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.


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 :-)

  • $\begingroup$ I get your point. But it would be very help full for me if you give some light on the above posted DN preconditioner. $\endgroup$
    – 420
    Dec 11 '20 at 4:38

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