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 |
Keywords:
Newton-Krylov; Fully coupled solvers; Schwarz domain decomposition; Two-level methods; Multilevel methods; Parallel methods; Stabilized finite element methodsReferences:
| [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; 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 · Zbl 0701.65072 |
| [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) · Zbl 0727.65105 |
| [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; 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; 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; 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, (Yao, Z.; Yuan, M.; Zhong, W., Computational Mechanics, Proceedings of WCCM VI (2004), Tsinghua Univ. Press: Tsinghua Univ. Press Springer) |
| [19] | Saad, Y., Iterative Solution Methods for Sparse Linear Systems (2003), SIAM · Zbl 1031.65046 |
| [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; 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; 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; 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 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: Academic Press London, UK · Zbl 0976.65106 |
| [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, (Computational Mechanics, vol. 29 (2001), Springer: Springer Berlin) · 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 |
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