I've the following problem.I've a sparse square Matrix $\bf M$. I can write $\bf M$ as: ${\mathbf M} = \begin{bmatrix}\mathbf A_{11} & \dots & \mathbf A_{1n} \\ \vdots & \ddots & \vdots \\ \mathbf A_{n1} & \dots & \mathbf A_{nn}\end{bmatrix}$ where each of the $\mathbf A_{ij}$ is a sparse (symmetric). It's like (but not exactly) the one we would get if we solve the Poisson equation in 2D (5 neighbours). I've following cases..
- When in total $\mathbf M$ is symmetric and positive definite
- When $\mathbf M$ is only symmetric but not positive definite.
Now my questions are: For case (1) and For case (2) separately: What kind of iterative method must I use to solve the linear system ${\mathbf M}\mathbf x=\mathbf b$? Currently I'm doing it with CG but I want to go for PCG. So can anyone comment on what kind of preconditioner is best for faster convergence suitable to the above problem? My only criterion is faster convergence.
Regards,