I am trying to figure out a way to bound the following expression $\| (I - \alpha^2 D)^{-1} \| $, where:
# $\alpha>0$
# $I$ is the identity matrix $(n\times n)$
# $D$ is the matrix obtained by discretizing the Laplace operator on a uniform grid (with given mesh size $\Delta x$ ) with central second order finite difference scheme, with periodic B.C.'s
# the norm we use is the operator from $R^n$ to $R^n$, with $R^n$ endowed with the norm defined by $\| v \|^2 := \sum v_i^2 \Delta x$
If I know that $\| (I - D)^{-1} \| \leq 1$, how does $\alpha$ affect my results?
Thanks for any hint/help/reference!