We present a hybrid geometric-algebraic multigrid approach for
solving Poisson's equation on domains with complex geometries.
The discretization uses a novel fourth-order finite
volume cut cell representation to discretize the
Laplacian operator on a Cartesian mesh.
This representation is based on a weighted least-squares fit to a
which is used to provide a conservative and accurate framework for
the multi-resolution discretization, despite the presence of cut cells.
We use geometric multigrid coarsening with an algebraic multigrid
bottom solver, so that the memory overhead of algebraic coarsening
is avoided until the geometry becomes under-resolved.
With tuning, the hybrid approach has the simplicity
of geometric multigrid while still retaining the
robustness of algebraic multigrid.
We investigate at what coarse level the transition should occur,
and how the order of accuracy of the prolongation operator affects
multigrid convergence rates.
We also present some converged
solutions as examples of how the use of adaptivity and
a cell connectivity graph can affect performance