Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 487273, 12 pages.

Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations

Hung-Yu Ke, Ren-Chuen Chen, and Chun-Hsien Li

Full-text: Open access

Abstract

We will combine linear successive overrelaxation method with nonlinear monotone iterative scheme to obtain a new iterative method for solving nonlinear equations. The basic idea of this method joining traditional monotone iterative method (known as the method of lower and upper solutions) which depends essentially on the monotone parameter is that by introducing an acceleration parameter one can construct a sequence to accelerate the convergence. The resulting increase in the speed of convergence is very dramatic. Moreover, the sequence can accomplish monotonic convergence behavior in the iterative process when some suitable acceleration parameters are chosen. Under some suitable assumptions in aspect of the nonlinear function and the matrix norm generated from this method, we can prove the boundedness and convergence of the resulting sequences. Application of the iterative scheme is given to a logistic model problem in ecology, and numerical results for a test problem with known analytical solution are given to demonstrate the accuracy and efficiency of the present method.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 487273, 12 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807729

Digital Object Identifier
doi:10.1155/2013/487273

Mathematical Reviews number (MathSciNet)
MR3100827

Zentralblatt MATH identifier
06950703

Citation

Ke, Hung-Yu; Chen, Ren-Chuen; Li, Chun-Hsien. Convergence Analysis of an Iterative Method for Nonlinear Partial Differential Equations. J. Appl. Math. 2013, Special Issue (2013), Article ID 487273, 12 pages. doi:10.1155/2013/487273. https://projecteuclid.org/euclid.jam/1394807729


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References

  • A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, NY, USA, 1979.
  • R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1962.
  • D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, NY, USA, 1971.
  • R. P. Agarwal and Y. M. Wang, “Some recent developments of Numerov's method,” Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 561–592, 2001.
  • D. Greenspan and S. V. Parter, “Mildly nonlinear elliptic partial differential equations and their numerical solution. II,” Numerische Mathematik, vol. 7, pp. 129–146, 1965.
  • C. V. Pao, “Block monotone iterative methods for numerical solutions of nonlinear elliptic equations,” Numerische Mathematik, vol. 72, no. 2, pp. 239–262, 1995.
  • C. V. Pao, “Numerical methods for quasi-linear elliptic equations with nonlinear boundary conditions,” SIAM Journal on Numerical Analysis, vol. 45, no. 3, pp. 1081–1106, 2007.
  • S. V. Parter, “Mildly nonlinear elliptic partial differential equations and their numerical solution. I,” Numerische Mathematik, vol. 7, pp. 113–128, 1965.
  • S. Heikkilä and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York, NY, USA, 1994.
  • C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
  • C. V. Pao and X. Lu, “Block monotone iterative method for semilinear parabolic equations with nonlinear boundary conditions,” SIAM Journal on Numerical Analysis, vol. 47, no. 6, pp. 4581–4606, 2010.
  • Y. M. Wang, “A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 12, pp. 3646–3660, 2011.
  • R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, NY, USA, 1992.
  • R. Kannan and M. B. Ray, “Monotone iterative methods for nonlinear equations involving a noninvertible linear part,” Numerische Mathematik, vol. 45, no. 2, pp. 219–225, 1984.
  • A. W. Leung and D. A. Murio, “Accelerated monotone scheme for finite difference equations concerning steady-state prey-predator interactions,” Journal of Computational and Applied Mathematics, vol. 16, no. 3, pp. 333–341, 1986.
  • K. Ishihara, “Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem,” Numerische Mathematik, vol. 43, no. 3, pp. 419–437, 1984.
  • C. A. Brebbia and S. Walker, Boundary Element Techniques in Engineering, Newnes-Butterworths, London, UK, 1980.
  • Y. Deng, G. Chen, W. M. Ni, and J. Zhou, “Boundary element monotone iteration scheme for semilinear elliptic partial differential equations,” Mathematics of Computation, vol. 65, no. 215, pp. 943–982, 1996.
  • J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.
  • X. Lu, “Combined iterative methods for numerical solutions of parabolic problems with time delays,” Applied Mathematics and Computation, vol. 89, no. 1–3, pp. 213–224, 1998.
  • R. C. Chen and J. L. Liu, “Monotone iterative methods for the adaptive finite element solution of semiconductor equations,” Journal of Computational and Applied Mathematics, vol. 159, no. 2, pp. 341–364, 2003.
  • R. C. Chen and J. L. Liu, “An iterative method for adaptive finite element solutions of an energy transport model of semiconductor devices,” Journal of Computational Physics, vol. 189, no. 2, pp. 579–606, 2003.
  • R. C. Chen and J. L. Liu, “A quantum corrected energy-transport model for nanoscale semiconductor devices,” Journal of Computational Physics, vol. 204, no. 1, pp. 131–156, 2005.
  • R. C. Chen, “An iterative method for finite-element solutions of the nonlinear Poisson-Boltzmann equation,” WSEAS Transactions on Computers, vol. 7, no. 4, pp. 165–173, 2008.
  • W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, Boston, Mass, USA, 1992.
  • A. A. Samarskii, The Theory of Difference Schemes, vol. 240, Marcel Dekker, New York, NY, USA, 2001.
  • K. W. Morton, Numerical Solution of Convection-Diffusion Problems, vol. 12, Chapman & Hall, London, UK, 1996.
  • Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2nd edition, 2003.