Unsymmetric conjugate gradient methods and sparse direct methods in finite element flow simulation. (English) Zbl 0697.76039

Summary: A series of numerical experiments on the Cray XMP/48 and on the Cray 2 investigate the robustness and economy of direct and unsymmetric conjugate gradient (CG) type methods for the solution of matrix systems arising from a 3D FEM discretization of fluid flow problems. Computations on a Boussinesq flow model problem with either ILU preconditioned or unpreconditioned on symmetric CG methods are presented. Such experiments seem to indicate that the unpreconditioned BICG method is robust for moderately nonlinear incompressible Navier-Stokes FEM discretizations and that the ILU preconditioned BICG method is very robust and more economic than an unsymmetric frontal solver when the generous memory of the Cray 2 is exploited to store both the matrix and its preconditioner. We cover some of the programming aspects of direct and iterative methods on a supercomputer and find that direct methods have advantages: the crucial CPU-consuming area of code is compact but overwhelming, and its percentage of total CPU usage is independent of the spectral properties of the matrix involved. An optimal implementation of the unsymmetric CG method is more difficult because its work is related to the spectral distribution of the matrix considered and because there is no single portion of the code that overwhelmingly dominates the CPU usuage.


76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI


[1] The Finite Element Method, 3rd edn, McGraw-Hill, New York, 1977.
[2] and , Finite Element Solution of Boundary Value Problems, Academic Press, New York, 1984.
[3] Peano, Comput. Math. Appl. 2 pp 211– (1976)
[4] and , ’Adaptive refinement, error estimates, multigrid solution and hierarchic finite element method concepts’, Chap. 2 of: Accuracy estimates and adaptive refinements in finite element computations (ARFEC), pp. 25-60, , , and (Eds.) John Wiley amp; Sons, 1986.
[5] ’New solution procedures for linear and non-linear finite element analysis’, in (ed.), MAFELAP 1984, Academic Press, New York, 1985, pp. 49-81.
[6] Hughes, Comput. Methods Appl. Mech. Eng. 59 pp 85– (1986)
[7] Babuška, Numer. Methods 20 pp 179– (1973)
[8] Brezzi, RAIRO 8 pp 129– (1974)
[9] and , ’Conjugate gradient-like algorithms solving nonsymmetric linear systems’, Yale Research Report YALEU/DCS/RR-283, 1983.
[10] and , ’Multigrid and conjugate gradient methods’, in and (eds.), Multigrid Methods for Integral and Differential Equations, 1985, IMA Conference Series, no. 3, Clarendon Press, Oxford, 1985.
[11] ’Iterative methods for problems in numerical analysis’, D. Phil. Thesis, Numerical Analysis Group, Oxford University Computing Laboratory, 1978.
[12] ’Krylov subspace methods on supercomputers’, Research Institute for Advanced Computer Science, NASA Ames Research Center, Technical Report 88, 40, 1988.
[13] ’Iterative methods for large sparse systems of equations’, D. Phil. Thesis, Numerical Analysis Group, Oxford University Computing Laboratory, 1982.
[14] personal communication.
[15] personal communication.
[16] Hood, Int. j. numer. methods eng. 10 pp 379– (1978)
[17] Irons, Int. J. numer. methods eng. 2 pp 5– (1970)
[18] ’Conjugate gradient methods for indefinite systems’, in (ed.), Proc. Dundee Conf. on Numerical Analysis, Lecture Notes in Mathematics, Vol. 506, Springer, Berlin, pp. 73-89, 1976.
[19] ’MA32-A package for solving sparse unsymmetric matrices using the frontal method’, Harwell Report AERE R-10079, HMSO, London, 1981.
[20] , and , ’Laser doppler measurements of laminar and turbulent flow in a pipe bend’, NASA Contract Report 3551, Contract NASW-3258, May 1982.
[21] ’Numerical techniques for. simulation of three dimensional swirling flow’, Ph.D. Thesis, Department of Civil Engineering, Univerisity College of Swansea, 1988.
[22] ’Finite element computation of flows through bends’, European Research Community on Flow and Combustion (ERCOFTAC) Bulletin, No. III, July 1989.
[23] ’Preconditioning of iterative methods for linearised or linear systems’, D. Phil. Thesis, Numerical Analysis Group, Oxford University Computing Laboratory, in preparation.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.