A Hybrid Multigrid Algorithm for Poisson’s equation using an Adaptive, Fourth Order Treatment of Cut Cells

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 cell-averaged discretization, 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 underresolved. 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 in cases with underresolved geometries.