J. Mandel, Balancing domain decomposition, Communications in Applied Numerical Methods 9 (1993) 233--241.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Examples of such methods are given in Bramble, Pasciak, and Schatz [5] Mandel [28,27,26] and Smith [38] see also the discussion in Dryja and Widlund [18] or section 2. 5 of Smith [37] For other recent work on Neumann Neumann preconditioners that incorporates a coarse solver, see Mandel [30,31]. It has been known since 1958, cf. Hestenes [24] that the rate of convergence of a preconditioned conjugate gradient method can be estimated in terms of the condition number of a generalized eigenvalue problem; see also Golub and Van Loan [22] The results for all good algorithms of the ....

....note that the problem defined in V 0 = V H reduces to a regular finite element problem defined on the coarse mesh. 17 6. A Method with a Coarse Space of Minimal Dimension. The algorithm in this section has much in common with a method developed and analyzed recently by Mandel and Brezina, cf. [30,32,31]. Essentially the same coarse space is used, but our results and algorithms differ in several ways. Thus, we are able to design algorithms with good bounds without imposing extra restrictions on the intersection of the boundaries of the individual substructures and that of the original region. We ....

Jan Mandel. Balancing domain decomposition. Technical report, Computational Mathematics Group, University of Colorado at Denver, 1992. Communications in Applied Numerical Methods, to appear.

....of the entire domain. Similarly to the one level preconditioner, we define a coarse mesh restriction operator R 0 and let B 0 = R T 0 # R 0 AR T 0 # ,1 R 0 . The two level additive Schwarz preconditioner is M = B 0 p X i=1 B i (2. 11) and the two level hybrid II preconditioner [22, 27] is given by M = B 0 #I,B 0 A# p X i=1 B i : 2.12) In the work reported here we found that the alternative form M = B 0 p X i=1 B i #I , AB 0 # (2.13) performed better. To avoid generating and storing a separate coarse mesh, we define coarse mesh basis functions as aggregates of ....

J. MANDEL, Balancing domain decomposition, Communications in Applied Numerical Methods 9, (1993), pp. 233--241.

....of the entire domain. Similarly to the one level preconditioner, we define a coarse mesh restriction operator R 0 and let B 0 = R T 0 i R 0 AR T 0 j Gamma1 R 0 . The two level additive Schwarz preconditioner is M = B 0 p X i=1 B i (2. 11) and the two level hybrid II preconditioner [22, 27] is given by M = B 0 (I Gamma B 0 A) p X i=1 B i : 2.12) In the work reported here we found that the alternative form M = B 0 p X i=1 B i (I Gamma AB 0 ) 2.13) performed better. To avoid generating and storing a separate coarse mesh, we define coarse mesh basis functions as ....

J. MANDEL, Balancing domain decomposition, Communications in Applied Numerical Methods 9, (1993), pp. 233--241.

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J. Mandel, Balancing domain decomposition, Communications in Applied Numerical Methods 9 (1993) 233--241.

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J. Mandel. Balancing domain decomposition. Communications in Applied Numerical Methods, 9:233--241, 1993.

....The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced in [19] is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size h. Computational experiments ....

....is then used as a preconditioner in the conjugate gradients method. It is well known that the absence of a coarse problem results in deterioration of convergence of the iterations with increasing number of subdomains [11, 14] The Balancing Domain Decomposition (BDD) was introduced by Mandel [19] by adding a coarse problem to an earlier method of De Roeck and Le Tallec [11] known as the Neumann Neumann method, based in turn on earlier work for the case of two subdomains [2] and on a closely related method of Glowinski and Wheeler for mixed problems [17] The development of BDD was ....

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J. Mandel, Balancing domain decomposition, Communications in Applied Numerical Methods, 9 (1993), pp. 233--241.

....of a domain decomposition technique. In the successive Schur complement matrices obtained, each block contains the internal nodes of a subdomain. The inverse and application of all blocks on the same level can be done in parallel. One distinction with traditional domain decomposition methods [12] is that all subdomains are constructed algebraically and exploit no physical information. In addition, the reduced system (coarse grid acceleration) is solved by a multi level recursive process akin to a multigrid technique. We define several measures to characterize the efficiency of BILUM (and ....

J. Mandel. Balancing domain decomposition. Communications in Applied Numerical Methods, 9:233--241, 1993.

.... paper provides a convergence theory for the two level Schwarz preconditioner first described in [8 10] The preconditioner is a two level additive Schwarz method [4, 5, 18] Its novel feature is the use of aggregation ideas from algebraic multigrid [1, 22] and balancing Neumann Neumann methods [13 15] to construct the coarse mesh. The use of aggregation arose from necessity. In the applications reported in [8 10] the subdomains were irregular, and a coarse mesh based on hat functions over the subdomains was impractical. For the same reason, we needed minimal overlap between subdomains. ....

J. MANDEL, Balancing domain decomposition, Communications in Applied Numerical Methods 9, (1993), pp. 233--241.

....this computation cannot be reduced to subdomain solves, but rather re uires the solution of a subproblem that is associated with all neighboring subdomains or an expensive explicit calculation of the Schur complement matrix. Balancing Domain Decomposition, introduced by the present author [12], only relies on subdomain solves to achieve the same asymptotic bounds on the condition number, but pays the price in less exibility. The method of Farhat and Roux [ is based on a agrange multiplier approach and also only re uires subdomain solves, but is not asymptotically optimal. Because of ....

J. Mandel, Balancing domain decomposition. Communications in Applied Numerical Methods, to appear.

....a fully algebraic method with a coarse space, somewhat similar in spirit to EBE, with much improved convergence rate (independent of the number of subdomains) proved under regularity free assumptions for problems on unstructured meshes. 2. 7 BDD as an Algebraic Method The BDD preconditioner [62] is based on the original Neumann Neumann preconditioner by De Roeck and Le Tallec [26] described in Section 2.5. The algorithm proposed in [26] suffers from the lack of global distribution of information resulting in dramatic deterioration of performance with increasing number of subdomains. As ....

....distribution of information resulting in dramatic deterioration of performance with increasing number of subdomains. As noted in Le Tallec [55] the method becomes impractical when applied to problems with the number of subdomains larger than about 16. In order to defeat this drawback, Mandel [62] added a coarse problem as follows. Let n i = dimV i , 0 m i n i ; and Z i be n i Theta m i matrices of full column rank such that Ker S (i) ae Range Z i ; i = 1; J (2.14) and let W ae V be defined by W = fv 2 V : v = J X i=1 N i D i u i ; u i 2 Range Z i g: The space W will ....

[Article contains additional citation context not shown here]

, Balancing domain decomposition, tech. report, Computational Mathematics Group, University of Colorado at Denver, 1992. Communications in Applied Numerical Methods, to appear.

....The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced in [19] is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size h. Computational experiments ....

....is then used as a preconditioner in the conjugate gradients method. It is well known that the absence of a coarse problem results in deterioration of convergence of the iterations with increasing number of subdomains [11, 14] The Balancing Domain Decomposition (BDD) was introduced by Mandel [19] by adding a coarse problem to an earlier method of De Roeck and Le Tallec [11] known as the Neumann Neumann method, based in turn on earlier work for the case of two subdomains [2] and on a closely related method of Glowinski and Wheeler for mixed problems [17] The development of BDD was ....

[Article contains additional citation context not shown here]

J. Mandel, Balancing domain decomposition, Communications in Applied Numerical Methods, 9 (1993), pp. 233--241.

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