Performance of fully coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport.

*(English)*Zbl 1110.76315Summary: This study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton-Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behaviour of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non-zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two-level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large-scale distributed-memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two- and three-level preconditioners are demonstrated to be scalable to 1024 processors.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

##### Keywords:

multilevel preconditioners; multigrid; finite element methods; Newton-Krylov; Schwarz domain decomposition preconditioners; graph partitioning
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\textit{P. T. Lin} et al., Int. J. Numer. Methods Eng. 67, No. 2, 208--225 (2006; Zbl 1110.76315)

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##### References:

[1] | Gropp, Parallel Computing 27 pp 337– (2001) · Zbl 0770.65085 |

[2] | , . Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press: Cambridge, 1996. · Zbl 0857.65126 |

[3] | , . Multigrid. Academic Press: London, 2001. |

[4] | Brezina, Computing 63 pp 233– (1999) |

[5] | Vanek, Numerische Mathematik 88 pp 559– (2001) |

[6] | Shadid, Journal of Computational Physics 205 pp 24– (2005) |

[7] | , , . An aggregation-based domain decomposition preconditioner for groundwater flow. Technical Report TR00-13, Department of Mathematics, North Carolina State University, 2000. |

[8] | Lasser, Mathematics of Computation 72 pp 1215– (2003) |

[9] | . Algebraic coarse grid operators for domain decomposition based preconditioners. In Parallel Computational Fluid Dynamics–Practice and Theory, , , , (eds). Elsevier: The Netherlands, 2002; 119–126. |

[10] | Domain decomposition preconditioners: theoretical properties, application to the compressible Euler equations, parallel aspects. Ph.D. Thesis, EPFL, Lausanne, Switzerland, 2003. |

[11] | Hughes, Computer Methods in Applied Mechanics and Engineering 59 pp 85– (1986) |

[12] | Tezduyar, Advances in Applied Mechanics 28 pp 1– (1992) |

[13] | Shadid, Parallel Computing 23 pp 1307– (1997) |

[14] | Brown, SIAM Journal on Scientific and Statistical Computing 11 pp 450– (1990) |

[15] | Shadid, International Journal of Computational Fluid Dynamics 12 pp 199– (1999) |

[16] | Eisenstat, SIAM Journal on Optimization 4 pp 393– (1994) |

[17] | Iterative Methods for Sparse Linear Systems. SIAM: Philadelphia, PA, 2003. |

[18] | . Domain Decomposition Methods for Partial Differential Equations. Oxford University Press: Oxford, 1999. · Zbl 0931.65118 |

[19] | Saad, Numerical Linear Algebra with Applications 1 pp 387– (1994) |

[20] | , , , . Aztec User’s Guide Version 2.0. Technical Report SAND 99-8801J, Sandia National Laboratories, 1999. |

[21] | Tuminaro, Communications in Numerical Methods in Engineering 18 pp 383– (2002) |

[22] | , . ML 3.1 Smoothed Aggregation User’s Guide. Technical Report SAND2004-4819, Sandia National Laboratories, September 2004. |

[23] | . Algebraic multigrid. Multigrid Methods, vol. 3, Frontiers in Applied Mathematics. 1987; 73–130. |

[24] | . Multilevel k-way partitioning scheme for irregular graphs. Technical Report 95-064, Army HPC Research Center, 1995. |

[25] | Robust iterative solvers on unstructured meshes. Ph.D. Thesis, University of Colorado, Denver, 1997. |

[26] | , . A field-of-value approach for smoothed aggregation preconditioners. Technical Report SAND2005-3333, Sandia National Laboratories, May 2005. |

[27] | Amesos 2.0 Reference Guide. Technical Report SAND2004-4820, Sandia National Laboratories, September 2004. |

[28] | , . SuperLU Users’ Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory, October 2003. |

[29] | , , , , , , , , , , , , , . An Overview of Trilinos. Technical Report SAND2003-2927, Sandia National Laboratories, 2003. |

[30] | Multigrid Methods, Pitman Research Notes in Mathematics Series, vol. 294. Longman Scientific: 1993. |

[31] | An overview of SuperLU: algorithms, implementation, and user interface. Technical Report LBNL-53848, 2003. |

[32] | Devine, Computing in Science and Engineering 4 pp 90– (2002) · Zbl 05091837 |

[33] | (ed.). Cubit 9.0 Users Manual. Sandia Report SAND94-1100 Rev. 4/2002, Sandia National Laboratories, June 2004. |

[34] | Chow, Numerical Linear Algebra with Applications 10 pp 401– (2003) |

[35] | , , , , , . MPSalsa: a finite element computer program for reacting flow problems part 1: theoretical development. Technical Report SAND98-2864, Sandia National Laboratories: Albuquerque, NM, 87185, January 1999. |

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.