Octree-based Finite Element Method for electromagnetic scattering problems

Currently Finite Element Methods (FEM) fall into one of two categories: FEMs based on structured meshes, and those based on unstructured meshes. Work by Hill, Farle, et. al has explored nonconformal unstructured meshes to aid in the application of adaptive mesh refinement (AMR) and geometric multigrid methods on tetrahedral meshes. Demkowicz, Kurtz, et. al have worked with nonconformal hexahedral meshes. This work proposes an octree mesh based FEM which enjoys fast, robust meshing, as well as simplified matrix assembly. There is the potential to reap large benefits when applied to AMR and multigrid methods, since the overhead of local refinements on unstructured meshes is avoided, and projections between differing levels of mesh refinement are simplified. The stair-casing prevalent in structured meshes, while still present, is mitigated by the nonconformal nature of the octree mesh.