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 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.