The four point explicit decoupled group (EDG) Method: a fast poisson solver
01 Jan 1991-International Journal of Computer Mathematics (Gordon and Breach Science Publishers)-Vol. 38, pp 61-70
TL;DR: This paper introduces a four point explicit decoupled group (EDG) iterative method as a new Poisson solver and is shown to be very much faster compared to existing explicit group (EG) methods.
Abstract: The aim of this paper is to introduce a four point explicit decoupled group (EDG) iterative method as a new Poisson solver. The method is shown to be very much faster compared to existing explicit group (EG) methods due to D. J. Evans and M. J. Biggins (1982) and W. Yousif and D. J. Evans (1985). Some numerical experiments are included to confirm our recommendation.
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TL;DR: A four points modified explicit group method for solving a two dimensional Poisson equation with Dirichlet boundary condition is introduced and is shown to be superior compared to the existing four points–explicit group and explicit decoupled group methods.
Abstract: In this paper, wc introduce a four points modified explicit group method for solving a two dimensional Poisson equation with Dirichlet boundary condition. The method is shown to be superior compared to the existing four points–explicit group and explicit decoupled group methods due to D. J. Evans and M. J. Biggins (1982) and A. R. Abdullah (1991), respectively, Some experiment results of the test problem are given in order to confirm our claim.
118 citations
TL;DR: This paper extends the 4-point explicit de-coupled group (EDG) iterative method to the 6 and 9-poinl EDG methods for the solution of elliptic partial differential equations and shows graphically the technique of implementing the new grouping.
Abstract: In this paper we extend the 4-point explicit de-coupled group (EDG) iterative method, Abdullah (1991), to the 6 and 9-poinl EDG methods for the solution of elliptic partial differential equations. We will show graphically the technique of implementing the new grouping. Performance results for the algorithms are presented and a comparison with the 4-point scheme confirm the new groups to be computationally superior. Further, the implementations of the parallel 4, 6 and 9-point EDG methods on the Sequent Balance 8000 multiprocessor arc discussed and results from experiments performed are presented.
89 citations
TL;DR: The rotated four-point explicit decoupled group (EDG) iterative method for solving two dimensional parabolic PDE is introduced and its performance is compared with some methods of natural ordering.
Abstract: In this paper, the rotated four-point explicit decoupled group (EDG) iterative method for solving two dimensional parabolic PDE is introduced. Its performance is compared with some methods of natural ordering.
77 citations
01 Jan 2004
TL;DR: In this article, the authors apply the Half-sweep Iterative Alternating Decomposition Explicit (HSIADE) method for solving one-dimensional diffusion problems and derive the formulation of the HSIADE method.
Abstract: The primary goal of this paper is to apply the Half-Sweep Iterative Alternating Decomposition Explicit (HSIADE) method for solving one-dimensional diffusion problems. The formulation of the HSIADE method is also derived. Some numerical experiments are conducted that to verify the HSIADE method is more efficient than the Full-Sweep method.
63 citations
16 Dec 2004
TL;DR: The primary goal of this paper is to apply the Half-Sweep Iterative Alternating Decomposition Explicit (HSIADE) method for solving one-dimensional diffusion problems.
Abstract: The primary goal of this paper is to apply the Half-Sweep Iterative Alternating Decomposition Explicit (HSIADE) method for solving one-dimensional diffusion problems. The formulation of the HSIADE method is also derived. Some numerical experiments are conducted that to verify the HSIADE method is more efficient than the Full-Sweep method.
62 citations
Cites methods from "The four point explicit decoupled g..."
...Then the approximated solution at remaining points can be computed directly, see Abdullah [ 1 ], Ibrahim & Abdullah [4], Yousif & Evans [6]....
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...The half-sweep iterative method is introduced by Abdullah [ 1 ] via the Explicit Decoupled Group (EDG) iterative method to solve two-dimensional Poisson equations....
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01 Jan 1971
TL;DR: The ASM preconditioner B is characterized by three parameters: C0, ρ(E) , and ω , which enter via assumptions on the subspaces Vi and the bilinear forms ai(·, ·) (the approximate local problems).
Abstract: theory for ASM. In the following we characterize the ASM preconditioner B by three parameters: C0 , ρ(E) , and ω , which enter via assumptions on the subspaces Vi and the bilinear forms ai(·, ·) (the approximate local problems). Assumption 14.6 (stable decomposition) There exists a constant C0 > 0 such that every u ∈ V admits a decomposition u = ∑N i=0 ui with ui ∈ Vi such that N ∑ i=0 ai(ui, ui) ≤ C 0 a(u, u) (14.29) Assumption 14.7 (strengthened Cauchy-Schwarz inequality) For i, j = 1 . . . N , let Ei,j = Ej,i ∈ [0, 1] be defined by the inequalities |a(ui, uj)| ≤ Ei,j a(ui, ui) a(uj, uj) ∀ ui ∈ Vi, uj ∈ Vj (14.30) By ρ(E) we denote the spectral radius of the symmetric matrix E = (Ei,j) ∈ RN×N . The particular assumption is that we have a nontrivial bound for ρ(E) to our disposal. Note that due to Ei,j ≤ 1 (Cauchy-Schwarz inequality), the trivial bound ρ(E) = ∥E∥2 ≤ √ ∥E∥1 ∥E∥∞ ≤ N always holds; for particular Schwarz methods one usually aims at bounds for ρ(E) which are independent of N . Ed. 2011 Iterative Solution of Large Linear Systems 14.2 Additive Schwarz methods (ASM) 159 Assumption 14.8 (local stability) There exists ω > 0 such that for all i = 1 . . . N : a(ui, ui) ≤ ω ai(ui, ui) ∀ ui ∈ Vi (14.31) Remark 14.9 The space V0 is not included in the definition of E ; as we will see below, this space is allowed to play a special role. Ei,j = 0 implies that the spaces Vi and Vj are orthogonal (in the a(·, ·) inner product). We will see below that small ρ(E) is desirable. We will also see below that a small C0 is desirable. The parameter ω represents a one-sided measure of the approximation properties of the approximate solvers ai . If the local solver is of (exact) Galerkin type, i.e, ai(u, v) ≡ a(u, v) for u, v ∈ Vi , then ω = 1 . However, this does not necessarily imply that Assumptions 14.6 and 14.7 are satisfied. Lemma 14.10 (P. L. Lions) Let PASM be defined by (14.23) resp. (14.24). Then, under Assumption 14.6, (i) PASM : V → V is a bijection, and a(u, u) ≤ C 0 a(PASM u, u) ∀ u ∈ V (14.32) (ii) Characterization of b(u, u) : b(u, u) = a(P−1 ASM u, u) = min { N ∑ i=0 ai(ui, ui) : u = N ∑ i=0 ui, ui ∈ Vi } (14.33) Proof: We make use of the fundamental identity (14.27) and Cauchy-Schwarz inequalites. Proof of (i): Let u ∈ V and u = ∑ i ui be a decomposition of the type guaranteed by Assumption 14.6. Then: a(u, u) = a(u, ∑ i ui) = ∑ i a(u, ui) = ∑ i ai(Pi u, ui) ≤ ∑ i √ ai(Pi u, Pi u) ai(ui, ui) = ∑ i √ a(u, Pi u) ai(ui, ui) ≤ √∑ i a(u, Pi u) √∑ i ai(ui, ui) = √ a(u, PASM u) √∑ i ai(ui, ui) ≤ √ a(u, PASM u)C0 √ a(u, u) This implies the estimate (14.32). In particular, it follows that PASM is injective, because with (14.32), PASM u = 0 implies a(u, u) = 0 , hence u = 0 . Due to finite dimension, we conclude that PASM is bijective. Proof of (ii): We first show that the minimum on the right-hand side of (14.33) cannot be smaller than a(P−1 ASM u, u) . To this end, we consider an arbitrary decomposition u = ∑ i ui with ui ∈ Vi and estimate a(P−1 ASM u, u) = ∑ i a(P −1 ASM u, ui) = ∑ i ai(PiP −1 ASM u, ui) ≤ √∑ i ai(PiP −1 ASM u, PiP −1 ASM u) √∑ i ai(ui, ui) = √∑ i a(P −1 ASM u, PiP −1 ASM u) √∑ i ai(ui, ui) = √ a(P−1 ASM u, u) √∑ i ai(ui, ui) In order to see that a(P−1 ASM u, u) is indeed the minimum of the right-hand side of (14.33), we define ui = PiP −1 ASM u . Obviously, ui ∈ Vi and ∑ i ui = u . Furthermore, ∑ i ai(ui, ui) = ∑ i ai(PiP −1 ASM u, PiP −1 ASM u) = ∑ i a(P −1 ASM u, PiP −1 ASM u) = a(P−1 ASM u, ∑ i PiP −1 ASM u) = a(P −1 ASM u, u) This concludes the proof. Iterative Solution of Large Linear Systems Ed. 2011 160 14 SUBSTRUCTURING METHODS The matrix P ′ ASM = B −1A from (14.23) is the matrix representation of the operator PASM . Since PASM is self-adjoint in the A -inner product (see Lemma 14.2), we can estimate the smallest and the largest eigenvalue of B−1A by: λmin(B −1A) = inf 0 ̸=u ∈V a(PASM u, u) a(u, u) , λmax(B −1A) = sup 0 ̸=u ∈V a(PASM u, u) a(u, u) (14.34) Lemma 14.10, (i) in conjunction with Assumption 14.6 readily yields λmin(B −1A) ≥ 1 C 0 An upper bound for λmax(B −1A) is obtained with the help of the following lemma. Lemma 14.11 Under Assumptions 14.7 and 14.8 we have ∥Pi∥A ≤ ω, i = 0 . . . N (14.35) a(PASM u, u) ≤ ω (1 + ρ(E)) a(u, u) for all u ∈ V (14.36) Proof: Again we make use of identity (14.27). We start with the proof of (14.35): From Assumption 14.8, (14.31) we infer for all u ∈ V : ∥Pi u∥2A = a(Pi u, Pi u) ≤ ω ai(Pi u, Pi u) = ω a(u, Pi u) ≤ ω ∥u∥A ∥Pi u∥A which implies (14.35). For the proof of (14.36), we observe that the space V0 is assumed to play a special role. We define
2,527 citations
TL;DR: This paper focuses on iterative methods for solving sparse linear systems derived from the discretisation of self-adjoint elliptic partial differential equations by finite difference/element techniques.
58 citations
TL;DR: A new explicit 4-pint block over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the discretization of self-adjoint elliptic partial differential equations.
Abstract: In this paper, a new explicit 4-pint block over-relaxation scheme is presented for the numerical solution of the sparse linear systems derived from the discretization of self-adjoint elliptic partial differential equations. A comparison with the implicit line and 2-line block SOR schemes for the model problem shows the new technique to be competitive
25 citations