An inexact Newton method with inner preconditioned CG for non-uniformly monotone elliptic problems
Abstract
The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.
Keyword : inexact Newton iteration, conjugate gradients, nonlinear elliptic problems, iterative methods
How to Cite
Borsos, B. (2021). An inexact Newton method with inner preconditioned CG for non-uniformly monotone elliptic problems. Mathematical Modelling and Analysis, 26(3), 383-394. https://doi.org/10.3846/mma.2021.12899
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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O. Axelsson. On mesh independence and Newton-type methods. Appl. Math., 38:249–265, 1993. https://doi.org/10.21136/AM.1993.104554
O. Axelsson and V.A. Barker. Finite Element Solution of Boundary Value Problems: Theory and Computation. 0898714990, Society for Industrial and Applied Mathematics, 2001. https://doi.org/10.1016/C2013-0-10322-4
O. Axelsson and W. Layton. A two-level method for the discretization of nonlinear boundary value problems. SIAM J. Numer. Anal., 33:2359–2374, 1996. https://doi.org/10.1137/S0036142993247104
O. Axelsson and J. Maubach. On the updating and assembly of the Hessian matrix in finite element methods. Comp. Meth. Appl. Mech. Engrg., 71:41–67, 1988. https://doi.org/10.1016/0045-7825(88)90095-3
Zh. Bai, B. Huang and W. Ge. The iterative solutions for some fourth-order pLaplace equation boundary value problems. Appl. Math. Letters, 19:8–14, 2006. https://doi.org/10.1016/j.aml.2004.10.010
K. Böhmer. Numerical methods for nonlinear elliptic differential equations. Oxford University Press, Oxford, 2010.
B. Borsos and J. Karátson. Variable preconditioning for strongly nonlinear elliptic problems. J. Comput. Appl. Math., 350:155–164, 2019. https://doi.org/10.1016/j.cam.2018.10.004
B. Borsos and J. Karátson. Quasi-Newton variable preconditioning for nonuniformly monotone elliptic problems posed in Banach spaces. IMA J. Numer. Anal., 2021. https://doi.org/10.1093/imanum/drab024
Ph. Ciarlet. Linear and nonlinear functional analysis with applications. SIAM, Philadelphia, 2013.
P. Deuflhard. Global inexact Newton methods for very large scale nonlinear problems. Impact Comput. Sci. Engrg., 3(4):366–393, 1991. https://doi.org/10.1016/0899-8248(91)90004-E
I. Faragó and J. Karátson. Numerical Solution of Nonlinear Elliptic Problems via Preconditioning Operators: Theory and Application. Advances in Computation, Volume 11, NOVA Science Publishers, New York, 2002.
I. Faragó and J. Karátson. Variable preconditioning via quasi-Newton methods for nonlinear problems in Hilbert space. SIAM J. Numer. Anal., 41(4):1242– 1262, 2003. https://doi.org/10.1137/S0036142901384277
R. Glowinski. Variational methods for the numerical solution of nonlinear elliptic problems. SIAM, Philadelphia, 2015. https://doi.org/10.1137/1.9781611973785
A. Hirn. Finite element approximation of singular power-law systems. Math. Comp., 82:1247–1268, 2013. https://doi.org/10.1090/S0025-5718-2013-02668-3
M. Křižek and P. Neittaanmäki. Mathematical and Numerical Modeling in Electrical Engineering: Theory and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
A. Mang and G. Biros. A semi-Lagrangian two-level preconditioned NewtonKrylov solver for constrained diffeomorphic image registration. SIAM J. Sci. Comput., 39(6):B1064–B1101, 2017. https://doi.org/10.1137/16M1070475
A. Rieder. Inexact Newton regularization using conjugate gradients as inner iteration. SIAM J. Numer. Anal., 43(2):604–622, 2005. https://doi.org/10.1137/040604029
T. Rossi and J. Toivanen. Parallel fictitious domain method for a non-linear elliptic Neumann boundary value problem. Czech-US Workshop in Iterative Methods and Parallel Computing, Part I (Milovy, 1997). Numer. Linear Algebra Appl, 6(1):51–60, 1999.
F. Xie, Q-B. Wu and P-F. Dai. Modified Newton-SHSS method for a class of systems of nonlinear equations. Comput. Appl. Math., 38(1):Paper No. 19, 2019. https://doi.org/10.1007/s40314-019-0793-9
J. Xu. A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput., 15:231–237, 1994. https://doi.org/10.1137/0915016