# 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 $$S_2= A_{\Gamma\Gamma}^{(2)}-A_{\Gamma I}^{(2)}(A_{II}^{(2)})^{-1}A_{I\Gamma}^{(2)}.$$

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 $$A_{\Gamma\Gamma}^{(2)}=\pmatrix{-2},A_{II}^{(2)}=\pmatrix{-2 &1\\ 1&-2},A_{\Gamma I}^{(2)}=\pmatrix{2 &0},A_{I\Gamma}^{(2)}=\pmatrix{1 &0}^t$$

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.