Methods and Algorithms for Scientific Computing

Nonlinear Preconditioning: How to Use a Nonlinear Schwarz Method to Precondition Newton's Method


For linear problems, domain decomposition methods can be used directly as iterative solvers but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration and thus converges much faster. We show in this paper that also for nonlinear problems, domain decomposition methods can be used either directly as iterative solvers or as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (restricted additive Schwarz preconditioned exact Newton), which is similar to ASPIN (additive Schwarz preconditioned inexact Newton) but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two-level nonlinear iterative domain decomposition method and a two level RASPEN nonlinear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a nonlinear diffusion problem.


  1. nonlinear preconditioning
  2. two-level nonlinear Schwarz methods
  3. preconditioning Newton's method

MSC codes

  1. 65M55
  2. 65F10
  3. 65N22

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G. W. Anders Logg, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lect. Notes Comput. Sci. Eng. 84, Springer, New York, 2012.
M. Benzi, M. J. Gander, and G. H. Golub, Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT, 43 (2003), pp. 881--900.
W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, 2nd ed., SIAM, Philadelphia, 2000.
F. Caetano, M. J. Gander, L. Halpern, and J. Szeftel, Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations, Netw. Heterog. Media, 5 (2010), pp. 487--505.
X.-C. Cai and M. Dryja, Domain decomposition methods for monotone nonlinear elliptic problems, Contemp. Math., 180 (1994), pp. 21--27.
X.-C. Cai, W. D. Gropp, D. E. Keyes, R. G. Melvin, and D. P. Young, Parallel Newton--Krylov--Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput., 19 (1998), pp. 246--265.
X.-C. Cai, W. D. Gropp, D. E. Keyes, and M. D. Tidriri, Newton-Krylov-Schwarz methods in CFD, in Numerical Methods for the Navier-Stokes Equations, Notes Numer. Fluid Mech. 47, Vieweg+Teubner, Berlin, 1994, pp. 17--30.
X.-C. Cai and D. E. Keyes, Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), pp. 183--200.
X.-C. Cai, D. E. Keyes, and D. P. Young, A nonlinear additive Schwarz preconditioned inexact Newton method for shocked duct flow, in Proceedings of the 13th International Conference on Domain Decomposition Methods, 2001, pp. 343--350.
Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, Comput. Sci. Eng., SIAM, Philadelphia, 2006.
S. Descombes, V. Dolean, and M. J. Gander, Schwarz waveform relaxation methods for systems of semi-linear reaction-diffusion equations, in Domain Decomposition Methods in Science and Engineering XIX, Springer, New York, 2011, pp. 423--430.
M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method, BIT, 37 (1997), pp. 296--311.
M. Dryja and O. B. Widlund, An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Tech. report 339, Department of Computer Science, Courant Institute, 1987.
E. Efstathiou and M. J. Gander, Why restricted additive Schwarz converges faster than additive Schwarz, BIT, 43 (2003), pp. 945--959.
L. El Alaoui, A. Ern, and M. Vohralík, Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2782--2795.
A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput., 35 (2013), pp. A1761--A1791.
C. Farhat and F. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Internat. J. Numer. Methods Engrg., 32 (1991), pp. 1205--1227.
P. Forchheimer, Wasserbewegung durch Boden, Z. Vereines Deutscher Ingenieuer, 45 (1901), pp. 1782--1788.
M. J. Gander, A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6 (1998), pp. 125--145.
M. J. Gander, Schwarz methods over the course of time, Electron. Trans. Numer. Anal., 31 (2008), pp. 228--255.
M. J. Gander and C. Rohde, Overlapping Schwarz waveform relaxation for convection dominated nonlinear conservation laws, SIAM J. Sci. Comput., 27 (2005), pp. 415--439.
F. Haeberlein, Time Space Domain Decomposition Methods for Reactive Transport -- Application to $CO_{2}$ Geological Storage, Ph.D. thesis, Université Paris-Nord, Paris XIII, 2011.
F. Haeberlein and L. Halpern, Optimized Schwarz waveform relaxation for nonlinear systems of parabolic type, in Domain Decomposition Methods in Science and Engineering XXI, Lect. Notes Comput. Sci. Eng. 98, Springer, New York, 2014, pp. 29--42.
F. Haeberlein, L. Halpern, and A. Michel, Schwarz Waveform Relaxation and Krylov Accelerators for Reactive Transport, Technical report hal-01384281; also available online from
L. Halpern and J. Szeftel, Nonlinear nonoverlapping Schwarz waveform relaxation for semilinear wave propagation, Math. Comp., 78 (2009), pp. 865--889.
F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), pp. 251--266.
V. E. Henson, Multigrid methods nonlinear problems: An overview, in Electronic Imaging 2003, International Society for Optics and Photonics, 2003, pp. 36--48.
M. Kaviany, Principles of Heat Transfer in Porous Media, Springer-Verlag, Berlin, 1991.
P.-L. Lions, On the Schwarz alternating method. I, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., SIAM, Philadelphia, 1988, pp. 1--42.
S.-H. Lui, On Schwarz alternating methods for nonlinear elliptic PDEs, SIAM J. Sci. Comput., 21 (1999), pp. 1506--1523.
S.-H. Lui, On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs, Numer. Math., 93 (2002), pp. 109--129.
L. Marcinkowski and X.-C. Cai, Parallel performance of some two-level ASPIN algorithms, in Domain Decomposition Methods in Science and Engineering, Springer, New York, 2005, pp. 639--646.
D. A. Nield and A. Bejan, Convection in Porous Media, Springer, New York, 2006.
J. O. Skogestad, E. Keilegavlen, and J. M. Nordbotten, Domain decomposition strategies for nonlinear flow problems in porous media, J. Comput. Phys., 234 (2013), pp. 439--451.
X.-C. Tai and M. Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems, SIAM J. Numer. Anal., 35 (1998), pp. 1558--1570.
J. C. Ward, Turbulent flow in porous media, J. Hydr. Div. ASCE, 90 (1964), pp. 1--12.
J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), pp. 1759--1777.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A3357 - A3380
ISSN (online): 1095-7197


Submitted: 2 July 2015
Accepted: 22 July 2016
Published online: 1 November 2016


  1. nonlinear preconditioning
  2. two-level nonlinear Schwarz methods
  3. preconditioning Newton's method

MSC codes

  1. 65M55
  2. 65F10
  3. 65N22



Funding Information

National Natural Science Foundation of China : 11501483

Funding Information

Research Grants Council, University Grants Committee : ECS/22300115

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