Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 40A / ii / Solution

Let $D=-10I/(3h^2)$ be the diagonal part of $A$. The weighted Jacobi method isEquivalently it is $u^{(r+1)}=u^{(r)}+\omega D^{-1}(b-A{u}^{(r)})$. In grid form, all neighbours on the right are taken from the old iterate:The boundary values stay zero at every step. The iteration matrix is $T_{\omega }=I-\omega D^{-1}A$.