Cai, Xiao-Chuan; Keyes, David E.; Marcinkowski, Leszek Nonlinear additive Schwarz preconditioners and application in computational fluid dynamics. (English) Zbl 1025.76040 Int. J. Numer. Methods Fluids 40, No. 12, 1463-1470 (2002). Summary: The focus of this paper is on the numerical solution of large sparse nonlinear systems of algebraic equations on parallel computers. Such nonlinear systems often arise from the discretization of nonlinear partial differential equations, such as the Navier-Stokes equations for fluid flows, using finite element or finite difference methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the nonlinearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel nonlinear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier-Stokes equations are reported. Cited in 31 Documents MSC: 76M99 Basic methods in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs Keywords:nonlinear preconditioning; inexact Newton methods; nonlinear additive Schwarz; multilevel methods; domain decomposition; nonlinear equations PDFBibTeX XMLCite \textit{X.-C. Cai} et al., Int. J. Numer. Methods Fluids 40, No. 12, 1463--1470 (2002; Zbl 1025.76040) Full Text: DOI References: [1] Numerical Methods for Unconstrained Optimization and Non-linear Equations. Prentice-Hall: Englewood Cliffs, NJ, 1983. [2] Kelley, SIAM Journal on Numerical Analysis 35 pp 508– (1998) [3] Young, Journal of Computational Physics 92 pp 1– (1991) [4] Eisenstat, SIAM Journal on Optimization 4 pp 393– (1994) [5] Cai, SIAM Journal on Scientific Computing (2002) [6] Cai, Contemporary Mathematics 180 pp 21– (1994) [7] On the non-linear domain decomposition method. BIT 1997; 296-311. · Zbl 0891.65126 [8] Eisenstat, SIAM Journal on Scientific Computing 17 pp 16– (1996) [9] Lanzkron, SIAM Journal on Scientific Computing 17 pp 538– (1996) [10] Cai, SIAM Journal on Scientific Computing 19 pp 246– (1998) [11] Numerical Computation of Internal and External Flows. Wiley: New York, 1990. [12] The Portable, Extensible Toolkit for Scientific Computing, version 2.1.0, http://www.mcs.anl.gov/petsc, code and documentation, 2001. [13] A non-linear additive Schwarz preconditioned inexact Newton method for shocked duct flow. In Domain Decomposition Methods in Science and Engineering, (eds). CIMNE, UPS: Barcelona, 2001. [14] Dryja, SIAM Journal on Scientific Computing 15 pp 604– (1994) [15] Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press: Cambridge, 1996. · Zbl 0857.65126 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.