The postdoctoral associate will work with researchers in Rensselaer's Scientific Computation Research Center (SCOREC) on the development of parallel adaptive partition of unity methods. These methods represent a new class of discretization techniques for solving partial differential equations, which do not require the traditional grids of disjoint, non-overlapping cells, as used in typical finite element, finite difference or finite volume methods. However, they do require a discretization of the domain from which an appropriate set of partition cells interact with themselves and the domain boundary. Key to the construction of the partitions is determining how the partition cells interact with themselves and the domain boundary. The ultimate effectiveness of these methods will depend on efficient construction and control of the partitions. Building on expertise on the development of scalable parallel algorithms for automated adaptive finite element analysis over general three-dimensional domains, the proposed research program will consider alternative methodologies for the automatic construction and adaptive solution, in parallel, of partition of unity discretizations for geometric domains defined in terms of solid models. Specific areas of development include: (i) examination of alternative methods to use distributed octree structures on parallel computers to define basic partitions over the domain of the octree, (ii) effective techniques to perform the required integrations over partition of unity cells, with particular concern for integrating those cells that intersect boundaries of the domain, and (iii) develop parallel adaptive partition of unity solution methodologies
