preconditioner for $u''(x)=sin(x)$

I am interested in finding preconditioner to solve the problem for one dimensional problem $u”(x)=sin(x), u(0)=u(1)=0$ using Dirichlet-Neumann method.

The preconditioner $M$ coming from Dirichlet-Neumann method is $theta^{-1} S_2$, where $theta$ is the relaxation parameter use in Dirichlet-Neumann method with $0<theta<1$ and $S_2= A_{GammaGamma}^{(2)}-A_{Gamma I}^{(2)}(A_{II}^{(2)})^{-1}A_{IGamma}^{(2)}.$

Here I use finite difference method for discretisation and take seven grid points ${ 1,2,…,7}$ with $u(0)=u_1=0, u(1)=u_7=0$. To use Dirichlet-Neumann method, I divide my domain to two non overlapping subdomain with node points from ${1,2,3,4}$ in $Omega_1$ and ${4, 5,6,7}$ in $Omega_2$ and the interface node $Gamma={4}$. With this in hand I calculate $A_{GammaGamma}^{(2)}=pmatrix{-2},A_{II}^{(2)}=pmatrix{-2 &1 1&-2},A_{Gamma I}^{(2)}=pmatrix{2 &0},A_{IGamma}^{(2)}=pmatrix{1 &0}^t$

my first question is that am I correct in finding various submatrices written above.

If I am right in finding those matrix how am I going to extend the matrix $theta^{-1} S_2$ from order one-by-one to five-by-five.(as order for the monodomain matrix on seven grid points is five)

Thanking you.