TY - JOUR

T1 - Comparison of eigenvalue ratios in artificial boundary perturbation and Jacobi preconditioning for solving Poisson equation

AU - Yoon, Gangjoon

AU - Min, Chohong

N1 - Funding Information:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2009-0093827 ). G. Yoon was supported by National Institute for Mathematical Sciences (NIMS, A22200000 ).
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2017/11/15

Y1 - 2017/11/15

N2 - 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.

AB - 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.

KW - Artificial boundary perturbation

KW - Condition number

KW - Finite difference method

KW - Jacobi preconditioner

KW - Poisson equation

KW - Shortley–Weller

UR - http://www.scopus.com/inward/record.url?scp=85029114124&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2017.08.013

DO - 10.1016/j.jcp.2017.08.013

M3 - Article

AN - SCOPUS:85029114124

VL - 349

SP - 1

EP - 10

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -