×

zbMATH — the first resource for mathematics

On backtracking failure in Newton-GMRES methods with a demonstration for the Navier-Stokes equations. (English) Zbl 1143.76489
Summary: In an earlier study of inexact Newton methods, we pointed out that certain counterintuitive behavior may occur when applying residual backtracking to the Navier-Stokes equations with heat and mass transport. Specifically, it was observed that a Newton-GMRES method globalized by backtracking (linesearch, damping) may be less robust when high accuracy is required of each linear solve in the Newton sequence than when less accuracy is required. In this brief discussion, we offer a possible explanation for this phenomenon, together with an illustrative numerical experiment involving the Navier-Stokes equations.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bischof, C.H., Incremental condition estimation, SIAM J. matrix anal. appl., 11, 312, (1990) · Zbl 0697.65042
[2] C. H. Bischof, and, P. T. P. Tang, Robust Incremental Condition Estimation, Tech. Resp. CS-91-133, LAPACK Working Note 33, Computer Science Department, University of Tennessee, 1991.
[3] Brown, P.N.; Walker, H.F., GMRES on (nearly) singular systems, SIAM J. matrix anal. appl., 18, 37, (1997) · Zbl 0876.65019
[4] Dembo, R.S.; Eisenstat, S.C.; Steihaug, T., Inexact Newton methods, SIAM J. numer. anal., 19, 400, (1982) · Zbl 0478.65030
[5] Dennis, J.E.; Schnabel, R.B., numerical methods for unconstrained optimization and nonlinear equations, Series in automatic computaion, (1983), Prentice Hall Englewood Cliffs · Zbl 0579.65058
[6] Eisenstat, S.C.; Walker, H.F., Globally convergent inexact Newton methods, SIAM J. optimization, 4, 393, (1994) · Zbl 0814.65049
[7] Eisenstat, S.C.; Walker, H.F., Choosing the forcing terms in an inexact Newton method, SIAM J. sci. comput., 17, 16, (1996) · Zbl 0845.65021
[8] Freund, R.W.; Golub, G.H.; Nachtigal, N.M., Iterative solution of linear systems, Acta numer., 1, 57, (1992) · Zbl 0762.65019
[9] Golub, G.H.; Van Loan, C.F., matrix computations, (1989), Johns Hopkins Press Baltimore · Zbl 0733.65016
[10] S. A. Hutchinson, L. Prevost, J. N. Shadid, C. Tong, and, R. S. Tuminaro, Aztec User’s guide, Version 2.0, Tech. Rep. Sand99-8801J, Sandia National Laboratories, Albuquerque, 1999.
[11] Pernice, M.; Walker, H.F., NITSOL: a Newton iterative solver for nonlinear systems, SIAM J. sci. comput., 19, 302, (1998) · Zbl 0916.65049
[12] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual method for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856, (1986) · Zbl 0599.65018
[13] A. G. Salinger, K. D. Devine, G. L. Hennigan, H. K. Moffat, S. A. Hutchinson, and, J. N. Shadid, MPSalsa, A Finite Element Computer Program for Reacting Flow Problems. Part 2—User’s Guide, Tech. Rep. SAND96-2331, Sandia National Laboratories, Albuquerque, 1996.
[14] Shadid, J.N., A fully coupled newton – krylov solution method for parallel unstructured finite element fluid flow, heat and mass transport simulations, Int. J. comput. fluid dyn., 12, 199, (1999) · Zbl 0969.76049
[15] 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, (1997) · Zbl 0898.76066
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.