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  • Numerical Methods  (3)
  • 1
    Electronic Resource
    Electronic Resource
    An organization of sparse gauss elimination for solving partial differntial equations on distributed memory machines (1993)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 9 (1993), S. 175-189 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A computational arrangement of Gauss elimination is presented for solving sparse, nonsymmetric linear systems arising from partial differential equation problems. It is particularly targeted for use on distributed memory message passing multiprocessor computers and it is presented and analyzed in this context. The objective of the algorithm is to exploit the sparsity (i.e., reducing computation, communication, and memory requirements) and to optimize the data structure manipulation overhead. The algorithm is based on the nested dissection approach, which starts with a large set of very sparse, completely independent subsystems and progresses in stages to a single, nearly dense system at the last stage. The computational efforts of each stage are roughly equal (almost exactly equal for model problems), yet the data structures appropriate for the first and last stages are quite different. Thus we use different types of data structures and algorithm components at different stages of the solution. The new organization is a combination of previous techniques including nested dissection, implicit block factorization, domain decomposition, fan-in, fan-out, up-looking, down-looking, and dynamic data structures. © 1993 John Wiley & Sons, Inc.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690090206
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  • 2
    Electronic Resource
    Electronic Resource
    An experimental performance analysis for the rate of convergence of collocation on general domains (1989)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 5 (1989), S. 45-52 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper presents an experimental performance analysis for the rate of convergence of collocation on general domains using a bicubic Hermite basis. Twenty domains are selected for the experiment from the population of PDEs on nonrectangular domains found in Realistic PDE Solutions for Non-rectangular Domains (C. J. Ribbens and J. R. Rice, CSD-TR 639, Department of Computer Sciences, Purdue University, 1986), including one rectangle for comparison. The result shows that the convergence of the ELLPACK module COLLOCATION behaves as O(h4) on all 19 of the nonrectangular domains. This set includes a large variety of nonrectangular domains (only two have reentrant corners). We conclude that, with very high probability, this collocation module has O(h4) convergence on general domains.The experiment is made by using the Performance Evaluation System (PES) of ELLPACK, which includes the population of PDEs on nonrectangular domains. Several performance analysis tools are used to analyze the rate of convergence, the most informative is a visual examination of the convergence behavior.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690050105
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  • 3
    Electronic Resource
    Electronic Resource
    The performance of numerical methods for elliptic problems with mixed boundary conditions (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 4 (1988), S. 347-361 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics ; Numerical Methods
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider solving linear, second order, elliptic partial differential equations with boundary conditions of types Dirichlet (DIR), mixed (MIX), and nearly Neumann (Neu) by using software modules that implement five numerical methods (one finite element and four finite differences). They represent both the new generation of improved methods and the traditional ones; they are: Hermite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (5P), ordinary finite differences with Dyaknov iteration (DY), DY with Richardson extrapolation to achieve fourth order convergence (D4), and ordinary finite differences with multigrid iteration (MG). We carry out a performance evaluation in which we measure the grid size and the computer time needed to achieve three significant digits of accuracy in the solution. We compute the changes in these two measures as we change boundary condition types from DIR to MIX and MIX to NEU and then test the following hypotheses: (i) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions; (ii) finite element collocation (HC) is least affected; (iii) the fourth order modules (HC and D4) are less affected than the other second order modules; and (iv) the traditional 5-point finite differences (5P) are most affected. We establish these hypotheses with high levels of confidence by using several sample problems. The most significant conclusion is that a high order collocation method is preferred for problems with general operators and derivatives in the boundary conditions. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time performance: MG (best), HC and D4, DY, and 5P (worst).
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/num.1690040407
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