Volume 7, Issue 3
Two-grid Algorithms for an Ordinary Second Order Equation with an Exponential Boundary Layer in the Solution

L. Vulkov & A. Zadorin

Int. J. Numer. Anal. Mod., 7 (2010), pp. 580-592

Published online: 2010-07

Preview Purchase PDF 2 3364
Export citation
  • Abstract
This paper is concerned with the solution of the nonlinear system of equations arising from the A.M. Il'in's scheme approximation of a model semilinear singularly perturbed boundary value problem. We employ Newton and Picard methods and propose a new version of the two-grid method originated by O. Axelsson [2] and J. Xu [19]. In the first step, the nonlinear differential equation is solved on a "coarse" grid of size H. In the second step, the problem is linearized around an appropriate interpolation of the solution computed in the first step and the linear problem is then solved on a fine grid of size $h \ll H$. It is shown that the algorithms achieve optimal accuracy as long as the mesh sizes satisfy $h = O(H^{2^m})$, m=1,2,..., where m is the number of the Newton (Picard) iterations for the difference problem. We count the number of the arithmetical operations to illustrate the computational cost of the algorithms. Numerical experiments are discussed.
  • Keywords

Nonlinear boundary value problem boundary layer Il'in scheme nonlinear system Newton method Picard method two-grid method

  • AMS Subject Headings

65L10 65N06 65N12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-7-580, author = {L. Vulkov and A. Zadorin}, title = {Two-grid Algorithms for an Ordinary Second Order Equation with an Exponential Boundary Layer in the Solution}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2010}, volume = {7}, number = {3}, pages = {580--592}, abstract = {This paper is concerned with the solution of the nonlinear system of equations arising from the A.M. Il'in's scheme approximation of a model semilinear singularly perturbed boundary value problem. We employ Newton and Picard methods and propose a new version of the two-grid method originated by O. Axelsson [2] and J. Xu [19]. In the first step, the nonlinear differential equation is solved on a "coarse" grid of size H. In the second step, the problem is linearized around an appropriate interpolation of the solution computed in the first step and the linear problem is then solved on a fine grid of size $h \ll H$. It is shown that the algorithms achieve optimal accuracy as long as the mesh sizes satisfy $h = O(H^{2^m})$, m=1,2,..., where m is the number of the Newton (Picard) iterations for the difference problem. We count the number of the arithmetical operations to illustrate the computational cost of the algorithms. Numerical experiments are discussed.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/739.html} }
TY - JOUR T1 - Two-grid Algorithms for an Ordinary Second Order Equation with an Exponential Boundary Layer in the Solution AU - L. Vulkov & A. Zadorin JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 580 EP - 592 PY - 2010 DA - 2010/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/739.html KW - Nonlinear boundary value problem KW - boundary layer KW - Il'in scheme KW - nonlinear system KW - Newton method KW - Picard method KW - two-grid method AB - This paper is concerned with the solution of the nonlinear system of equations arising from the A.M. Il'in's scheme approximation of a model semilinear singularly perturbed boundary value problem. We employ Newton and Picard methods and propose a new version of the two-grid method originated by O. Axelsson [2] and J. Xu [19]. In the first step, the nonlinear differential equation is solved on a "coarse" grid of size H. In the second step, the problem is linearized around an appropriate interpolation of the solution computed in the first step and the linear problem is then solved on a fine grid of size $h \ll H$. It is shown that the algorithms achieve optimal accuracy as long as the mesh sizes satisfy $h = O(H^{2^m})$, m=1,2,..., where m is the number of the Newton (Picard) iterations for the difference problem. We count the number of the arithmetical operations to illustrate the computational cost of the algorithms. Numerical experiments are discussed.
L. Vulkov & A. Zadorin. (1970). Two-grid Algorithms for an Ordinary Second Order Equation with an Exponential Boundary Layer in the Solution. International Journal of Numerical Analysis and Modeling. 7 (3). 580-592. doi:
Copy to clipboard
The citation has been copied to your clipboard