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 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

A fourth-order finite volume embedded boundary (EB) method is presented for the unsteady Stokes equations. The algorithm represents complex geometries on a Cartesian grid using EB, employing a technique to mitigate the ``small cut-cell"" problem without mesh modifications, cell merging, or state redistribution. Spatial discretizations are based on a weighted least-squares technique that has been extended to fourth-order operators and boundary conditions, including an approximate projection to enforce the divergence-free constraint. Solutions are advanced in time using a fourth-order additive implicit-explicit Runge-Kutta method, with the viscous and source terms treated implicitly and explicitly, respectively. Formal accuracy of the method is demonstrated with several grid convergence studies, and results are shown for an application with a complex bio-inspired material. The developed method achieves fourth-order accuracy and is stable despite the pervasive small cells arising from complex geometries.

Over the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, providing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available. The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, their design trade-offs and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah.