Finite/Spectral Element Navier-Stokes Methods on Vector Hypercubes and Geometry-Defining Processor Reconfigurable Lattices

Abstract

In this paper we present a high-efficiency medium-grained parallel spectral (high-order finite) element method for numerical solution of incompressible fluid flow problems in general domains. The method is based upon: naturally concurrent iterative procedures; geometry-based distribution of work amongst processors; nearest-neighbor sparsity and high-order substructuring for minimum communication; general locally-structured/globally-unstructured parallel constructs; and efficient embedding of vector reduction operations for inner product and norm calculations. A detailed analysis is presented for the computational complexity of the method on a model algorithm-native distributed-memory parallel processor, and a comparison is given of the communication requirements of high-order spectral element methods and low-order finite element substructure techniques.The parallel spectral element method is implemented on two particular distributed-memory architecture/hardware realizations. The first system considered is the fast, general-purpose Intel vector hypercube. The generality, high efficiency, and good absolute performance of the spectral element-Intel hypercube algorithm-architecture coupling is demonstrated by the solution of several complex-geometry Stokes and unsteady Navier-Stokes problems; serial-supercomputer speeds are obtained at a fraction of serial-supercomputer cost. The second system considered is an experimental special-purpose architecture for partial differential equations, reconfigurable-lattice Geometry-Defining Processors (GDPs). The reconfigurable-lattice GDP system is a scalable, cost-efficient architecture that is functionally equivalent to the optimal algorithm- native parallel processor by virtue of geometry-based reconfigurability and a specialized bus structure. The results presented here for GDPs are part real (hardware) and part (Intelhypercube) emulated.KeywordsHamiltonian PathSpectral ElementDomain Decomposition MethodParallel ProcessorSpectral Element MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.