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

Question

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.

Answer 1

In general, the appropriate preconditioners for elliptic problems such as yours are multigrid methods. In this 1d case, however, the simplest discretizations lead to tri-diagonal matrices and in that case the Thomas algorithm can be used to solve the problem directly without too much trouble. So you don't even need a preconditioner if you use a three-point stencil. If you have a higher-order discretization, one can generate a a two-level scheme whereby you first move from the higher-order discretization to a three-point stencil, which you then solve exactly using the Thomas algorithm.

Using a domain-decomposition method seems like an inefficient approach, though of course it is also harmless: Solving such simple 1d problems is without much challenge on today's computers :-)