# AMLI-cycle multigrid for non-symmetric linear systems with M-matrices

## Nadir Bayramov

**Jan. 28, 2014, 12:45 p.m. S2 059**

We consider time-dependent convection-diffusion problem with natural Robin boundary conditions. An exponential fitting (edge-averaged) finite element scheme is applied to discretize this problem (in space).

Earlier we have derived uniform 1st order approximation property of this scheme. Here we focus on the solution of the resulting linear system at any time step (after backward Euler time discretization) which corresponds to convection-diffusion-reaction equation. We are trying to apply preconditioner based on non-linear AMLI-cycle multigrid, which has already been studied for elliptic problems. The open question is how to choose appropriate coarsening on each level. Since for the given problem the system matrix is an M-matrix, we can use an approach based on graph matching that leads to optimal convergence behavior for (symmetric) graph Laplacian. But the possibility of finding the optimal coarsening for our case and related convergence results are open questions and are being studied.