AMG-preconditioned CG — benchmark notes

The elliptic pressure system $A\mathbf{x} = \mathbf{b}$ dominates implicit reservoir simulation. Unpreconditioned CG converges in iterations $\propto \sqrt{\kappa(A)}$, and $\kappa(A)$ blows up with grid size and permeability contrast.

Algebraic multigrid (AMG) as a preconditioner flattens the spectrum, giving near mesh-independent convergence:

$$\|\mathbf{x}_k - \mathbf{x}^*\|_A \le 2\left(\frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1}\right)^{k}\|\mathbf{x}_0 - \mathbf{x}^*\|_A$$

with $\kappa$ the preconditioned condition number $\kappa(M^{-1}A)\ll\kappa(A)$.

Caution

Placeholder — benchmark numbers (iteration counts, wall time vs grid size) to be added. Runs on the HPC box.