On the rate of convergence of the $k \times k$ block, $k$ -line iterative methods: $k \rightarrow \infty $

Autor: Seymour V. Parter, Chang-Ock Lee
Rok vydání: 1995
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Zdroj: Numerische Mathematik. 71:59-90
ISSN: 0945-3245
0029-599X
DOI: 10.1007/s002110050136
Popis: Some years ago there was a great interest in the asymptotic (as h → 0) rates of convergence of block iterative methods for elliptic difference equations ([V], [HV], [P1], [PS1], [PS2]). The k x k block iterative methods and the k-line iterative methods - for k a fixed finite integer - were analyzed in [BBP], [PS2], [P1], [P2]. These works dealt with rather general problems; general domains and general (variable coefficients, non-symmetric, etc.) elliptic operators which were definite (i.e. Reλ > 0 for all eigenvalues λ). The recent development of massively parallel computers and the associated interest in Domain Decomposition Methods ([M]) has led to an interest in the case where (1.1) k = k(h) → ∞ as h → 0. In this paper we limit ourselves to the model problem; the finite difference discretization of the Dirichlet problem for Poisson's equation in the unit square. That is: (1.2a) Δu = f in Ω = [0,1] x [0,1], (1.2b) u = 0 on ∂Ω We then obtain asymptotic formulae for the dominant eigenvalues p(k x k) and p(kL) of the Jacobi k x k block iterative method and the Jacobi k-Line iterative methods respectively. These estimates are given in the form (1.3) ρ 1 - μh 2 .
Databáze: OpenAIRE
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