The purpose of this paper is to supply a summary of library subroutines and functions for parallel MIMD computers. The subroutines have been developed at the University of Chemnitz during a period of the last five years. In detail, they are concerned with vector operations, inter-processor communication and simple graphic output to workstations. One of the most valuable features is the machine-independence of the communication subroutines proposed in this paper for a hypercube topology of the parallel processors (excepting a kernel of only two primitive system-dependend operations). They were implemented and tested for different hardware and operating systems including transputer, nCube, KSR, PVM. The vector subroutines are optimized by the use of C language and enrolled loops (BLAS1-like). The paper includes hints for using the libraries with both Fortran and C programs.

This paper concerns the solution of plate bending problems in domains composed of rectangles. Domain decomposition (DD) is the basic tool used for both the parallelization of the conjugate gradient method and the construction of efficient parallel preconditioners. A so-called Dirich- let DD preconditioner for systems of linear equations arising from the fi- nite element approximation by non-conforming Adini elements is derived. It is based on the non-overlapping DD, a multilevel preconditioner for the Schur-complement and a fast, almost direct solution method for the Dirichlet problem in rectangular domains based on fast Fourier transform. Making use of Xu's theory of the auxiliary space method we construct an optimal preconditioner for plate problems discretized by conforming Bogner-Fox-Schmidt rectangles. Results of numerical experiments carried out on a multiprocessor sys- tem are given. For the test problems considered the number of iterations is bounded independent of the mesh sizes and independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requiresO(h^-2 ln h^-1 ln epsilon^-11) arithmetical operations per processor in order to solve the finite element equations with the relative accuracy epsilon.

The investigation of new parallel algorithms for MIMD computers requires some postprocessing facilities for quickly evaluating the behavior of those algorithms We present two kinds of visualization tool implementations for 2D and 3D finite element applications to be used on a parallel computer and a host workstation.

We present a MPI based software library for computing the fast Fourier transforms on massively parallel, distributed memory architectures. Similar to established transpose FFT algorithms, we propose a parallel FFT framework that is based on a combination of local FFTs, local data permutations and global data transpositions. This framework can be generalized to arbitrary multi-dimensional data and process meshes. All performance relevant building blocks can be implemented with the help of the FFTW software library. Therefore, our library offers great flexibility and portable performance. Likewise FFTW, we are able to compute FFTs of complex data, real data and even- or odd-symmetric real data. All the transforms can be performed completely in place. Furthermore, we propose an algorithm to calculate pruned FFTs more efficiently on distributed memory architectures. For example, we provide performance measurements of FFTs of size 512^3 and 1024^3 up to 262144 cores on a BlueGene/P architecture.

In the last years multilevel preconditioners like BPX became more and more popular for solving second-order elliptic finite element discretizations by iterative methods. P. Oswald has adapted these methods for discretizations of the fourth order biharmonic problem by rectangular conforming Bogner-Fox-Schmidt elements and nonconforming Adini elements and has derived optimal estimates for the condition numbers of the preconditioned linear systems. In this paper we generalize the results from Oswald to the construction of BPX and Multilevel Diagonal Scaling (MDS-BPX) preconditioners for the elasticity problem of thin smooth shells of arbitrary forms where we use Koiter's equations of equilibrium for an homogeneous and isotropic thin shell, clamped on a part of its boundary and loaded by a resultant on its middle surface. We use the two discretizations mentioned above and the preconditioned conjugate gradient method as iterative method. The parallelization concept is based on a non-overlapping domain decomposition data structure. We describe the implementations of the multilevel preconditioners. Finally, we show numerical results for some classes of shells like plates, cylinders, and hyperboloids.

The parallel version of precondition techniques is developed for matrices arising from the Galerkin boundary element method for two-dimensional domains with Dirichlet boundary conditions. Results were obtained for implementations on a transputer network as well as on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited for a MIMD computer. A comparison of numerical results for iterative and direct solution methods is presented and underlines the superiority of iterative methods for large systems.

The parallel version of precondition iterative techniques is developed for matrices arising from the panel boundary element method for three-dimensional simple connected domains with Dirichlet boundary conditions. Results were obtained on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited also in three-dimensional case for implementation on a MIMD computer and that they are much more efficient than usual direct solution techniques.

The parallel version of precondition iterative techniques is developed for matrices arising from the panel boundary element method for three-dimensional simple connected domains with Dirichlet boundary conditions. Results were obtained on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited also in three-dimensional case for implementation on a MIMD computer and that they are much more efficient than usual direct solution techniques.

In the paper, the parallelization of multi-grid methods for solving second-order elliptic boundary value problems in two-dimensional domains is discussed. The parallelization strategy is based on a non-overlapping domain decomposition data structure such that the algorithm is well-suited for an implementation on a parallel machine with MIMD architecture. For getting an algorithm with a good paral- lel performance it is necessary to have as few communication as possible between the processors. In our implementation, communication is only needed within the smoothing procedures and the coarse-grid solver. The interpolation and restriction procedures can be performed without any communication. New variants of smoothers of Gauss-Seidel type having the same communication cost as Jacobi smoothers are presented. For solving the coarse-grid systems iterative methods are proposed that are applied to the corresponding Schur complement system. Three numerical examples, namely a Poisson equation, a magnetic field problem, and a plane linear elasticity problem, demonstrate the efficiency of the parallel multi- grid algorithm.

The experimental program ¨SPC-PM Po 3D¨ is part of the ongoing research of the Chemnitz research group Scientific Parallel Computing (SPC) into finite element methods for problems over three dimensional domains. The package in its version 2.0 is documented in two manuals. The User's Manual provides an overview over the program, its capabilities, its installation, and handling. Moreover, test examples are explained. The aim of the Programmer's Manual is to provide a description of the algorithms and their realization. It is written for those who are interested in a deeper insight into the code, for example for improving and extending. In Version 2.0 the program can solve the Poisson equation and the Lam'e system of linear elasticity with in general mixed boundary conditions of Dirichlet and Neumann type. The domain $\Omega\subset\R^3$ can be an arbitrarily bounded polyhedron. The input is a coarse mesh, a description of the data and some control parameters. The program distributes the elements of the coarse mesh to the processors, refines the elements, generates the system of equations using linear or quadratic shape functions, solves this system and offers graphical tools to display the solution. Further, the behavior of the algorithms can be monitored: arithmetic and communication time is measured, the discretization error is measured, different preconditioners can be compared. We plan to extend the program in the next future by including a multigrid solver, an error estimator and adaptive mesh refinement, as well as the treatment of coupled thermo-elastic problems. The program has been developed for MIMD computers; it has been tested on Parsytec machines (GCPowerPlus-128 with Motorola Power PC601 processors and GCel-192 on transputer basis) and on workstation clusters using PVM. The special case of only one processor is included, that means the package can be compiled for single processor machines without any change in the source files.