Simultaneous refinement and coarsening for adaptive meshing

In the numerical simulation of the combustion process and microstructural evolution, we need to consider the adaptive meshing problem for a domain that has a moving boundary. During the simulation, the region ahead of the moving boundary needs to be refined (to satisfy stronger numerical conditions), and the submesh in the region behind the moving boundary should be coarsened (to reduce the mesh size). We present a unified scheme for simultaneously refining and coarsening a mesh. Our method uses sphere packings and guarantees that the resulting mesh is well-shaped and is within a constant factor of the optimal possible in the number of mesh elements. We also present several practical variations of our provably good algorithm.