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The objective of this paper is to propose communication procedures suitable for unstructured finite element solvers implemented on distributed-memory parallel computers such as the Connection Machine CM-5 system. First, a data-parallel implementation of the recursive spectral bisection (RSB) algorithm proposed by Pothen et al. is presented. The RSB algorithm is associated with a node renumbering scheme which improves data locality of reference. Two-step gather and scatter operations taking advantage of this data locality are then designed. These communication primitives make use of the indirect addressing capability of the CM-5 vector units to achieve high gather and scatter bandwidths. The performance of the proposed communication strategy is illustrated on large-scale three-dimensional fluid dynamics problems.

An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practical applications. Advanced multigrid topics, such as the use of algebraic multigrid methods, and the combination of multigrid techniques with adaptive meshing strategies are dealt with in subsequent sections. These represent current areas of research, and the unresolved issues are discussed. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to

A parallel implementation of an implicit finite element formulation for incompressible fluids on a distributed-memory massively parallel computer is presented. The dominant issue that distinguishes the implementation of finite element problems on distributed-memory computers from that on traditional shared-memory scalar or vector computers is the distribution of data (and hence workload) to the processors and the nonuniform memory hierarchy associated with the processors, particularly the nonuniform costs associated with on-processor and o#-processor memory references. Accessing data stored in a remote processor requires computing resources an order of magnitude greater than accessing data locally in a processor. This distribution of data motivates the development of alternatives to traditional algorithms and data structures designed for shared-memory computers, which must now account for distributed-memory architectures. Data structures as well as data decomposition and dat...

This survey paper assesses the status of compressible Euler and Navier-Stokes solvers on unstructured grids. Different spatial and temporal discretization options for steady and unsteady flows are discussed. The integration of these components into an overall framework to solve practical problems is addressed. Issues such as grid adaptation, higher order methods, hybrid discretizations and parallel computing are briefly discussed. Finally, some outstanding issues and future research directions are presented.

Efficient data motion is critical for high performance computing on distributed memory architectures. The value of some techniques for efficient data motion is illustrated by identifying generic communication primitives. Further, the efficiency of these primitives is demonstrated on three different applications using the finite element method for unstructured grids and sparse solvers with different communication requirements. For the applications presented, the techniques advocated reduced the communication times by a factor of between 1.5 -- 3. 1 Introduction The finite element method is a popular technique for solving boundary and initial value problems. Moderate sized engineering problems have been successfully simulated using this technique. The primary bottleneck for the simulation of large problems has been available computational resources. With the advent of massively parallel architectures, simulating significantly larger 1 Also affiliated with the Division of Applied Scien...

Contents 1 Summary 2 2 Governing equations 2 3 Spatial discretization methods 3 4 Steady state solution techniques 6 4.1 Explicit schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 4.1.1 Acceleration techniques : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 4.2 Implicit schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 4.2.1 Direct methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 4.2.2 Standard iterative methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 4.2.3 Line-implicit methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 4.2.4 Incomplete LU factorization methods : : : : : : : : : : : : : : : : : : : : : : 11 4.2.5 Advanced iterative methods : : : : : : : : : : : : : : : : : : : : : : : : : : : :<

In this paper we investigate the application of Krylov methods to compressible ows, and the e ect of implicit boundary conditions on the implicit solution of nonlinear problems. Two defect-correction procedures, namely, Approximate Factorization (AF) for structured grids, and ILU/GMRES for general grids are considered. Also, considered here, is Newton-Krylov matrix-free methods that we combine with the use of mixed discretization schemes in the implicitly de ned Jacobian and its preconditioner. Numerical experiments that show the performance of our approaches are then presented. This work was supported by the National Aeronautics and Space Administration under

We investigate in this paper the application of Schwarz-based algorithms to compressible flows. First, we study the combination of these methods with defect-correction procedures. We then study the effect on the Schwarz-based methods of replacing the explicit treatment of the boundary conditions by an implicit one. In the last part of this paper we study the combination of these methods with Newton-Krylov matrixfree methods. Numerical experiments that show the performance of our approaches are then presented.

Massively parallel processors introduces new demands on software systems with respect to performance, scalability, robustness and portability. The increased complexity of the memory systems and the increased range of problem sizes for which a given piece of software is used, poses serious challenges to software developers. The Connection Machine Scientific Software Library, CMSSL, uses several novel techniques to meet these challenges. The CMSSL contains routines for managing the data distribution and provides data distribution independent functionality. High performance is achieved through careful scheduling of operations and data motion, and through the automatic selection of algorithms at run--time. We discuss some of the techniques used, and provide evidence that CMSSL has reached the goals of performance and scalability for an important set of applications. 1.1 INTRODUCTION The main reason for large scale parallelism is performance. In order for massively parallel ar...