An efficient domain‐decomposition pseudo‐spectral method for solving elliptic differential equations

Mai‐Duy, N.; Tran-Cong, T.

doi:10.1002/cnm.987

Cited by 9 publications

(3 citation statements)

References 10 publications

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“…There are some new developments for handling various types of partial differential equations. The DD methods developed for elliptic problems include pseudo-spectral method [1], additive/multiplicative Schwarz methods [2,3] and the finite element tearing and interconnecting (FETI) method [4,5]. For parabolic problems, the DD methods include an explicit/implicit Galerkin method [6], a stabilized explicit Lagrange multiplier method [7], Dawson's method [8,9], the explicit prediction and implicit correction (EPIC) method [10], the stabilized explicit/implicit domain decomposition (SEIDD) method [11], the alternating explicit/implicit domain decomposition (AEIDD) method [12], the implicit prediction and implicit correction (IPIC) method [13] and the modified implicit prediction (MIP) method [14,15].…”

Section: Introductionmentioning

confidence: 99%

Second‐order treatment of the interface of domain decomposition method for parabolic problems

Jun

Mai

2010

Numerical Linear Algebra App

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Estimations for the values on interface lines are necessary in a domain decomposition method. However, the accuracy of the estimations is of the first order for most of unconditionally stable domain decomposition schemes. In this paper, a second order of accuracy for the estimations on interface lines is presented. With the new scheme, the optimal number of decomposed subdomains has been observed and proposed. Moreover when the SOR iterative method is used, the optimal over-relaxation parameter is also studied. A matrix A of order N has Property A if there exist two disjoint subsets S 1 and S 2 of W , the set of the first N positive integers, such that S 1 + S 2 = W and such that if i = j and if either a i, j = 0 or a j,i = 0, then i ∈ S 1 and j ∈ S 2 or else i ∈ S 2 and j ∈ S 1 .

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Section: Introductionmentioning

confidence: 99%

Second‐order treatment of the interface of domain decomposition method for parabolic problems

Jun

Mai

2010

Numerical Linear Algebra App

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“…However, the disadvantage of the implicit time stepping is that it is a unconditionally stable scheme and it involves solving a large sparse linear system for each time step. In order to obtain the solution efficiently, domain decomposition methods developed for elliptic problems can be applied, such as pseudo-spectral method [1], preconditioning [2][3][4], additive/multiplicative Schwarz methods [5][6][7][8], methods based on the Lagrange multiplier technique [9], the finite element tearing and interconnecting (FETI) method [10], and many variants of the FETI method [11,12]. Recently, applications of domain decomposition method for special structure problems have also been developed, see [13,14].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of the rectangular domain decomposition method

Jun

Mai

2008

Commun. Numer. Meth. Engng.

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SUMMARYWhen solving parabolic partial differential equations using finite difference non-overlapping domain decomposition methods, one often uses the stripwise decomposition of spatial domain and it can be extended to the rectangular decomposition without further analysis. In this paper, we analyze the rectangular decomposition when the modified implicit prediction (MIP) algorithm is used. We show that the performance of the rectangular decomposition and the stripwise decomposition is different. We compare spectral radius, maximum error, efficiency, and total operations of the rectangular and the stripwise decompositions. We investigate the accuracy of the interface of the rectangular decomposition and the effects of the correction phase of the rectangular decomposition. Numerical experiments have been done in both two and three spatial dimensions and show that the rectangular decomposition is not better than the stripwise decomposition.

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“…When applying the integral collocation formulation for the solution of differential equations, with RBFs or Chebyshev polynomials, the constants of integration have been found to be very useful. They provide an alternative way, which is very effective, for the implementation of multiple boundary conditions [23][24][25] and also allow a higher-order smoothness of the approximate solution across the subdomain interfaces [26]. It will be shown that, in the context of fictitious-domain techniques, the constants of integration can be utilized for the purpose of imposing the prescribed conditions on the actual boundary.…”

Section: Introductionmentioning

confidence: 99%

An integral‐collocation‐based fictitious‐domain technique for solving elliptic problems

Mai‐Duy

See

Tran-Cong

2007

Commun. Numer. Meth. Engng.

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SUMMARYThis paper presents a new fictitious-domain technique for numerically solving elliptic second-order partial-differential equations (PDEs) in complex geometries. The proposed technique is based on the use of integral collocation schemes and Chebyshev polynomials. The boundary conditions on the actual boundary are implemented by means of integration constants. The method works for both Dirichlet and Neumann boundary conditions. Several test problems are considered to verify the technique. Numerical results show that the present method yields spectral accuracy for smooth (analytic) problems.

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An efficient domain‐decomposition pseudo‐spectral method for solving elliptic differential equations

Cited by 9 publications

References 10 publications

Second‐order treatment of the interface of domain decomposition method for parabolic problems

Second‐order treatment of the interface of domain decomposition method for parabolic problems

Numerical analysis of the rectangular domain decomposition method

An integral‐collocation‐based fictitious‐domain technique for solving elliptic problems

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