An octree-based approach for fast elliptic solvers

We discuss an octree-based approach for the solution of elliptic partial differential equations, especially Poisson's equation and the convection diffusion equation. The discretization is derived from a starting discretization on a very fine octree grid. For the actual computation a discretization on a much coarser grid is generated by an accumulation process based on hierarchical transformation and partial elimination of unknowns. We also describe an efficient multigrid solver which takes advantage of the underlying octree structure. It is based on recursive substructuring of the domain and is very similar to the accumulation process. By adding additional unknowns to the coarse grids the resulting solver is robust even for the convection diffusion equation.