D. Yang. A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems. IMA J. Numer. Anal., 16:75-91, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....involved. The main objective of this paper is to better understand IR methods for model problems where direct analysis can be made. In particular, we analytically estimate values for the parameters involved in two recently proposed and analyzed IR methods. Namely, we consider an averaging scheme [17,24,25] (denoted by AVE in the sequel) and a Robin type IR scheme [10] denoted by ROB) In [17] Fourier analysis is applied for the theoretical analysis and shows that the fast convergence rate of the AVE method in the case of constant coecents and rectangular subdomains. the theoretical results are ....

....a Robin type IR scheme [10] denoted by ROB) In [17] Fourier analysis is applied for the theoretical analysis and shows that the fast convergence rate of the AVE method in the case of constant coecents and rectangular subdomains. the theoretical results are veri ed by the experimental ones. In [24], a convergence analysis of the AVE method is carried out at di erential level using Hilbert space techniques. Numerical experiments verify the fast convergence of the method and its stability with respect to di erent decompositions and di erent problems, using constant relaxation parameters. In ....

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D. Yang. A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems. IMA J. Numer. Anal., 16:75-91, 1996.

....and elegant simplicity in implementation, great savings in computer storage (in 3D, even small overlapping of subdomains could cause a lot more storage) and direct applicability to transmission problems. In this direction, several other types of domain decomposition have been considered [6, 11, 10, 12]. Their modifications can apply directly to domain decompositions with cross points and with long and narrow subdomains. In [3] we successfully implemented a variant of the methods [12] for selfadjoint and non selfadjoint elliptic partial differential equations with variable coefficients and full ....

....problems. In this direction, several other types of domain decomposition have been considered [6, 11, 10, 12] Their modifications can apply directly to domain decompositions with cross points and with long and narrow subdomains. In [3] we successfully implemented a variant of the methods [12] for selfadjoint and non selfadjoint elliptic partial differential equations with variable coefficients and full diffusion tensor, in the case of 100 subdomains with 81 cross points in 2 D. The variant also works well for long and narrow subdomains with length of a subdomain being 40 times as ....

D. Q. Yang, A parallel iterative nonoverlapping domain decomposition method for elliptic interface problems, to appear.

....and elegant simplicity in implementation, great savings in computer storage (in 3D, even small overlapping of subdomains could cause a lot more storage) and direct applicability to transmission problems. In this direction, several other types of domain decomposition have been considered [6, 11, 10, 12]. Their modifications can apply directly to domain decompositions with cross points and with long and narrow subdomains. In [3] we successfully implemented a variant of the methods [12] for selfadjoint and non selfadjoint elliptic partial differential equations with variable coefficients and full ....

D. Q. Yang, A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems, IMA J. Numer. Anal. 16(1996) 75-91.

....for parabolic and hyperbolic problems. The nonoverlapping domain decomposition method considered herein is closely related to and based on the ones presented by Funaro, Quarteroni and Zanolli [12] Marini and Quarteroni [18, 19] Lions [16] Rice, Vavalis and the author [21] and the author [27, 31, 32]. In Funaro, Quarteroni and Zanolli [12] and Marini and Quarteroni [18, 19] Neumann data are passed from one subdomain to its adjacent, while Dirichlet data are passed from its adjacent to itself. Convergence analysis of this method can be easily made for general elliptic problems [18] This ....

....role in computation. However, the convergence of this method is very slow unless some parameter is carefully chosen. The idea of this method has been used by Despr es [6] and Douglas et al. [9] to construct mixed finite element domain decomposition methods. In Rice, Vavalis and Yang [21] and Yang [31], Dirichlet data are passed at odd iterations and Neumann data at even iterations. This method requires that the information at the previous iteration level be passed to the subdomain problems at the current iteration level. Thus it can be efficiently implemented on computers with parallel ....

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Yang, D. Q. A parallel iterative nonoverlapping domain decomposition method with interface relaxation for elliptic problems. (preprint)

....for parabolic and hyperbolic problems. The nonoverlapping domain decomposition method considered herein is closely related to and based on the ones presented by Funaro, Quarteroni and Zanolli [12] Marini and Quarteroni [18, 19] Lions [16] Rice, Vavalis and the author [21] and the author [27, 31, 32]. In Funaro, Quarteroni and Zanolli [12] and Marini and Quarteroni [18, 19] Neumann data are passed from one subdomain to its adjacent, while Dirichlet data are passed from its adjacent to itself. Convergence analysis of this method can be easily made for general elliptic problems [18] This ....

Yang, D. Q. 1994 Parallel iterative nonoverlapping domain decomposition methods for elliptic and time-dependent problems. Presented at the SIAM Annual Meeting, San Diego, California, July 25-29.

....procedure in the subdomains and explicit flux calculation on the inter domain boundaries. The grids on the subdomains need not match up in such a way that they are restrictions of a global regular finite element grid over the whole physical domain, as opposed to other domain decomposition methods [9, 12, 13, 19]. This gives great flexibility for applying grid refinement or uniform fine grids in subdomains that contain local fronts or layers, and grid derefinement or uniform coarse grids in subdomains over which the solution changes slowly. The organization of the paper is as follows. In x2 we present our ....

....the diffusion coefficient as in equation (2.1) and C a constant. Note that unconditionally stable schemes are impossible since explicit flux computation is used on the interface. In contrast to the so called blockwise implicit schemes discussed above, iterative domain decomposition methods, e.g. [4, 9, 12, 19], can be considered. These iterative methods do not impose any stability condition and restrictions on the diffusion coefficient, but require a certain number of iterations at each time step, possibly with some efficient preconditioners. Since a quite good initial guess at each time step can be ....

D. Q. Yang, A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems, IMA J. Numer. Anal. 16 (1996), 75-91. Department of Mathematics, Wayne State University, Detroit, MI 48202. E-mail address: dyang@na-net.ornl.gov

....the method given in [3] can not be parallelized. Therefore, the method described in this paper can be efficiently implemented on parallel computers. A convergence analysis of this domain decomposition method for elliptic problems with variable coefficients and more than two subdomains is given in [8]. Mixed finite element approximations of this method are analyzed in [2] ....

D. Q. Yang, A parallel iterative nonoverlapping domain decomposition method with interface relaxation for elliptic problems. (preprint) 7

....with numerous others in their references. In this paper, we conduct numerical experiments for a nonoverlapping domain decomposition method for elliptic problems, selfadjoint or not, with Dirichlet boundary conditions. Our domain decomposition procedure is similar to the one considered elsewhere[19], but here we introduce underrelaxation on the interface of subdomains and an averaging mechanism at cross points of subdomains to ensure convergence. Each full iteration of the domain decomposition procedure contains a Dirichlet sweep, in which we solve Dirichlet subdomain problems, and a Neumann ....

....of iterations is getting slightly larger when the grid gets finer. The number of iterations required for the process to stop depends on the condition number of the problem and is different for different examples. Also, note that independence of the number of iterations upon the grid size was proved[19] only for strip domain decompositions. Table 5: Numerical results with 5 2 5 subdomains: the number of iterations needed to satisfy Eq. 13) Example 1 Example 2 Example 3 Example 4 Grid = 1 40 2 1 40 58 19 68 50 Grid = 1 80 2 1 80 70 25 82 74 12 Table 6: Numerical results with ....

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D. Q. Yang, A parallel iterative nonoverlapping domain decomposition method with interface relaxation for elliptic problems, in preparation. 15

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D. Q. Yang, A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems, IMA J. Numer. Anal., to appear.

....to ensure the convergence. Examples are Funaro, Quarteroni, and Zanolli [19] Marini and Quarteroni [32, 33] Lions [31] Despr es [14] Douglas, Paes Leme, Roberts, and Wang [15] Kim [23] Quarteroni [34] Le Tallec and Tidriri [25] Benamou and Despres [3] Engquist and Zhao [18] and Yang [40, 41, 42]. In these methods, Robin or alternating DirichletNeumann interface conditions are applied. These methods are especially desirable for the so called interface or transmission problems that have different characteristics on different regions of the domain. In Le Tallec and Tidriri [25] the ....

....at each iteration level. Indeed, this algorithm is motivated by the ones proposed by Funaro, Quarteroni, and Zanolli [19] Marini and Quarteroni [32, 33] Lions [31] Despr es [14] Rice, Vavalis, and Yang [36] Quarteroni [34] Benamou and Despres [3] Douglas and Yang [16] and Yang [40, 42, 41], where regular problems with = 0 and j = 0 were considered. Note that the scheme (2.6) 2.9) is a formal idea and its real meaning should be understood in variational forms, which will be given in the next section. The reason is that u k A j Gamma is not well defined for u 2 H ....

D. Q. Yang, A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems, IMA J. Numer. Anal. 16(1996) 75-91.

....to ensure and accelerate the convergence of the iterative procedure. This algorithm is motivated by the ones proposed by Funaro, Quarteroni, and Zanolli [4] Marini and Quarteroni [7, 8] Lions [6] Despr es [2] Rice, Vavalis, and Yang [10] Quarteroni [9] Douglas and Yang [3] and Yang [11, 12], where regular problems with = 0 and j = 0 were considered. This algorithm is different from others in that Dirichlet and Neumann subdomain problems were solved at the same iteration levels but on different subdomains in [4, 7, 8, 11] and Robin subdomain problems were solved in [6, 2, 12] For ....

....and Yang [10] Quarteroni [9] Douglas and Yang [3] and Yang [11, 12] where regular problems with = 0 and j = 0 were considered. This algorithm is different from others in that Dirichlet and Neumann subdomain problems were solved at the same iteration levels but on different subdomains in [4, 7, 8, 11] and Robin subdomain problems were solved in [6, 2, 12] For problems with continuous solution and coefficients, this algorithm reduces to [10, 3] where convergence results for general coefficients were not provided. In [13] a detailed analysis for the algorithm will be made at the differential ....

D. Q. Yang, A parallel iterative nonoverlapping domain decomposition procedure for elliptic problems, IMA J. Numer. Anal. 16 (1996), 75--91.

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