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.