moreover, we say T h is quasi-uniform if it is shape regular and satis es

h 1 h ; with 0 and 1 xed positive constants. As quasi-uniformity is too restrictive for unstructured meshes, the present paper will assume only that all the ne and coarser triangulations used are shape regular, not necessarily quasi-uniform. Let V h be a piecewise linear nite element subspace of H01 ( ) de ned on T h with the set of basis functions fhi gni=1 , and Oih =supp hi . Later on we will use the following simple fact: if T h is shape regular, there exist a constant 0 > 0 and an integer 1 > 0, both depending only on 0 in (15) and independent of h so that, for 1 i n, diam Oih 0 h ; 8 h Oih ; card f h 2 T h ; h Oih g 1 : (16) Decompose the domain into p non-overlapping subdomains ~ k (1 k p), then extend each ~ k to a larger one k such that the distance between @ k and @ ~ k is bounded from below by k > 0. We assume that @ k does not cut through any element h 2 T h . For the subdomains meeting the boundary @ we cut o the part of k which is outside

. We allow each k to be of quite dierent size and of quite dierent shape from other subdomains, but we make the following assumption:

(A1)

Any point x 2 belongs to at most q0 subdomains of f k gpk=1 with q0 > 0 an integer.

De ne the subspaces fV k gpk=1 of V h corresponding to the subdomains f k gpk=1 by

V k = fv 2 V h ; v = 0 on n k g:

(17)

To develop a two level method, we introduce a coarse grid T H which forms a 0 -shape regular triangulation of , but otherwise has nothing to do with T h , i.e., none of the nodes of T H need to be nodes of T h . Let 0 be the coarse grid domain, i.e. 0 = [ H 2T H H , and fqiH g the set of nodes of T H .


12

n 0

@

0

0 n

Figure 1: The ne domain and non-matching coarse domain 0 Denote by V 0 (resp. V^ 0 ) the subspace of H01 ( 0 ) (resp. H 1 ( 0 )) consisting of piecewise linear functions de ned on T H with f kH ; qiH 2 0 g (resp. f kH ; qiH 2 0 g) the set of nodal basis functions. We remark that V 0 need not be piecewise linear as is V h . Our linearity assumption on V 0 is just for the sake of simplicity. Note that 0 usually does not match with , and V 0 6 V h , cf. Fig. 1. We need to impose a few reasonable assumptions on the coarse grid 0 :

(A2) (A3) (A4)

For any H 2 T H , the measure of all h 2 T h : h \ @ H 6= ;, is bounded by the measure of H with a constant factor. For any node qih 2 @ \ ( 0 n ), one can construct two 0 -shape regular simplices îh and ~ih such that qih 2 îh , îh ~ih \ 0 , diam(îh ) diam(~ih ) diam(Oih ), and one face of ~ih lies in Rd n 0 .

n 0 [qkH 2@ 0 O^kH ; O^kH = BqkH (diam OkH ) \ , where OkH = supp kH .

Note that (A3) means the coarse grid part outside the ne grid is of ne element sizes, while (A4) says the ne grid part outside the coarse grid is of coarse element sizes, cf. Fig. 2. Here Bp (r) is a ball centered at point p with radius r. 3.2 H 1 -stability and L2 -optimal approximation of two linear interpolants Since V 0 6 V h for our interest, the convergence proofs for overlapping domain decomposition methods require the existence of an operator I0 : V 0 ! V h to satisfy the H 1 -stability and L2-optimal approximation properties. We introduce notations: for h 2 T h and H 2 T H ,

N ( h ) = [ h \ h6=; h ; hk = max h ; Bk = [ H \ k 6=; H ; h k N ( H ) = [ H \ H 6=; H ; Hk = max H ; Sk = [ H Bk N ( H ): H B 0

0

0

0

k


~ih

13

n 0

qih îh

@

0

0 n

qkH

Figure 2: The shadow part belonging to ( n 0 ) is covered by balls BqkH (rk ), rk =diam OkH as stated in (A4). qih is a ne node in @ \ ( 0 n ), and îh and ~ih are two simplices stated in (A3). Note that N ( h ) (resp. N ( H )) is the union of closest neighboring ne (resp. coarse) elements to h (resp. H ). Bk is the union of all coarse elements having nonempty intersection with the subdomain k . We assume that (A5) hk < Hk ; and cardf H 2 T H ; H Bk g q0 for 1 k p with q0 > 0 an integer. (A6) Any point x 2 0 belongs to at most q0 subdomains of fSk gpk=1 .

Remark 3 (A5) means that the minimum number of coarse elements whose union covers the subdomain k is less than a xed constant q0 .

Standard nite element interpolant De ne 0h : V 0 ! V h and h : V 0 ! H 1 ( ) to be the standard nite element interpolants, i.e.

0h u =

X

qih 2

u(qih )hi ; h u =

X

qih 2

u(qih )hi : 8 u 2 V 0 ;

(18)

Note the only dierence in the de nitions of 0h and h is that the former excludes the boundary nodes but the latter includes the boundary nodes. The following lemma gives the completely local properties of these two interpolants which are sharper than ones in Chan-Smith-Zou [12], where only global bounds are derived:

14


Lemma 4

With the assumptions (A2)-(A3) and (A5), for any u 2 V 0 and s; t = 0; 1, s t, t?s hk 1?t t?s hk 1?t ju ? h ujs; k < hk ( H ) jujt;Bk ; 1 k p; hk ( H ) jujt;kH < k

k

holds with kH 2 T H such that juj1;1;kH = max H Bk juj1;1; H , and 0

0

t?s ju ? 0h ujs; k < hk jujt;Bk ; 1 k p:

Proof As u 2 V 0 , so u 2 W 1;1 (Rd ). For any h 2 T h , one gets (cf. Theorem 3.1.5 in [17]) 2(1?s) d 2 ju ? h uj2s; h < (19) h h juj1;1; h : Using the analog of (16) for T H , we have for any H Bk that X h \ H 6=; h k

2(1?s) ju ? h uj2s; h < hk

X h \ H 6=; h k

hd juj21;1; h (by (19))

2(1?s) d 2 2(t?s) hk 2(1?t) juj2 (by (A2) & inverse ineq); < t;kH hk Hk juj1;1;kH < h k ( Hk )

now the inequality for h follows from (A5) and the relation:

ju ? h uj2s; k

X

X

ju ? h uj2s; h :

H Bk h H = h k \

6

;

For 0h : let Dh be the set of boundary nodes of h D = @ \ ( 0 n ), we can write

T h which also belong to 0 , i.e.,

u ? 0h u = (u ? h u) + ^ h u; ^ h u = We have j^ h uj2s; k <

< <

X

qih 2 k \Dh X

q 2 k \Dh h i

X

qih 2 k \Dh

ju(qih )hi j2s;Oih < h?k 2s kuk20;îh

X

qih 2 k \Dh X

q 2 k \Dh h i

X

q 2Dh h i

u(qih )hi :

(20)

jhi j2s;Oih kuk20;1;îh (by (A3))

h?k 2s kuk20;~ih (inverse ineq & (A3))

h2(k t?s) juj2t;~ih ((A3) & Poincare ineq for t = 1);

this with (20) and the triangle inequality implies the required result for 0h .


15

Clement's interpolant

We now introduce a locally de ned interpolant RH0 proposed by Clement in [18]. Operators with similar properties to RH0 can be found in Scott-Zhang [32], see also Xu-Zou [39].

De nition 5

The mapping RH0 : L2 ( 0 ) ! V 0 and RH : L2 ( 0 ) ! V^ 0 are de ned by

RH0 u =

X

qiH 2 0

Qi u(qiH ) iH ; RH u =

X

qiH 2 0

Qi u(qiH ) iH ; 8 u 2 L2 ( 0 );

where Qi u 2 P1 (OiH ), OiH = supp iH , satis es

(Qi u; p)0;OiH = (u; p)0;OiH ; 8 p 2 P1 (OiH )

Recall that V 0 and V^ 0 are de ned in Section 3.1. Clement [18] proved

Lemma 6 For any u 2 H 1 ( 0 ) and s; t = 0; 1; s t, ju ? RH ujs; H < Ht?s jujt;N ( H ) 8 H 2 T H : For the interpolant RH0 , we have

Lemma 7

Under the assumption (A4), for any u 2 V h and s; t = 0; 1, s t, t?s ju ? RH0 ujs; k < Hk jujt;Sk (1 k p); jRH0 ujs; H < jujs;N ( H ) ;

if Bk \ @ 0 = ; and H 2 T H : H \ @ 0 = ;; and X 2(t?s) 2(t?s) 2 2 ju ? RH0 uj2s; k < Hk jujt;Sk + H Hk jujt;OîH (1 k p);

jR0

uj2

H s; H

<

juj2

s;N ( H ) +

qi

X

qiH 2@ 0 \H

juj2s;OîH ;

if Bk \ @ 0 6= ;Pand H 2 T H : H \ @ 0 6= ;. Here OîH is de ned in (A4) and the above summand qiH is made for all qiH 2 @ 0 : OiH \ k 6= ;.

Proof The rst case is an immediate consequence of Lemma 6 and the fact RH0 u = RH u on H Bk . Next we show only the case that Bk \ @ 0 6= ;, the result for H 2 T H : H \ @ 0 = 6 ; can be proved analagously. Obviously, ju ? RH0 uj2s; k = ju ? RH0 uj2s; k \ 0 + ju ? RH0 uj2s; k \( n 0 ) : (21)

16


One obtains by (A4) and the Poincare inequality that

ju ? RH0 uj2s; k \( n 0 ) = juj2s; k \( n 0 ) X

<

qjH 2@ 0

X

juj2s;O^jH \ k

qjH 2@ 0 (diam OjH )2(t?s) juj2t;O^jH \ k

2(t?s) 2 < Hk jujt; k :

P Using u ? RH0 u = u ? RH u + qiH 2@ 0 Qi u(qiH ) iH , we have

ju ? RH0 uj2s; k \ 0 <

X

Bk H

ju ? RH uj2s; H +

X

q

H i

jQi u(qiH ) iH j2s; k \ 0 ;

where the second summand is made for all qiH 2 @ 0 : OiH \ k 6= ;. To estimate this term, for any such coarse node qiH , one derives

jQi u(qiH ) iH j2s; k \ 0 < < <

X

H O X

H i

j iH j2s;OiH kQi uk20;1; H H?2s kQi uk20; H H?2s kQiuk20;OiH (inverse ineq)

H OiH H?2s kuk20;OiH H?2s kuk20;OîH ((A4) & Qi 's de nition) Hk2(t?s) juj2t;OîH (Poincare ineq);

this with Lemma 6 gives

2(t?s) 2 ju ? RH0 uj2s; k \ 0 < Hk jujt;Sk +

X

qiH

Hk2(t?s) juj2t;OîH ;

Lemma 7 follows from (21) and the above.

4 Two level additive Schwarz method for elliptic problems In this and the next two sections, we apply the general theory of Section 2 to second order elliptic and parabolic problems. For simplicity, we restrict ourselves only to pure Dirichlet or pure Neumann boundary conditions. First, in this section, we consider the following self-adjoint elliptic problem:

?

d X

@ (a @u ) + b u = f; in

ij @x @x j i;j =1 i

(1)

with Dirichlet boundary condition: u = 0 on @ . Here Rd (d = 2; 3) as described in Section 3.1, (aij (x)) is symmetric, uniformly positive de nite, and b(x) 0 in . The purpose of this section is to present more simpli ed proofs than our old ones in [15, 12] by means of the present framework and also give an improved local bound on the condition number. The previous bounds of the condition numbers are global.


17

The weak formulation of the above problem is: Find u 2 H01 ( ) such that

A (u; v) = (f; v); 8 v 2 H01 ( )

with

A (u; v) =

Z X d

@u @v + b uvdx: aij @x j @xi i;j =1

All notations of this section are inherited from Section 3. The nite element problem is: Find u 2 V h such that

A (u; v) = (f; v); 8 v 2 V h : (2) Based on the local nite element spaces V k (1 k p) and the coarse space V 0 = V H

de ned in Section 3.1, we shall use the abstract theory of Section 2 to construct the two level overlapping Schwarz methods on unstructured meshes. We de ne scalar products (; )k = (; )0; k on V k for 1 k p and (; ) = (; )0;

on V h , and then de ne an SPD operator A on V h and a coarse operator A0 by (Au; v) = A (u; v); 8 u; v 2 V h ; (A0 u; v)0 = A 0 (u; v); 8 u; v 2 V 0 and local operators Ak , 1 k p by (Ak u; v)k = A k (u; v); 8 u; v 2 V k : Since V k V h , 1 k p, we de ne Ik : V k ! V h to be the natural injection operator. Note V 0 6 V h , de ne I0 : V 0 ! V h to be the natural interpolant 0h in (18). One may also use other choices of I0 , e.g., the Clement interpolant Rh0 . Choose the local solvers Rk , 0 k p to be exact solvers, i.e. Rk?1 = Ak . Then the preconditioner M in (1) for A is: p X M = Ik A?k 1 Qk : k=0

In order to apply Theorem 1 for the estimate of (MA), we need a partition lemma for the nite element space V h :

V h = I0 V 0 + V 1 + + V p ; whose proof will be given at the end of the section.

Lemma 8

With the assumptions (A1) - (A6)P, for any u 2 V h , therePexist uk 2 V k (1 k p) and u0 = RH0 u 2 V 0 such that u = pk=0 Ik uk = I0 RH0 u + pk=1 uk and p X

2

Hk )juj2 ; juk j21; < (1max k p k2 1;

k=1 ju0 js; 0 < jujs; ; s = 0; 1:

p X

k=1

2

hk )kuk2 ; kuk k20; < 0;

(1max kp 2 k

The following theorem gives the bound of the condition number (MA).

(3) (4)

18


Theorem 9

Under the assumptions (A1) - (A6), we have

2

Hk : (MA) < 1max kp 2 k

Note that from Theorem 9 one can expect an optimal condition number if the local overlap k is proportional to the size Hk for each k (1 k p).

Proof

It suces by Theorem 1 to verify (P1) - (P3). ForP (P1): by Lemma 8, 8 u 2 V h, there exist uk 2 V k and u0 = RH0 u such that u = I0 u0 + pk=1 uk . We derive p X k=0

(Rk?1 uk ; uk )k = A 0 (u0 ; u0 ) +

p X k=1

2 A k (uk ; uk ) < ku0 k1; 0 +

p X

k=1

kuk k21;

Hk2 )kuk2 < ( max Hk2 )(Au; u); < ( max 1; 1kp 2 1kp k2 k 2 2 that indicates K0 < max1kp Hk =kk. For (P2): we get for any uk 2 V (0 k p), 2 2 (AIk uk ; Ik uk ) < kIk uk k1; = kuk k1; < (Auk ; uk ) = (Ak uk ; uk )k (k 6= 0); (AI0 u0 ; I0 u0) < kI0 u0k21; <

<

p X k=1

p X k=1

2 ku0k21;Bk < ku0k1; 0 < (A0 u0 ; u0)0 (Lemma 4 & (A6));

that implies !0 < 1. h For (P3): 8 u 2 V and uk 2 V k , p X

k=1

(Au; Ik uk ) =

kI0 u0 k21; k (by = [pk=1 k )

p X k=1

p X

A k (u; uk ) < ( p X

k=1

A k (u; u)) 21 (

p X k=1

A k (uk ; uk )) 12

1 1 < (Au; u) 2 ( (Auk ; uk )) 2 (by (A1));

k=1

thus 0 < 1. That completes the proof of Theorem 9 by using Theorem 1. Proof of Lemma 8: (4) follows directly from Lemma 7. We now prove (3). For any

u 2 V h , choose u0 = RH0 u and let v0 = I0 u0 . Then Lemmas 4 and 7 implies for s = 0; 1

that

ju ? v0 j2s; k ju ? u0 j2s; k + ju0 ? I0 u0j2s; k 2(1?s) (juj2 + X juj2 ) + h2(1?s) ju j2 : < H 0 1;Bk 1;Sk k k 1;O^ iH H qi

The rest of the proof is quite routine, we refer to Chan-Zou [15].


19

5 Two level additive Schwarz method for parabolic problems Consider the following non-selfadjoint parabolic problems:

@u + Lu = f in (0; T ); @t

where Rd (d = 2; 3) as described in Section 3.1, and

Lu = ?

d d @u + d u) + X @u + cu @ (X a b ij i i i=1 @xi i=1 @xi j =1 @xj

d X

with (aij (x; t)) symmetric, uniformly positive de nite and continuous on [0; T ], the functions bi (x; t); di (x; t) and c(x; t) continuous on [0; T ]. The initial and boundary conditions are u(x; 0) = u0(x) on ; u = 0; on @ (0; T ): After discretizing the variational problem corresponding to the above parabolic problem by using some implicit nite dierence schemes in time with a time step size and the nite element space V h in space, the resulting discrete system may be formulated: Find u 2 V h such that E (u; v) A (u; v) + B (u; v) = (f; v); 8 v 2 V h : (1) where Z Z d X @u @v dx + Z c uv dx aij @x A (u; v) = uv dx + @x and

i

i;j =1

B (u; v) =

j

Z X d

i=1

@u v + d u @v ) dx: (bi @x i @x i

i

It is known that (1) has a unique solution for sucient small (relative to the coecients) time step , which will be assumed implicitly throughout this section. Based on the local nite element spaces V k (1 k p) and the coarse space V 0 de ned in Section 3.1, we now construct the two level additive type preconditioners for the non-symmetric operator E by using the perturbation theory of Section 2.3. De ne two non-symmetric operators E and B and a symmetric operator A on V h by (Eu; v) = E (u; v); (Bu; v) = B (u; v); (Au; v) = A (u; v); 8 u; v 2 V h where and in the sequel, (; ) = (; )0; and (; )k = (; )0; k for 1 k p, and de ne local operators Ak for 1 k p and the coarse operator A0 by (Ak u; v)k = A k (u; v); 8 u; v 2 V k : Let the operators Ik : V k ! V h be de ned as in Section 4, and Rk = A?k 1 : V k ! V k , then (8) gives preconditioner M for the non-symmetric operator E :

M=

p X

k=0

Ik A?k 1 Qk :


20

Remark 4

Note that in the de nition of the preconditioner M above we use SPD solvers both for the coarse space V 0 and for local subspaces V k (1 k p), although the original operator E to be preconditioned is non-symmetric.

As stated in Section 2.3, instead of solving Eu Au + Bu = f from (1), one may solve the following preconditioned system MEu = Mf by GMRES method, the convergence rate is (1 ? 12 = 22 )1=2 , where 1 and 2 can be bounded as follows:

Theorem 10

In addition to (A1) - (A6), we assume further that

k2 : < min k H2 k

Then we have

(u; MEu)A > min k2 ; = max kMEukA < 1: 1 = min 2 u6=0 kuk u6=0 (u; u)A 1kp Hk2 A

(2)

Proof

Denote k k2A = A (; ) = (A; ), k k2A; k = A k (; ) (1 k p). By Theorem 3, it suces to verify (P1) - (P5) and (11). (P1) - (P3) can be shown in the same as in Theorem 9 with minor natural modi cation. Thus we know 1 K0 < max1kp Hk2 =k2 , !0 1 and 0 1. To prove (P5), we rst see by the de nition of A, B and Cauchy-Schwarz inequality that (Au; u) < kuk20; + juj21; < (Au; u); 8 u 2 H 1 ( ); hence for any u; v 2 H 1 ( ), using Cauchy-Schwarz inequality again indicates

j(Bu; v)j = p

Z X d

i=1

p

p

p

p

@u )(b v) + (d u)( @v ) dx < kuk kvk ; ( @x A A i i @xi i

that shows (P5) with 1 < . The same reasoning gives

p h k j(Bu; uk )j < kukA; k kuk kA; k ; 8 u 2 V ; uk 2 V ; 1 k p;

so we can derive p X k=1

(Bu; Ik uk ) =

<

p X

p B (u; uk ) <

p X

kukA; k kuk kA; k (by (3)) k=1 p p 1=2 X 1=2 p X A k (u; u) A k (uk ; uk ) (Schwarz ineq) k=1 k=1 k=1

(3)


21

p 1=2 p X < ( u ; u ) A (by (A1)); k

k k k=1

therefore (P4) holds with 1 < . From above, it is easily seen that (11) holds if < min1kp k2 =Hk2 . Now by Theorem 3, 2 k2 ; 2! 1=2 (2 + + )1=2 < 1; 1 (1 4?K1 ) > min 0 1 2 0 0 1kp Hk2 0 that ends the proof of Theorem 10.

6 Multilevel additive Schwarz method for elliptic problems For the simplicity of exposition, we consider only two dimensional polygonal domain

and the elliptic problem

?

2 X

@ (a @u ) + b u = f; in

ij @x @x j i;j =1 i

with Neumann boundary condition 2 X

i;j =1

@u n = 0; aij @x i j

on @ ;

where b > 0 and continuous on while (aij ) is symmetric, uniformly positive de nite and continuous on . It is straightforward to generalize the results of this section to three dimensions. We remark that in Sections 4-5 where the Dirichlet boundary conditins are considered, we allow that the coarse domain may partly cover the ne domain or partly be contained in the ne domain. But for the Neumann boundary conditions in this section, we require the coarser grid domains completely cover the ne grid domain. We refer to Chan-Go-Zou [9] for some numerical experiments in general case. The nite element problem is formulated as: Find u 2 V h such that A (u; v) = (f; v); 8 v 2 V h ; (1) where V h H 1 ( ) consists of piecewise linear functions de ned on T h . In this section, we will use the theory of Section 2.2 to construct multilevel additive type preconditioners for the operator A corresponding to A (; ). Let fT l gLl=0 be a not necessarily nested sequence of shape regular triangulations on

with hl the maximum diameter of all elements in T l . T L = T h is the nest triangulation on which the nite element space V h and the nite element problem (1) are de ned. Denote the coarser domains corresponding to the coarser triangulations T l , 0 l L ? 1 by l . As we indicated above, the coarse grid domain is required to cover the original ne grid domain (cf. Fig. 3), i.e. we assume

L = l = [ l 2T l l ; 0 l < L:


22

l

Figure 3: The ne domain and l-th coarse domain l For 0 l < L, let V l H 1 ( l ) be the piecewise linear nite element space de ned l ml l on T l , and fqil gm set of nodal points of the triangulation T l and the i=1 and fi gi=1 the l set of nodal basis functions of V , resp. We will de ne some linear coarse-to- ne grid transfer mappings such that spaces V l are subspaces of V h under these mappings. For that purpose, we always assume that the dimensions of all coarser spaces V l (l < L) are less than the dimension of the ne space V L. To de ne the multilevel additive preconditioner, we use the idea of the two-level algorithm on each level, i.e. decompose each domain l for 1 l L to overlapping subdomains of l . Thus we assume that for each level l = 1; 2; ; L, without the coarsest l level, f lk gNk=1 is an overlapping domain decomposition of l , obtained by extending a l given nonoverlapping subdomain covering f ~ lk gNk=1 of l such that dist(@ ~ lk ; @ lk \ l ) l > 0, 1 k Nl . l is called the l-th level overlapping ratio. Here the boundaries of the subdomains lk are required to align with the boundaries of the l-th level elements in T l . We make the following very reasonable hypotheses about the triangulations: (H1) For any coarse element l 2 T l, 0 l < L, let m( l ) be the measure of all h 2 T h : h \ @ l 6= ;, then m( l ) < j l j.

(H2) (H3)

l Any point in l is covered by at most q0 subdomains of f lk gNk=1 . l l l l Any coarse boundary element in T (l < L), i.e. \ @ 6= ;, has a signi cant part inside the original domain . More accurately, one can construct a 0 -shape regular simplex 0l such that 0l l \ and diam 0l diam l .

For each coarse space V l (1 l p), we de ne a mapping Il : V l ! V h to be the standard nite element interpolant h , i.e. X u(qih )hi ; 8 u 2 V l ; ; (2) Il u h u = qih 2

and for each subdomain lk on l-th level, we de ne a local subspace by Vkl = fv 2 V l ; v = 0 on @ lk \ l g V l


23

and a prolongation operator Ikl : Vkl ! V h to be Il , but Ikl 's adjoint Qlk : V h ! Vkl by (Qlk u; vkl )l = (u; Ikl vkl ); 8 u 2 V h ; vkh 2 Vkl where (; )l = (; )0; l is the scalar product in L2( l ). Furthermore, we de ne local operators Alk : Vkl ! Vkl by (Alk u; v)l = A lk (u; v); 8 u; v 2 Vkl ;

and let Rkl = (Alk )?1 for simplicity of exposition. Then we may construct the additive Schwarz preconditioner as in (5) by

M=

Nl L X X l=0 k=1

Ikl Rkl Qlk

Nl L X X l=0 k=1

Ikl (Alk )?1 Qlk :

To estimate the condition number (MA), we need only to verify the conditions

(P10 )-(P30) by Theorem 2. For this purpose, we will use the following three lemmas.

The rst one indicates that the standard nodal value interpolation h is H 1 stable and has the L2 optimal approximation when it is restricted in the coarser nite element subspaces, whose proof is the same as the one for Lemma 4.

Lemma 11

Under the assumption (H1), for any coarser triangulation T l , 0 l < L, and any u 2 V l , we have

jh ujs; < jujs; l ; s = 0; 1; ku ? h uk0; < h juj1; l : The second lemma gives the properties of the Clement's interpolant, cf. De nition 5, Lemma 6 and Clement [18].

Lemma 12

Let Rl : L2( ) ! V l be the Clement's interpolant operator corresponding to T l , 0 l < L, then for s = 0; 1; t = 1; 2; s t and any u 2 H t ( l ), we have t?s ju ? Rl ujs; l < hl jujt; l :

(3)

Moreover, let V^ l H 2 ( l ) be a higher order nite element space de ned on T l (e.g. Argyris and Clough-Tocher nite elements) and R^l : L2 ( l ) ! V^ l the corresponding Clement's interpolant, then R^l satis es (3) and jR^l uj2; l < juj2; l :

De ne an orthogonal projection P l : H 1 ( ) ! Il V l with Il = h by

A (P l u; v) = A (u; v); 8 u 2 H 1 ( ); v 2 Il V l :

Note that Il = h is de ned in (2).


24

Lemma 13

Suppose that the domain is convex, then for 0 l < L, h kv ? P l vk0; < hl kvk1; ; 8 v 2 V :

Proof

We apply the Aubin-Nitsche trick to prove Lemma 13, and rst show that for any u 2 H 2 ( ), there exists ul 2 V l such that

ku ? Il ul k1; < hl kuk2; :

(4)

Let ^ be an open domain in Rd large enough such that 0 ^ . It is well-known (cf. Stein [35]) that there exists a linear continuous extension operator E : H 2 ( ) ! H 2 ( ^ ) such that Eu = u in and

kEuk2; ^ < kuk2; :

(5)

We show the function ul = Rl Eu 2 V l satis es (4). The triangle inequality gives ku ? Il ul k1; kEu ? Rl Euk1; + kRl Eu ? R^l Euk1; + kR^l Eu ? Il Rl Euk1; : By Lemmas 11-12, we obtain kEu ? Rl Euk1; < hl jEuj2; l , and

kRl Eu ? R^l Euk1; < ^ kRl Eu ? Il Rl Euk1; < <

kEu ? Rl Euk1; + kEu ? R^l Euk1; (by Lemma 12) hl jEuj2; l < hl juj2; (by (5)); kR^ l Eu ? Il R^l Euk1; + kIl (R^l Eu ? Rl Eu)k1;

hl jR^l Euj2; l + kR^l Eu ? Rl Euka; l (by the interpolation result & Lemma 11)

< hl kEuk2; l < hl kuk2; (by (5));

which proves (4). Now the Aubin-Nitsche trick will give the nal result. For any v 2 V h , let w 2 H 1 ( ) be the solution of the problem:

A (w; u) = (v ? P l v; u); 8 u 2 H 1 ( ): and wh the nite element solution of w in Il V l : A (wh ; u) = (v ? P l v; u); 8 u 2 Il V l : Since is convex, we know w 2 H 2 ( ), and kwk2; < kv ? P l vk0; (cf. Grisvard [25]). Hence the previous result (5) says there exists wl 2 V l such that kw ? Il wl ka; < hl kwk2; ;


25

from this and the de nitions of w and P l , we obtain kv ? P l vk20; = A (w; v ? P l v) = A (w ? wh ; v ? P l v)

< kw ? wh kA; kv ? P l vkA; kw ? Il wl kA; kvkA;

l < hl kwk2; kvkA; < hl kv ? P vk0; kvkA; ; that implies Lemma 13. Here k kA; is the A (; )-induced norm. Lemma 4 says the interpolant h is L2 - and H 1 -stable in the coarse nite element spaces. In the following lemma, we show that the inverse of h is also L2 - and H 1 stable. Due to technicalities in our proof of the lemma, we classify the triangulations of all coarser levels into two groups: triangulations \ far away " from and \ very close " to the ne triangulation. That is, for some positive interger l0 < L, the triangulations T l with 0 l l0 (resp. l0 < l < L) are regarded as far away from (resp. very close to) the ne triangulation. For these two groups of coarser meshes, we assume that

(H4)

(H5)

The set of nodes of each coarse triangulation which is very close to the ne triangulation is a subset of the nodes of the ne triangulation. And for any coarse element from these triangulations, say l in T l (l0 < l < L), all ne elements which have non-empty intersections with l are roughly of the same sizes as l . More accurately, for any ne element h of T h satisfying h \ l 6= ;, there are two positive constants 0 and 1 such that 0 diam( h ) diam( l ) 1 diam( h ): For each coarse triangulation T l which is far away from the ne one, i.e. 0 l l0 , and any ne element h in T h , if h has a non-empty intersection with some coarse element in T l , say l 2 T l , then there exists another ne element 0h (possibly h itself) which is completely located inside l and only an O(h) (or element-size) distance away from h , i.e. dist( h ; 0h ) 2 diam( h ) with 2 a positive constant.

Remark 5

Asssumptions (H4) and (H5) are both practically reasonable. (H4) means that element sizes of the ne triangulation and those coarser triangulations which are close enough to the ne one are of the same magnitude locally (not globally). (H5) means that each coarse element of those triangulations which are far away from the ne one contains one or more ne elements. We only require that the sets of nodes of the coarse triangulations which are close enough to the ne triangulation are subsets of ne nodes; the corresponding coarse domains are allowed to be non-nested to the ne domain. Many existing coarse grid generating algorithms possess this property, see [1, 2] and [9, 11, 13]. In practice, l0 = L ? 3 or L ? 4 would be enough, i.e. the third or fourth coarsening of the ne triangulation. The two assumptions (H4) and (H5) are posed because of our technical reasons. Other less restrictive assumptions are possible.

Now we can prove

26


Lemma 14

With the asumptions (H1), (H4) and (H5), we have for 0 l < L and s = 0; 1 that l l l jul js; l < jh u js; ; 8 u 2 V :

(6)

Proof

We prove the lemma in two steps: l l0 and l0 < l. (a) For the case l l0 : let h be any ne element in T h , we have 2 l 2 kulk20; h < h ku k0;1; h h2 jul (x0 )j2 ;

(7)

where x0 2 h . As x0 2 l , x0 must be in some element l of T l . By (H5), there exists another ne element 0h inside l such that dist( h ; 0h ) 2 diam( h ). Let zi , i = 1; 2; 3 be the vertices of 0h , by Taylor expansion we have

ul (x0 ) = ul (z1 ) + rul (x) (x0 ? z1 ); x 2 0h : Squaring both sides and integrating over h gives h2 jul (x0 )j2 h2 jul (z1 )j2 + h2 krul k20;0h : Using the shape regularity we know kh ul k20;0h h2 inequality and h ul = ul on 0h implies

P3

l 2 i=1 (u (zi )) , this with the inverse

l 2 l 2 l 2 h2 jul (x0 )j2 < kh u k0;0h + ku k0;0h kh u k0;0h :

Thus we obtain by using this last relation and summing over all h in (7) that

kulk20; < kh ul k20; :

(8)

Now consider any boundary element l in T l , i.e. l \ @ l 6= ;. By noting the linearity of ul in l and (H3), we obtain

kul k2s; l < kul k2s;0l ; for s = 0; 1:

(9)

This with (8) indicates (6) holds with s = 0. To show (6) holds with s = 1, taking any ne element h of T h , we know from the standard interpolation result (cf. [17]) that

jul ? h ul j21; h < h2 jul j21;1; h h2 jrul (x0 )j2 ; x0 2 h :

(10)

As above, we can nd an element l of T l and another ne element 0h inside l such that dist( h ; 0h ) 2 diam( h ). Since ul is linear in l and h ul = ul on l , we obtain by (10) and inverse inequality that ?2 l 2 2 l2 ?2 l 2 jul ? h ul j21; h < h ju j1;1;0h < h ku k0;0h = h khu k0;0h :

(11)

Noting (ul ? h ul ) does not change by replacing ul by ul + c for any constant c, we derive

jul ? h ul j21; h < jh ul j21;0h ;

27


this combining with triangle inequality and (9) implies (6) for s = 1. (b) For the case l0 < l L. Taking any coarse element l , we have by (H4) and the shape regularities of T h and T l that X X X kh ul k20; h > (diam( h ))2 (ul (xi ))2 h \ l 6=;

h \ l 6=;

l 2 > (diam( ))

X

xi 2N ( h ) l 2 (ul (xi ))2 > ku k0; l :

xi 2N ( l )

Here N ( h ) is the set of nodes of h , the same for N ( l ). This with (9) shows (6) for s = 0. For s = 1: the shape regularities of T l and T h , and the assusmption (H4) imply

jul j21; l <

< <

X

(ul (xi ) ? ul (xj ))2

xi ;xj 2N ( l ) X

X

(ul (xi ) ? ul (xj ))2

h \ l 6=; xi ;xj 2N ( h ) X jh ul j21; h ; h \ l 6=;

combining with (9) implies (6) for s = 1. This completes the proof of Lemma 14. We are now in a position to estimate the condition number (MA). For convex domain , we have

Theorem 15

With the assumptions (H1) - (H5), where = max1lL (hl + hl?1 )=l .

2 2 (MA) < L ;

Proof

It suces by Theorem 2.2 to verify (P10 ) - (P30 ). (P20 ) is obvious with !0 < 1 from the l l de nition of Ik , Rk and Lemma 11. To prove (P30 ), for any ulk 2 Vkl de ned on lk , let ^ lk = supp Ikl ulk lk \ . Recall l Ik = h , then (H2) and the Cauchy-Schwarz inequality leads to Nl L X X l=0 k=1

(Au; Ikl ulk ) =

Nl L X X l=0 k=1

A ^ lk (u; Ikl ulk )

Nl L X X l=0 k=1

A ^ lk (u; u)

1=2 1=2 < L (A (u; u) )

Nl L X 1=2 X l=0 k=1

Nl L X X l=0 k=1

A ^ lk (Ikl ulk ; Ikl ulk ) 1

A (Ikl ulk ; Ikl ulk ) 2 ;

1=2


28

which says (P30 ) holds with 0 = L. Finally we verify (P10 ). To do so, we need a proper decomposition for any nite element function u in V h . As in our earlier report [14], we use the following decomposition: u0 = P 0 u; u1 = P 1 (u ? u0); u2 = P 2 (u ? u0 ? u1 );

ul = P l (u ? u0 ? u1 ? ? ul?1 );

uL = P h (u ? u0 ? u1 ? ? uL?1) = u ? u0 ? u1 ? ? uL?1: P It is readily seen that ul 2 Il V l and u = Ll=0 ul : P Let wl = u ? il?=01 ui for 1 l L but w0 = u. Then ul = P l wl and wl = wl?1 ? ul?1 = wl?1 ? P l?1 wl?1 : By the de nition of P l , we derive for 0 l L that kul kA; kwl kA; kwl?1 kA; kw1 kA; kukA; ;

(12) (13)

and we get

kul k0; = kP l wl k0; kwl k0; + kwl ? P l wl k0; (triangle ineq) = kwl?1 ? P l?1 wl?1 k0; + kwl ? P l wl k0; (by (12)) < (14) (hl?1 + hl )kukA; (Lemma 13); 1 l L; ku0k0; < (15) ku0 kA; kukA; : As ul 2 Il V l , we can write ul = Il vl , vl 2 V l . We further decompose vl . It is l known (cf. Dryja-Widlund [21], Bramble et al. [5]) that there exists a partition fkl gNk=1 P N l of unity for l related to the subdomains f lk g such that k=1 kl (x) = 1 on l and for 1 k Nl , ?1 supp kl lk [ @ ; 0 kl 1; krkl kL ( i ) < (16) l : 1

Using this partition of unity, we can decompose vl as Nl Nl X X l l l v = l (k v ) vkl ; k=1 k=1

vkl 2 Vkl ;

where l is the standard nodal value interpolant corresponding to V l . This gives

u = u0 +

L X l=1

ul = u0 +

Nl L X X l=1 k=1

Il vkl :

By the standard proof (cf. Xu [36], or Chan-Zou [15]), we have Nl X k=1

?2 kvkl k2a; l < (jvl j21; l + l kvl k20; l ):

(17)


29

Thus, we deduce from (16)-(17) that Nl L X X l=0 k=1

(Alk vkl ; vkl ) = A 0 (v0 ; v0 ) +

< <

L X

Nl L X X l=1 k=1

A lk (vkl ; vkl )

(jvl j21; l + l?2 kvl k20; l ) (set 0 = 1 & by (17))

l=0 L X l=0

(jul j21; + l?2 kul k20; ) (by Lemma 14)

2 < L (Au; u) (by (13)-(15) );

which shows that (P10 ) holds with K0 = 2 L. Now Theorem 15 follows from Theorem 1.

Remark 6

The condition number bound given in Theorem 15 grows like L2 . It is known that in the structured case, one can remove this dependence on L and obtain optimal condition number (cf. Zhang [42] and Oswald [31]). At this point, we do not know how to obtain a similar optimal bound for our unstructured case.

Remark 7

The convexity assumption on the domain is only needed in the proof of Lemma 13 for the technical requirement by use of Aubin-Nitsche trick. We don't know if Lemma 13 holds for non-convex domain. But our numerical experiments demonstrated very satisfactory results also for non-convex domains, e.g. airfoil-shaped domains in [11, 12, 9, 10].

Acknowledgements The authors would like to express their sincere thanks to one anonymous referee for many detailed and constructive comments, and also to Jinchao Xu for his helpful suggestions, especially for the second part of the proof of Lemma 14. Most of this work was done during the visit of the rst author at Department of Mathematics, The Chinese University of Hong Kong.

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