Sharp bounds of the inverse matrices resulted from five-point stencil in solving Poisson equations

In this paper, we derive the least upper bound (in the infinity norm) and the greatest lower bound of a class of the inverse matrices resulted from the five-point stencil in solving the Poisson equations on the unit square. The obtained bounds are sharp and provide more accurate convergence estimati...

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Bibliographic Details
Published in:Linear algebra and its applications 2014-03, Vol.444, p.231-245
Main Authors: Xiao, Mingqing, Xu, Jianhong
Format: Article
Language:English
Subjects:
Five-point stencil
Infinity norm
Matrix inverse
Poisson equations
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Summary:In this paper, we derive the least upper bound (in the infinity norm) and the greatest lower bound of a class of the inverse matrices resulted from the five-point stencil in solving the Poisson equations on the unit square. The obtained bounds are sharp and provide more accurate convergence estimation than the current one in literature. Our approach is based on a matrix theoretic setting which can capture the characteristics of this type of matrices. As an application, we apply the result to the unbiased random walk in a unit square with an absorbing boundary and give the least upper bound of the mean first passage time for an inside particle to reach the boundary.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2013.11.029