M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element methods. Comm. Pure Appl. Math., 48:121--155, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....operators R j and R j copy (scaled) nodal data from the interface x c to the nodes on the boundary of the subdomain and vice versa. Hence the ordinary additive Schwarz implementation can be used. A coarse grid can be added to the Neumann Neumann preconditioner without changes of the algorithm [DW95] just like in the overlapping Schwarz case adding one subspace j = 0. Coupling a coarse grid equation in a multiplicative symmetric way instead, called balancing [Man93] requires some extensions resulting in a mixture of the additive and the multiplicative Schwarz algorithm. One can use the ....

Dryja M. and Widlund O. B. (1995) Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48: 121--155.

....we have (weak) regular splittings [33] 47] For examples of splittings for which the inequality (39) holds see [35] Another situation worth mentioning where (39) holds is when A i is semidefinite and the inexact solver is definite. This process is usually called regularization; see, e.g. [14], 30] In [20] it is shown that the damped additive Schwarz iterations with inexact local solves converge in the M matrix case under the condition (37) and 1=q. Furthermore, it is shown that the induced splittings corresponding to (35) and (36) A = M Gamma N = M Gamma N are weak ....

M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Communications in Pure and Applied Mathematics, 48:121--155, 1995.

....[17,34,35] The parameter H represents the diameter of a subdomain in a domain decomposition method. Multigrid and domain decomposition methods are examples of preconditioners that meet these requirements. They have also been successfully implemented on parallel machines; see e.g. for details [16,17,18,19,24,23,30,36,39,44,45], and the references therein. We also require that C is a good preconditioner for the pressure mass matrix M p , i.e. 9m 0 ; m 1 0 m 2 0 p t Cp p t M p p m 2 1 p t Cp 8p 2 M (22) and we finally assume that C is spectrally equivalent to C, i.e. 9c 0 ; c 1 0 c 2 0 p t Cp ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute, March 1993. To appear in Comm. Pure Appl. Math.

....The lack of generous overlap results in bounds for the condition number that are not optimal, but which can be made independent of the jumps of the coecients. In a Neumann Neumann method, the degrees of freedom of the local spaces are related to the entire boundaries of the substructures; see [5, 9, 20, 21, 8, 12, 28, 10, 22, 31]. For an introduction to iterative substructuring methods, we refer to [31] and to the references therein. The study and analysis of preconditioners for N ed elec and Raviart Thomas approximations is quite new. Extensive work has started only in the past three years, in order to extend classical ....

....f 2 L 2( n . The generalization to the case of the X h( Neumann boundary conditions) does not present any particular diculty. In particular, we remark that if Neumann conditions are considered on some part of the boundary N N has to be added to ; see de nition (6) and, e.g. [12]. We now introduce a Schur complement formulation of problem (15) We refer to [31, Ch. 4] for a general discussion of Schur complement methods, and to Section 7 for some implementation issues. Let T i be a substructure. Let A and A (i) be the matrices of the bilinear forms a( and a T i ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic nite element problems. Comm. Pure Appl. Math., 48(2):121{ 155, February 1995.

.... A( Delta; Delta) Combining these two formulas with Assumptions (1) 3) we get min (T as ) C Gamma2 0 ; max (T as ) ae(E) 1) 2 The next convergence result on the unaccelerated multiplicative Schwarz method is due to Bramble, Pasciak, Wang, and Xu [22] see also Dryja and Widlund [39]. Theorem 2.7 kT ms kA 1 Gamma (2 Gamma ) 1 2 2 ae(E) 2 C 2 0 1=2 ; where : maxf1; g. The symmetric multiplicative Schwarz operator T sms1 can also be analyzed; cf. Dryja and Widlund [39] 27 Theorem 2.8 (T sms1 ) 1 2 2 ae(E) 2 )C 2 0 2 Gamma ; ....

....is due to Bramble, Pasciak, Wang, and Xu [22] see also Dryja and Widlund [39] Theorem 2.7 kT ms kA 1 Gamma (2 Gamma ) 1 2 2 ae(E) 2 C 2 0 1=2 ; where : maxf1; g. The symmetric multiplicative Schwarz operator T sms1 can also be analyzed; cf. Dryja and Widlund [39]. 27 Theorem 2.8 (T sms1 ) 1 2 2 ae(E) 2 )C 2 0 2 Gamma ; where : maxf1; g. For a more detailed discussion on Schwarz methods, we refer to Smith, Bjrstad, and Gropp [81] and Oswald [65] In our applications, we will assume that the parameters C 0 ; ae(E) and are ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2), February 1995.

....we have (weak) regular splittings [29] 41] For examples of splittings for which the inequality (33) holds see [31] Another situation worth mentioning where (33) holds is when A i is semidefinite and the inexact solver is definite. This process is usually called regularization; see, e.g. [13], 27] In [19] it is shown that the damped additive Schwarz iterations with inexact local solves converge in the M matrix case under the condition (31) and 1=q. Furthermore, it is shown that the induced splittings corresponding to (29) and (30) A = M Gamma N = M Gamma N are ....

M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Communications in Pure and Applied Mathematics, 48:121--155, 1995.

.... preconditioner with a coarse space was introduced in [27] under the name Balancing Domain Decomposition and further studied in [12, 28, 29] Our abstract framework and analysis are related to the representation of another Neumann Neumann method with a coarse space as an abstract Schwarz method in [16]. For earlier work on the Neumann Neumann and similar preconditioners without a coarse space, see [1, 13, 14, 21, 32] A domain decomposition method in a sense dual to Neumann Neumann is obtained by enforcing intersubdomain continuity by Lagrange multipliers [19, 35, 17] That method, known as ....

....complement operator S = N X i=1 R T i S i R i ; RR n2635 16 Patrick Le Tallec , Jan Mandel. Marina Vidrascu and the interface right hand side F = N X i=1 R T i Tr GammaT i L i : With this notation, and by elimination of U i , the original problem (22) reduces to the form [16, 23] Su = F in V: 25) 3.2. The abstract Neumann Neumann preconditioner. For the abstract problem given by (25) it seems natural to precondition the sum S = P R T i S i R i by a weighted sum of the inverses M Gamma1 = P D i S Gamma1 i D T i , as proposed among other in [1, 7, 32] We ....

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M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math, 48 (1995), pp. 121-- 155.

....of edges of Omega i . We obtain (50) by noting that each term of the sum is bounded by juj 2 W 1 a;h( Omega i ) 7. The Neumann Neumann Basis. In this section, we consider a NeumannNeumann coarse space. This is the P 1 nonconforming version of a coarse space studied in Dryja and Widlund [8], and Mandel and Brezina [9] However, here we use an approximate harmonic extension inside the substructures. We note that the coarse spaces considered by these authors differ only in how certain weights are chosen. Mandel and Brezina use weights that are convex combinations of the coefficient ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute, March 1993. Submitted to Comm. Pure Appl. Math.

....we have (weak) regular splittings [29] 41] For examples of splittings for which the inequality (33) holds see [31] Another situation worth mentioning where (33) holds is when A i is semide nite and the inexact solver is de nite. This process is usually called regularization; see, e.g. [13], 27] In [19] it is shown that the damped additive Schwarz iterations with inexact local solves converge in the M matrix case under the condition (31) and 1=q. Furthermore, it is shown that the induced splittings corresponding to (29) and (30) A = M N = M N are weak regular. ....

M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic nite element problems. Communications in Pure and Applied Mathematics, 48:121-155, 1995.

....# log d h #w# 2 L # (0,d) But by Lemma 4.5, #w# L # (0,d) # log d h 1 2 #v# 1 2,## . The desired estimate then follows. LEMMA 4.13. Let n = 3. For any u # V h(## vanishing on one edge of# , #u# 0,# # d log d h 1 2 u 1,# . Proof. The proof follows [40]. Without loss of generality, assume# = 0, d) 3 and that the edge is on the z axis. Let # s =# # z = s . By the assumption, for z # (0, d) u vanishes at least at one point in # z . Thus for any constant c, max (x,y)##z u(x, y, z) # max (x,y)##z u(x, y, z) c c # 2 ....

....(from the DD) shares a common face with # We shall make special remarks for more general cases. 890 JINCHAO XU AND JUN ZOU 6.2. Neumann Neumann methods. In this subsection, we discuss the methods based on modifying the local operator S i . The presentation here mainly follows Dryja and Widlund [40]. The modification, first proposed in [40] is based on the following scaled full H 1 inner product: u, v) 1,# i = #u, #v) 0,# i h 2 0 (u, v) 0,# i # u, v # H 1(# i ) Correspondingly, a nonsingular operator S i : V h (# i ) ## V h (# i ) can be defined as follows: # ....

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M. Dryja and O. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121--155.

....(0;d) But by Lemma 4.5, kwk L 1 (0;d) log d h ) 1=2 kvk 1=2; Omega ; the desired estimate then follows. Lemma 4.13. Let n = 3. For any u 2 V h( Omega Gamma vanishing on one edge of Omega , kuk 0; Omega d log d h 1=2 juj 1; Omega : Proof. The proof follows [35]. Without loss of generality, we assume Omega = 0; d) 3 and the edge is on the z axis. Let Delta s = Omega fz = sg. By the assumption, for z 2 (0; d) u vanishes at least at one point in Delta z . Thus for any constant c, max (x;y)2 Delta z ju(x; y; z)j max (x;y)2 Delta z ju(x; y; z) ....

....the domain decomposition) shares a common face with Omega Gamma We shall make special remarks for more general cases. 6.2. Neumann Neumann methods. In this subsection, we discuss the methods based on modifying the local operator S i . The modification, first proposed by Dryja and Widlund [35], is based on the fol lowing scaled full H 1 inner product (u; v) 1; Omega i = ru; rv) 0; Omega i h Gamma2 0 (u; v) 0; Omega i 8 u; v 2 H 1( Omega i ) Correspondingly, a nonsingular operator S i : V h ( Gamma i ) 7 V h ( Gamma i ) can be defined as follows h S i u; ....

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M. Dryja and O. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48:121--155, 1995.

....growth in p. Mathematics Subject Classification (1991) 41A10, 65N30, 65N55 1. Introduction An overlapping domain decomposition method for p version finite elements was analyzed in Pavarino [19] using the framework provided by the additive Schwarz method of Dryja and Widlund [13] [14]. Due to the generous overlap between subdomains, we were able to show that the condition number of the iteration operator is bounded by a constant independent of the degree p, the mesh size H and the number of subdomains N . Here we consider a variant of the method based on local refinement. Our ....

Dryja, M., Widlund, O.B. (1993): Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute. To appear in Comm. Pure Appl. Math.

....with diagonal element equal to the inverse of the number of subdomains in which the unknown belongs. The definition of the matrices D i should reflect the coefficient variation of the elliptic problem across the interface. There are various way to take into account this variation, see for example Dryja and Widlund [1993]. In general, the matrices D i are defined such that they can be easily computed automatically. The experience shows that diagonal matrix are sufficient in most of the cases. 3.2 Definition of the matrices Z i , i = 1; k. The preconditioner requires to find a solution u i of the problem S i ....

Dryja, M.; Widlund, O. [1993]: Schwarz method of Neumann-Neumann type for the three-dimensional elliptic finite element problems, to appear.

.... of PMPF satis es (PMPF ) C 1 log H h 3 ; where C is a positive constant independent of h; H . This result is similar to estimates for other non overlapping domain decomposition methods; see Dryja, Smith, and Widlund [7] for iterative substructuring methods, Dryja and Widlund [8] for Neumann Neumann algorithms, and Mandel [22] and Mandel and Brezina [23] for balancing algorithms. 4 The FETI method for mortars The FETI algorithm can also be applied when mortar nite elements are considered on The price we have to pay for the inherent exibility of the mortar nite ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic nite element problems. Comm. Pure Appl. Math., 48(2):121-155, February 1995.

.... with a coarse space was introduced in [27] under the name Balancing Domain Decomposition and further studied in [12, 29, 28] Our abstract framework and analysis are related to the representation of other Neumann Neumann methods with coarse spaces considered as an abstract Schwarz method in [16]. For earlier work on the Neumann Neumann and similar preconditioners without a coarse space, see [1, 13, 14, 21, 33] A domain decomposition method, in a sense dual to Neumann Neumann, is obtained by enforcing intersubdomain continuity by Lagrange multipliers [19, 36, 17] That method, known as ....

.... 2 V; the global Schur complement operator by S = N X i=1 R T i S i R i ; and the interface right hand side by F = N X i=1 R T i Tr GammaT i L i : With this notation, and after the elimination of U i , the original problem (19) reduces to the form S u = F in V; 22) see, e.g. [16, 23]. Revised January 1997 10 3.2. The abstract Neumann Neumann preconditioner. For the abstract problem given by (22) it seems natural to precondition the sum S = P R T i S i R i by a weighted sum of the inverses, as proposed, e.g. in [1, 7, 33] M Gamma1 = X D i S Gamma1 i D T i ....

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M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math, 48 (1995), pp. 121--155.

....and solve a similar problem on each subdomain (alternatively or in parallel) with boundary information about the solution from the neighboring subdomains. This general idea has been applied to various elliptic problems (for a recent review on Schwarz methods we refer to Dryja and Widlund [11]) Since time discretization of parabolic problems leads to certain elliptic problems on the consecutive time level, domain decomposition algorithms can be applied to this class of problems as well. However, discretizations of parabolic equations have one very important feature: a large parameter ....

M. Dryja and O. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Communications on Pure and Applied Mathematics, XLVIII, 1 -- 35 (1995).

....= I Gamma n X i=0 P i )e k ; where e k = u Gamma u k is the error of the k th approximation. Similarly, the reduction of the error of one step of the multiplicative method is governed by the equation e k 1 = I Gamma P n ) I Gamma P 1 ) I Gamma P 0 )e k : We refer to [25] and references included there for abstract analysis. Here we only summarize the main results. They are due to a number of authors: Dryja and Widlund [25, 24] Nepomnyaschikh [55] Bramble, Pasciak, Wang, Xu [9, 70] Lions [44] Bj rstad and Mandel [6] and others. Let C 0 be the constant such ....

....of the multiplicative method is governed by the equation e k 1 = I Gamma P n ) I Gamma P 1 ) I Gamma P 0 )e k : We refer to [25] and references included there for abstract analysis. Here we only summarize the main results. They are due to a number of authors: Dryja and Widlund [25, 24]; Nepomnyaschikh [55] Bramble, Pasciak, Wang, Xu [9, 70] Lions [44] Bj rstad and Mandel [6] and others. Let C 0 be the constant such that for all u 2 V , there exists a representation u = P n i=0 u i ; u i 2 V i , and n X i=0 a i (u i ; u i ) C 0 a(u; u) Let B = b ij ] be the matrix ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math, 48:121--155, 1995.

....of technical lemmas, but the principal argument is complete. For related work on the Neumann Neumann preconditioner, we refer to Glowinski and Wheeler [8] For a somehow different formulation of the Neumann Neumann problem with similar bounds for second order problems we refer to Dryja and Widlund [7]. The BDD method was also applied to mixed problems by Cowsar, Mandel, and Wheeler [4] 2. Finite Element Plate Model Let Omega ae IR 2 be a bounded polygonal domain decomposed into nonoverlapping subdomains Omega 1 ; Omega k . There is given a conforming triangulation fTg of Omega ....

Maksymilian Dryja and Olof B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Tech. Report 626, Department of Computer Science, Courant Institute, March 1993, Submitted to Comm. Pure Appl. Math.

....gradient method for problem (1.4) Then we obtain oe(M Gamma1 A) oe(CA) oe(T ) i.e. the convergence properies of the method may be studied as the spectrum of the sum of the approximate projections T i . This path has been taken by Bj rstad and Mandel [5] Dryja, Smith and Widlund in [30] [32] and in a different setup by Xu [94] The following theorem summarizes the results of their research. Theorem 2.1.1 (Dryja, Widlund [32] Let T = P J i=1 T i and (i) there exists a constant C 0 such that for all u 2 V there exists a decomposition u = P J i=0 u i ; u i 2 V i , such that J X ....

....be studied as the spectrum of the sum of the approximate projections T i . This path has been taken by Bj rstad and Mandel [5] Dryja, Smith and Widlund in [30] 32] and in a different setup by Xu [94] The following theorem summarizes the results of their research. Theorem 2.1. 1 (Dryja, Widlund [32]) Let T = P J i=1 T i and (i) there exists a constant C 0 such that for all u 2 V there exists a decomposition u = P J i=0 u i ; u i 2 V i , such that J X i=0 a i (u i ; u i ) C 2 0 a(u; u) ii) there exists a constant such that for i = 0; J , a(u; u) a i (u; u) 8u 2 V i ....

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M. Dryja and O. B. Widlund, Schwarz methods of NeumannNeumann type for three-dimensional elliptic finite element problems, Tech. Report 626, Department of Computer Science, Courant Institute, March 1993. To appear in Comm. Pure Appl. Math.

....of technical lemmas, but the principal argument is complete. For related work on the Neumann Neumann preconditioner, we refer to Glowinski and Wheeler [8] For a somehow different formulation of the Neumann Neumann problem with similar bounds for second order problems we refer to Dryja and Widlund [7]. The BDD method was also applied to mixed problems by Cowsar, Mandel, and Wheeler [4] 2. Finite Element Plate Model Let Omega ae IR 2 be a bounded polygonal domain decomposed into nonoverlapping subdomains Omega 1 ; Omega k . There is given a conforming triangulation fTg of Omega ....

Maksymilian Dryja and Olof B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Tech. Report 626, Department of Computer Science, Courant Institute, March 1993, submitted to Comm. Pure Appl. Math.

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M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2):121--155, February 1995.

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M. DRYJA AND O. B. WIDLUND, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121--155.

.... and f(v) fv dx for f 2 L For simplicity, let be a bounded polygonal region in with a diameter of size O(1) The extension of the results to can be carried out easily by using the theory developed here in this paper and the well known three dimensional additive Schwarz techniques; [9, 10, 12]. Let T be a shape regular, quasi uniform triangulation, of size O(h) and V H 0( the nite element space consisting of continuous piecewise linear functions associated with the triangulation. We are interested in solving the following discrete problem associated with (1.3) Find u 2 V ....

....i=1;N ;k=1;n The matrix form of the coarse projection operator e P 0 is 0 = e R e R 0 A; 4.3) where e A 0 = e R 0 A e R 0 is an N N matrix. We remark that e A 0 is more sparse than coarse space matrices that appear in other methods such as Neumann Neumann or FETI type algorithms [12, 13, 18, 23], since only connections with the neighboring subdomains appear in the stencils associated to a coarse basis function. Another feature of this coarse space problem is that the computation of the right hand side, i.e. e R 0 Au, for some u, can be done inside each i ; this is a clear advantage ....

M. Dryja and O. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic nite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-155.

....problem is reduced to a problem for the subdomain interface variables, by using a Schur complement procedure. This interface problem is then solved by a preconditioned Krylov subspace method. The scalability of the nonoverlapping substructuring methods has also been established, cf. 19] and [21]. The Finite Element Tearing and Interconnecting (FETI) methods form a special family of domain decomposition methods. They are iterative substructuring methods of dual type. The rst FETI method was proposed in [29] for solving positive de nite elliptic partial di erential equations. In this ....

....the same. The inf sup stability of this new mixed nite elements is also proved by using a macroelement methods, like in the proof of Theorem 4. The only thing di erent is that, instead of using Lemma 8 in the proof, we use the following lemma, which can be found in a somewhat di erent form in [21]. Lemma 19 De ne an interpolation operator I : W W by: I w (x) ik ij (x) 8w 2 W : We then have 1=2 ( C(1 log(H=h) jw j 1=2 ( kw I w k 2 ( CHjw j ( where C is a positive constant independent of H and h. 66 By using this lemma, we can ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic nite element problems. Comm. Pure Appl. Math., 48(2):121-155, February 1995.

....overall our algorithm is very ecient, parallel, and robust. The results on general heterogeneous problems are also supported in part by our theory. Neumann Neumann methods were rst introduced and analyzed for second order elliptic problems; see Cowsar, Mandel, and Wheeler [11] Dryja and Widlund [12], Mandel [24] Mandel and Brezina [25] and Pavarino [28] More recently, this family of methods has been extended to plate and shell problems, see Le Tallec, Mandel, and Vidrascu [21] to convection di usion problems, see Achdou, Le Tallec, Nataf, and Vidrascu [1] and Alart, Barboteu, Le Tallec, ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic nite element problems. Comm. Pure Appl. Math., 48(2):121{ 155, February 1995.

....# 1 #, the scaled additive smoothers R j = #R a j satisfy the hypotheses of Theorem 3.1 with # = # #. ii) The multiplicative smoothers R j = R m j satisfy the hypotheses of Theorem 3.1 with # = # 2 #. Results of this type can be found in many places, for example in [4, Chapters 3 and 5] [5], 13, Chapter 5] and [15] Therefore we merely sketch a proof here. The first hypothesis of Theorem 3.1 for the additive smoother follows from (3.1) with x k = y k = P k j x and Schwarz s inequality. It is well known that the left hand side of (3.2) is precisely equal to (R a j 1 x, ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann--Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math 48 (1995), 121--155.

....of the mortar method, the problem is approximated by the finite element method with piecewise linear functions on nonmatching meshes. The domain decomposition method is of iterative substructuring type and is described as an additive Schwarz method (ASM) using the general framework of ASMs; see [DW95, Ben95a]. It is applied to the Schur complement of our discrete problems, i.e. interior variables of all subregions are first eliminated using a direct method. In this paper, we consider the mortar element method in the geometrically conforming case only. The region Omega is a union of simplices Omega i ....

....CH 2 ffib(u; u) 19) where ffi is given by (12) and C is constant independent of H, h i and ae i . A proof of this lemma is a slighted modification of the proof of Lemma 9 in [DSW96] 5 Proof of Theorem 1.3. 1 Using the general theorem of ASMs, we need to check three key assumptions; see [DW95] and [Ben95a] 94 DRYJA Assumption (ii) It is shown that ae( C in view of Lemma 4.3. Assumption (iii) Of course, 1 for b 0 (u; u) u 2 V 0 and b (W i ) k (u; u) u 2 V (W i ) k . We now show that for u 2 V (F ) m b(u; u) Cb (F ) m (u; u) 20) Let fl i(m) ffi j(m) be the ....

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Dryja M. and Widlund O. (1995) Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48(2): 313--348.

....FETI methods previously developed. While the global part of the preconditioner for a strictly dual FETI method is directly associated with the dual variables, that of a FETI DP method is not. We note that primal iterative substructuring methods have been studied quite extensively, see, e.g. 6] [8], and [5] well before a similarly complete, and quite challenging, mathematical theory was developed for the FETI methods, see [15] 20] and [13] FETI algorithms using inexact subdomain solvers have also been developed and analyzed by two of the authors in [12] We note that primal iterative ....

....The sets of nodes on ; i ; and are denoted by h ; i;h ; and h ; respectively. As in previous work on Neumann Neumann and FETI algorithms, a crucial role is played by the weighted counting functions i ; which are associated with the individual subdomain boundaries i ; cf. [5, 8, 14, 19]. In this paper they will be used in the de nition of certain diagonal scaling matrices. These functions are de ned, for 2 [1=2; 1) and for x 2 h [ h , by a sum of contributions from i , and its relevant next neighbors i (x) 8 : X j2Nx j (x) x 2 i;h j;h ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic nite element problems. Comm. Pure Appl. Math., 48(2):121{ 155, February 1995.

....and Valli [32] for a general introduction to domain decomposition methods. Most of the theoretical and numerical work for Neumann Neumann methods has been carried out for second order elliptic problems; see Mandel [25] Mandel and Brezina [26] Cowsar, Mandel and Wheeler [9] Dryja and Widlund [10], and Pavarino [29] More recently, this family of methods has been extended to plate and shell problems, see Le Tallec, Mandel, and Vidrascu [22] to convection di usion problems, see Alart, Barboteu, Le Tallec, and Vidrascu [3] and Achdou, Le Tallec, Nataf, and Vidrascu [1] and to vector eld ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic nite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-155.

.... FETI as well as of dual primal FETI preconditioners which have a rate of convergence which is bounded independently of possible jumps of the coe#cients of an elliptic model problem often considered in the theory of Neumann Neumann and other iterative substructuring algorithms; see, e.g. DW95, DSW94, MB96] and the references therein. Our new results become possible because of special scalings. One of them, for the preconditioner, is closely related to an important algorithmic idea used in the best of the Neumann Neumann methods. The other scaling a#ects the choice of the projection ....

....i , on ## i , and on # are denoted by # i,h , ## i,h , and # h , respectively. As in previous work on Neumann Neumann algorithms, a crucial role is played by the weighted counting functions i # # W , which are associated with the individual subdomain boundaries ## i ; cf. e.g. DSW96, DW95] In this paper they will be used primarily in the definition of certain diagonal scaling matrices. These functions are defined, for # # [1 2, #) and for x # # h # ## h , by a sum of contributions from# i , and its relevant next neighbors i (x) # # # # # # # # j#Nx # # j (x) x ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2):121--155, February 1995.

....explain the success of Rnquist s algorithm, but we believe that this attempt brings a much greater understanding of the mechanisms involved, as well as some promising new methods that will be implemented and compared with Rnquist s in the near future. 3 The Schwarz framework of Dryja and Widlund [52] accommodates, in an elegant fashion, the use of overlapping spaces in the design of preconditioners. The preconditioner is viewed as consisting of different modules, each of which is often associated with a geometrical object, or with a simplified global model. We try to stress, throughout the ....

....j ; u j ) 1=2 8u i 2 V h i ; 8u j 2 V h j : Let ae(E) be the spectral radius of the matrix E, with entries E ij . Theorem 2.3.1 Assume that H1, H2, and H3 hold. Then, C 2 0 a(u; u) a(T as u; u) ae(E) 1) a(u; u) 8u 2 V h( Omega Gamma : The proof of this lemma can be found, e.g. in [52]. We state now a result for the operator Tms . Let e k = u k Gamma u be the error at step k of the iterative algorithm (2.6) It is clear that an upper bound for the relative decrease of the a norm of the residual at each iteration is given by an upper bound on the a norm of the error ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2):121--155, February 1995.

.... introduce a new one parameter family of FETI preconditioners and prove a bound on the rate of convergence which is independent of possible jumps of the coe#cients of an elliptic model problem previously considered in the theory of Neumann Neumann and other iterative substructuring algorithms; see [11, 8, 23, 30, 31]. In fact, we have found it possible to reduce the analytic core of the theory for the new class of FETI methods to a variant of an estimate # SCAI Institute for Algorithms and Scientific Computing, GMD German National Research Center for Information Technology, Schloss Birlinghoven, ....

....stressing the similarities to the analysis of the FETI algorithms. In an appendix, that concludes our paper, we collect some auxiliary technical results which are needed in the proofs of Lemmas 6 and 8; they are borrowed almost directly from Dryja, Smith, and Widlund [8] and from Dryja and Widlund [11, 10]. 2. Elliptic model problem, finite elements, and geometry. Let# # R 3 , be a bounded, polyhedral region, let ## D # ## be a closed set of positive measure, and let ## N : ## ## D be its complement. We impose Dirichlet and Neumann FETI AND NEUMANN NEUMANN METHODS 3 boundary ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2):121-- 155, February 1995.

....We solve (2.5) with a preconditioned conjugate gradient method, using an additive Schwarz method determined by a finite family of subspaces V s whose sum spans V h , and bilinear forms b s ( s defined on V s V s . Using the Schwarz framework described in Dryja and Widlund [15], we define approximate projections T s : V h # V s , by b s (T s u, v s ) a # (u, v s ) #v s # V s . ETNA Kent State University etna mcs.kent.edu Mario A. Casarin and Olof B. Widlund 83 The preconditioned operator T is given by T = #T 0 X s#1 T s , where # is a positive ....

....all u # V h , there exists u s s , u s # V s , with X s b s (u s , u s ) # C 2 0 a # (u, u) where u = X s u s , and let # be a constant such that a # (u, u) # #b s (u, u) #u # V s . Since our algorithm is a two level algorithm, the third hypothesis of Theorem 2. 2 in [15] is trivially satisfied. This theorem then provides a bound on the condition number of T : #(T ) # CC 2 0 #. 3.1) In subsection 3.3, we will introduce our algorithm and establish bounds for C 0 and #. 3.2. A hierarchical basis. Before we can introduce our preconditioner in detail, we need to ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-- 155.

....solve (6) with a preconditioned conjugate gradient method, using an additive Schwarz method determined by a finite family of subspaces fV s g whose sum spans V h , and bilinear forms fb s ( Delta; Delta)g s defined on V s Theta V s . Using the Schwarz framework described in Dryja and Widlund [14], we define approximate projections T s : V h V s ; by b s (T s u; v s ) a Gamma (u; v s ) 8v s 2 V s : 8 The preconditioned operator T is given by T = ffT 0 X s1 T s ; where ff is a positive parameter that is used to tune the algorithm; see Section 4. Let C 2 0 be a constant ....

.... for all u 2 V h , there exists fu s g s , u s 2 V s , with X s b s (u s ; u s ) C 2 0 a Gamma (u; u) where u = X s u s ; and let be a constant such that a Gamma (u; u) b s (u; u) 8u 2 V s : Since our algorithm is a two level algorithm, the third hypothesis of Theorem 2:2 in [14] is trivially satisfied. This theorem then provides a bound on the condition number of T : T ) CC 2 0 : 17) In subsection 3.3, we will introduce our algorithm and establish bounds for C 0 and : 3.2. A hierarchical basis. Before we can introduce our preconditioner in detail, we need to ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-- 155.

....fluids, will be the subject of part II of this work [30] The main result of this paper is a polylogarithmic bound for the condition number of the iteration operator with the wire basket preconditioner. As in the scalar case, this result is obtained by working within the Schwarz framework; see [35, 12, 13]. The paper is organized as follows. In Section 2, we briefly describe the system of 2 linear elasticity. In Section 3, we discretize the problem by the spectral element method and Gauss Lobatto Legendre quadrature. In preparation for the definition of the wire basket preconditioner, we ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-- 155. 17

....and other preconditioners for Stokes problems. In the Stokes case, we can use any other scalar substructuring preconditioner for each component of u in (31) instead of using the wire basket preconditioner. For example, we could use a Neumann Neumann preconditioner; see Dryja and Widlund [18] for a detailed analysis of this family of preconditioners for h Gammaversion finite elements and Pavarino [29] for an extension to spectral elements. We then obtain a Laplacian based Neumann Neumann preconditioner b S Gamma with a scalar Neumann Neumann preconditioner b SNN in each diagonal ....

.... Omega j of each element. Here R Omega j are restriction matrices returning the degrees of freedom associated with the boundary of Omega j , D j are diagonal matrices and y denotes an appropriate pseudo inverse for the singular Schur complements associated with the interior elements; see [18, 29] for more details. The polylogarithmic bound proven in the scalar case, carries over to the case now under consideration: c(1 log n) Gamma2 u T Gamma b S Gamma u Gamma u T Gamma 2 6 4 SH 0 0 0 SH 0 0 0 SH 3 7 5u Gamma Cu T Gamma b S Gamma u Gamma 8u Gamma 2 V n ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121-- 155.

.... Gamma1 i (OE) if i (OE) 6= 0; y i (OE) 0 otherwise : The preconditioner is given by S Gamma1 prec = X i N i S (i) y N i ; where the N i are diagonal matrices and the diagonal entry corresponding to the basis function OE is y i (OE) For previous work, see Dryja and Widlund [29], Bourgat, Glowinski, Le Tallec, and Vidrascu [12] Le Tallec, De Roeck, and Vidrascu [38] Le Tallec and De Roeck [21] 42 Cowsar, Mandel, and Wheeler [20] Kuznetsov, Manninen, and Vassilevski [36] Mandel [43] and Mandel and Brezina [45, 44] A major technical difficulty stems from the fact ....

....are singular. The components corresponding to tetrahedra that touch the Dirichlet part of the boundary, are not singular. There are several ways to deal with this. 1. We can work with pseudoinverses S (i) y of the local Schur complements. 2. We can follow the approach taken in Dryja and Widlund [29], and solve local problems using a different elliptic operator, for the substructures that do not touch the boundary: a i (u; u) Z Omega i (ru) 2 dx 1 H 2 i Z Omega i u 2 dx: Dryja and Widlund [29] contains a detailed discussion concerning those substructures that touch the ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math., 48(2):121--155, February 1995.

.... introduced in [76] have also been discussed in the conforming case in the recent work by Dryja, Smith, and Widlund [35] We also introduce approximate harmonic extensions, and fi Neumann Neumann coarse spaces (generalizations of the Neumann Neumann coarses space introduced by Dryja and Widlund [41], and Mandel and Brezina [55] These topics are also discussed in the conforming case in Chapter 4. Finally, at the end of Chapter 5, we use our results of Chapter 4 to analyze some multilevel for hybrid mixed finite element methods that are insensitive to the jumps of the coefficients across ....

.... report Multilevel Schwarz Methods for Elliptic Problems with Discontinuous Coefficients in Three Dimensions completed in March 1994 in joint work with Maksymilian Dryja and Olof Widlund [34] This work follows earlier work on iterative substructuring methods [35] and Neumann Neumann type methods [41]. It is also focused on making the performance of the algorithms independent of the jumps in the coefficients. We explore the use of nonstandard, exotic coarse spaces, 4 and also derive a new condition on the coefficients, quasi monotonicity, for which we can establish the same basic results for ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute, March 1993. To appear in Comm. Pure Appl. Math.

....1 structures, and the number of levels. We consider two classes of the methods, additive and multiplicative. The multiplicative methods are variants of the multigrid V cycle method. In our design and analysis, we use a general Schwarz method framework developed in Dryja and Widlund [11,12,15], and Dryja, Smith, and Widlund [10] for the additive variant, and Bramble, Pasciak, Xu, and Wang [3] for the multiplicative ones. Among the particular cases, discussed in this paper, are the BPX algorithm, cf. Bramble, Pasciak and Xu [4] and the multilevel Schwarz method with one dimensional ....

....can have large variations only across the interfaces of these substructures. We then design methods with several coarse spaces, sometimes known as exotic coarse spaces; cf. Widlund [25] Some are new and others have previously been discussed; see Dryja, Smith and Widlund [10] Dryja and Widlund [15], and Sarkis [20] One of our main results is that the condition number of the resulting systems can be estimated from above by C (1 ) 2 with C independent of the jumps of coefficients, of the number of substructures, and also of . For multiplicative variants such as the V cycle multigrid, ....

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Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute, March 37 1993. To appear in Comm. Pure Appl. Math.

....energy a(u; u) An upper bound on a(u i ; u i ) a(u i ; u i ) u i 2 V i ; also enters the bound on (K Gamma1 J K) if inexact solvers are used for some or all of the subspaces. In this study, we use the block Jacobi framework but there is also a more general theory; cf. Dryja and Widlund [14]. Thus, any block Jacobi method can be viewed as an additive Schwarz method based on a direct sum of subspaces. There are also GaussSeidel like, multiplicative, as well as hybrid Schwarz algorithms; cf. Dryja, Smith, and Widlund [11] for a general discussion. 3. A choice of subspaces. A method ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute, March 1993. Submitted to Comm. Pure Appl. Math.

....of the use of such coarse problems in the multigrid context. The other major family of domain decomposition methods uses overlapping subregions; see Dryja and Widlund [14, 15] for a discussion of recent work. For further comments and an overview of the literature, we refer to our recent papers [12, 13, 30]. We note that the principal goal of domain decomposition theory is to provide a good upper bound on the condition number of the preconditioned operator. It is well known that the number of conjugate gradient iterations grows in proportion to p ; see, e.g. Golub and Van Loan [19] All the ....

.... energy a(u; u) If inexact solvers are used for some or all of the subspaces, upper bounds on a(u i ; u i ) a i (u i ; u i ) u i 2 V i ; also enters the bound on (K Gamma1 J K) In this study, we use the block Jacobi framework but there is also a more general theory; cf. Dryja and Widlund [13]. Any block Jacobi method can be viewed as an additive Schwarz method based on a direct sum of subspaces. There are also GaussSeidel like, multiplicative, as well as hybrid Schwarz algorithms; see Dryja, Smith, and Widlund [12] for a general discussion. It follows from this general theory that ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems, Tech. Rep. 626, Department of Computer Science, Courant Institute, March 1993. To appear in Comm. Pure Appl. Math.

....a(u; u) An upper bound on a(u i ; u i ) a i (u i ; u i ) u i 2 V i ; also enters the bound on ( K Gamma1 J K) if inexact solvers are used for some or all of the subspaces. In this study, we use the block Jacobi framework but there is also a more general theory; see Dryja and Widlund [22]. Thus, any block Jacobi method can be viewed as an additive Schwarz method based on a direct sum of subspaces. There are also GaussSeidel like, multiplicative, as well as hybrid Schwarz algorithms; cf. Dryja, Smith, and Widlund [20] for a general discussion. Using the estimates of this paper, we ....

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of Neumann-Neumann type for threedimensional elliptic finite element problems. Technical Report 626, Department of Computer Science, Courant Institute, March 1993. To appear in Comm. Pure Appl. Math.

.... Omega ih n Omega when the sum extends over all subdomains Omega jh that share x as a common node. This property is important for our analysis. A somewhat similar coarse space defined in terms of discrete harmonic functions v i with data on Omega i equal to our OE i (x) was used in [3] for an analysis of the Neumann Neumann algorithm. 3 Analysis With our new choice of subspaces we can take the bilinear forms b i (u; v) V i Theta V i R; i = Gamma1; 0; 1; N all equal to the form a(u; v) defined by (1) Similarly, our linear projection operators T i : V h V i ; ....

.... (32) In a similar way we estimate the terms u j (OE j Gamma OE BF j ) in (30) Next, if Omega i belongs to NBE , that is, OE i = OE BE i , we have a slightly weaker estimate than (31) for the term u i u 2 i c(1 log H i h i ) H i juj 2 H 1( Omega i ) 33) by Lemma 6 in [3]. The difference (OE i Gamma OE BE i ) which is nonzero only on an edge contributes a factor H i (see (22) so that ae i j u i (OE i Gamma OE BE i )j 2 H 1( Omega i ) c H i h i a i (u; u) where we have replaced (1 log(H i =h i ) by its upper bound H i =h i . We again estimate the ....

M. Dryja and O. B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math., 48 (1995), pp. 121--155.

No context found.

M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element methods. Comm. Pure Appl. Math., 48:121--155, 1995.

No context found.

M. Dryja and O. B. Widlund. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element methods. Comm. Pure Appl. Math., 48:121--155, 1995.

No context found.

M. Dryja and O. B. Widlund, Schwarz methods of neumann-neumann type for three-dimensional elliptic finite element problems, tech. rep., Courant Institute of Mathematical Science, New York University, New York, NY, 1993. (To appear). 44

No context found.

Dryja M, Widlund OB. Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Communications on Pure Applied Mathematics 1995; 48(2):121--155.

No context found.

Maksymilian Dryja and Olof B. Widlund. Schwarz methods of NeumannNeumann type for three-dimensional elliptic nite element problems. Comm. Pure Appl. Math., 48(2):121-155, February 1995.

No context found.

M. Dryja and O. Widlund, Schwarz Methods of Neumann-Neumann type for ThreeDimensional Elliptic Finite Element Problems. Comm. Pure Appl. Maths Vol XLVIII p121-155 (1995).

No context found.

M. Dryja and O. B. Widlund, Schwarz methods of Neumann--Neumann type for three-- dimensional elliptic finite element problems, Comm. Pure Appl. Math. (submitted).

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