Convergence Analysis for the Multiplicative Schwarz Preconditioned Inexact Newton Algorithm

The multiplicative Schwarz preconditioned inexact Newton (MSPIN) algorithm, based on decomposition by field type rather than by subdomain, was recently introduced to improve the convergence of systems with unbalanced nonlinearities. This paper provides a convergence analysis of the MSPIN algorithm. Under reasonable assumptions, it is shown that MSPIN is locally convergent, and desired superlinear or even quadratic convergence can be obtained when the forcing terms are picked suitably.

[1]  Xiao-Chuan Cai,et al.  A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations , 2005 .

[2]  Hengbin An,et al.  On Convergence of the Additive Schwarz Preconditioned Inexact Newton Method , 2005, SIAM J. Numer. Anal..

[3]  David E. Keyes,et al.  A Nonlinear Additive Schwarz Preconditioned Inexact Newton Method for Shocked Duct Flow , 2000 .

[4]  S. H. Lui,et al.  Nonlinearly Preconditioned Newton ’ s Method , 2003 .

[5]  J. Nordbotten,et al.  Two-Scale Preconditioning for Two-Phase Nonlinear Flows in Porous Media , 2016, Transport in Porous Media.

[6]  Eirik Keilegavlen,et al.  Domain decomposition strategies for nonlinear flow problems in porous media , 2013, J. Comput. Phys..

[7]  Barbara Wohlmuth,et al.  A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction , 2009 .

[8]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[9]  Lulu Liu,et al.  Field-Split Preconditioned Inexact Newton Algorithms , 2015, SIAM J. Sci. Comput..

[10]  Homer F. Walker,et al.  NITSOL: A Newton Iterative Solver for Nonlinear Systems , 1998, SIAM J. Sci. Comput..

[11]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[12]  Homer F. Walker,et al.  Choosing the Forcing Terms in an Inexact Newton Method , 1996, SIAM J. Sci. Comput..

[13]  David E. Keyes,et al.  Nonlinearly Preconditioned Inexact Newton Algorithms , 2002, SIAM J. Sci. Comput..

[14]  D. Keyes,et al.  Nonlinear additive Schwarz preconditioners and applications in computational fluid dynamics , 2002 .

[15]  W. Hackbusch,et al.  On the nonlinear domain decomposition method , 1997 .

[16]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[17]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[18]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[19]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[20]  Homer F. Walker,et al.  Globally Convergent Inexact Newton Methods , 1994, SIAM J. Optim..