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Comparison of eigenvalue ratios in artificial boundary perturbation and Jacobi preconditioning for solving Poisson equation

Title
Comparison of eigenvalue ratios in artificial boundary perturbation and Jacobi preconditioning for solving Poisson equation
Authors
Yoon G.Min C.
Ewha Authors
민조홍
SCOPUS Author ID
민조홍scopus
Issue Date
2017
Journal Title
Journal of Computational Physics
ISSN
0021-9991JCR Link
Citation
vol. 349, pp. 1 - 10
Keywords
Artificial boundary perturbationCondition numberFinite difference methodJacobi preconditionerPoisson equationShortley–Weller
Publisher
Academic Press Inc.
Indexed
SCI; SCIE; SCOPUS WOS scopus
Abstract
The Shortley–Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O(1/(h⋅hmin) to O(h−3) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O(1/(h⋅hmin) to O(h−2), but also keeps the sharp second order convergence. © 2017 Elsevier Inc.
DOI
10.1016/j.jcp.2017.08.013
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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