×

zbMATH — the first resource for mathematics

Performance of fully coupled domain decomposition preconditioners for finite element transport/reaction simulations. (English) Zbl 1087.76069
Summary: We describe an iterative linear system solution methodology used for parallel unstructured finite element simulation of strongly coupled fluid flow, heat transfer, and mass transfer with nonequilibrium chemical reactions. The nonlinear/linear iterative solution strategies are based on a fully coupled Newton solver with preconditioned Krylov subspace methods as the underlying linear iteration. Our discussion considers computational efficiency, robustness and a number of practical implementation issues. The evaluated preconditioners are based on additive Schwarz domain decomposition methods which are applicable for totally unstructured meshes. A number of different aspects of Schwarz schemes are considered including subdomain solves, use of overlap and the introduction of a coarse grid solve (a two-level scheme). As we will show, the proper choice among domain decomposition options is often critical to the efficiency of the overall solution scheme. For this comparison we use a particular spatial discretization of the governing transport/reaction partial differential equations (PDEs) based on a stabilized finite element formulation. Results are presented for a number of standard 2D and 3D computational fluid dynamics (CFD) benchmark problems and some large 3D flow, transport and reacting flow application problems.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
76V05 Reaction effects in flows
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
PDF BibTeX Cite
Full Text: DOI
References:
[1] Benzi, M., Preconditioning techniques for large linear systems: a survey, Jcp, 182, 418-477, (2002) · Zbl 1015.65018
[2] Brown, P.N.; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. sci stat. comp., 11, 3, 450-481, (1990) · Zbl 0708.65049
[3] X.-C, Cai, An additive Schwarz algorithm for nonselfadjoint elliptic equations, in: Tony F. Chan, Roland Glowinski, Jacques Periaux, Olof B. Widlund (Eds.), Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, 1989, pp. 232-244
[4] Cai, X.-C.; Gropp, W.D.; Keyes, D.E., Convergence rate estimate for a domain decomposition method, Num. math., 61, 2, 153-169, (1992)
[5] de Vahl Davis, G.; Jones, I.P., Natural convection in a square cavity: a comparison exercise, Int. J. num. meth. fluids, 3, 227-248, (1983) · Zbl 0538.76076
[6] Eisenstat, S.C.; Walker, H.F., Globally convergent inexact Newton methods, SIAM J. optim., 4, 2, 393-422, (1994) · Zbl 0814.65049
[7] Farhat, C.; Maman, N.; Brown, G.W., Mesh partitioning for implicit computations via iterative domain decomposition - impact and optimization of the subdomain aspect ratio, Int. J. num. meth. eng., 38, 6, 989-1000, (1995) · Zbl 0825.73780
[8] Ghia, U.; Ghia, K.N.; Shin, C.T., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. comput. phys., 48, 387-411, (1982) · Zbl 0511.76031
[9] Gropp, W.D.; Kaushik, D.K.; Keyes, D.E.; Smith, B.F., High-performance parallel implicit CFD, Parallel comput., 27, 337-362, (2001) · Zbl 0971.68191
[10] W.D. Gropp, B. Smith, Scalable, extensible, and portable numerical libraries, in: Proceedings of the Scalable Parallel Libraries Conference 87-93, IEEE, Los Alamitos, CA, 1994
[11] Hendrickson, B., Load balancing fictions, falsehoods and fallacies, Appl. math. model., 25, 2, 99-108, (2000) · Zbl 1076.65537
[12] Hendrickson, B.; Kolda, T.G., Graph partitioning models for parallel computing, Parallel comput., 26, 12, 1519-1534, (2000) · Zbl 0948.68130
[13] B. Hendrickson, R. Leland, The Chaco User’s Guide, Version 2.0., Sandia National Laboratories Technical Report, SAND95-2344, Albuquerque, NM, 1995
[14] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comp. meth. appl. mech. eng., 59, 85-99, (1986) · Zbl 0622.76077
[15] R.S. Tuminaro, M. Heroux, S.A. Hutchinson, J.N. Shadid, Official Aztec User’s Guide Version 2.1, Sandia National Laboratories Technical Report, SAND99-8801J 1999
[16] Knoll, D.A.; McHugh, P.R., Newton-Krylov methods applied to a system of convection-diffusion-reaction equations, Comput. phys. commun., 88, 141-160, (1995) · Zbl 0923.76199
[17] Kommu, S.; Sinha, D.; Knieling, J., A theoretical/experimental study of silicon epitaxy in horizontal single-wafer chemical vapor deposition reactors, J. elect. chem. soc., 147, 4, 1538-1550, (2000)
[18] Lin, P.T.; Sala, M.; Shadid, J.N.; Tuminaro, R.S., Performance of a geometric and an algebraic multilevel preconditioner for incompressible flow with transport, ()
[19] Saad, Y., Iterative solution methods for sparse linear systems, (2003), SIAM
[20] Saad, Y., Krylov subspace methods on supercomputers, SIAM J. sci. stat. comput., 10, 6, 1200-1232, (1989) · Zbl 0693.65028
[21] Y. Saad, Gen-Ching Lo, S. Kuznetsov, PSPARSLIB Users Manual: A Portable Library of parallel Sparse Iterative Solvers, 18 January 1998
[22] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[23] Schreiber, R.; Keller, H.B., Driven cavity flows by efficient numerical techniques, J. comput. phys., 49, 310-333, (1983) · Zbl 0503.76040
[24] J.N. Shadid, A comparison of parallel preconditioners for solution of unstructured finite element fluid flow, heat and mass transfer simulations, Proceedings of the Fourth Japan-US Symposium on Finite Element Methods in Large-Scale Computational Fluid Dynamics, Nihon University, Tokyo, Japan, 1998, April 2-4
[25] Shadid, J.N., A fully-coupled Newton-Krylov solution method for parallel unstructured finite element fluid flow, heat and mass transfer simulations, Int J. CFD, 12, 199-211, (1999) · Zbl 0969.76049
[26] Shadid, J.; Hutchinson, S.; Hennigan, G.; Moffat, H.; Devine, K.; Salinger, A.G., Efficient parallel computation of unstructured finite element reacting flow solutions, Parallel comput., 23, 1307-1325, (1997) · Zbl 0894.68019
[27] J.N. Shadid, A.G. Salinger, R.C. Schmidt, T.M. Smith, S.A. Hutchinson, G.L. Hennigan, K.D. Devine, H.K. Moffat, MPSalsa Version 1.5: A Finite Element Computer Program for Reacting Flow Problems: Part 1 - Theoretical Development, Sandia National Laboratories Technical Report, SAND98-2864, Albuquerque, NM, 1999
[28] Shadid, J.N.; Tuminaro, R.S.; Walker, H.F., An inexact Newton method for fully-coupled solution of the Navier-Stokes equations with heat and mass transport, J. comput. phys., 137, 155-185, (1997) · Zbl 0898.76066
[29] Shadid, J.N.; Tuminaro, R.S., A comparison of preconditioned nonsymmetric Krylov methods on a large-scale MIMD machine, SIAM J. sci. comput., 15, 2, 440-459, (1994) · Zbl 0798.65047
[30] Shakib, F.; Hughes, T.J.R.; Johan, Z.A., New finite-element formulation for computational fluid-dynamics. 10. the compressible Euler and Navier-Stokes equations, Comp. meth. appl. mech. eng., 89, 1-3, 141-219, (1991)
[31] Silvester, D.; Wathen, A., Fast iterative solution of stabilized Stokes systems - part II: using general block preconditioners, SIAM J. num. anal., 31, 5, 1352-1367, (1994) · Zbl 0810.76044
[32] Smith, B.F.; Bjorstad, P.E.; Gropp, W.D., Domain decomposition: parallel multilevel methods for elliptic partial differential equations, (1996), Cambridge University Press Cambridge · Zbl 0857.65126
[33] Tezduyar, T.E., Stabilized finite element formulations for incompressible flow computations, Adv. appl. mech., 28, 1-44, (1992) · Zbl 0747.76069
[34] Trottenberg, U.; Oosterlee, C.; Schuller, A., Multigrid, (2001), Academic Press London, UK
[35] Tuminaro, R.S.; Tong, C.H.; Shadid, J.N.; Devine, K.D.; Day, D.M., On a multilevel preconditioning module for unstructured mesh Krylov solvers: two-level Schwarz, Commun. num. meth. eng., 18, 383-389, (2002) · Zbl 0999.65101
[36] Vincent, C.; Boyer, R., A preconditioned conjugate gradient Uzawa-type method for the solution of the Stokes problem by mixed Q1-P0 stabilized finite elements, Int. J. num. meth. fluids, 14, 289-298, (1992) · Zbl 0745.76046
[37] Waisman, H.; Fish, J.; Tuminaro, R.S.; Shadid, J.N., The generalized global basis (GGB) method, Int. J. num. meth. eng., 61, 8, 1243-1269, (2004) · Zbl 1075.74685
[38] Wesseling, P., Principles of computational fluid dynamics, () · Zbl 0709.76098
[39] Wille, S.O., A preconditioned alternating inner-outer iterative solution method for the mixed finite element formulation of the Navier-Stokes equations, Int. J. num. meth. fluids, 18, 1135-1151, (1994) · Zbl 0806.76047
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.