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

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

Published online: 2010-07

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• 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 $h<<H$. 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.

65L10, 65N06, 65N12

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@Article{IJNAM-7-580, author = {L. G. and Vulkov and and 21348 and and L. G. Vulkov and A. I. and Zadorin and and 20960 and and A. I. 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 $h<<H$. 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 - Vulkov , L. G. AU - Zadorin , A. I. 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, boundary layer, Il'in scheme, nonlinear system, Newton method, Picard method, 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 $h<<H$. 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. G. Vulkov & A. I. 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:
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