@Article{IJNAM-7-3,
author = {Vulkov, L., G. and Zadorin, I., A.},
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 = {<p style="text-align: justify;">This paper is concerned with the solution of the nonlinear system of
equations arising from the A.M. Il&#39;in&#39;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 &quot;coarse&quot; 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 $h&lt;&lt;H$.
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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2010-IJNAM-739},
url = {https://global-sci.com/article/83609/two-grid-algorithms-for-an-ordinary-second-order-equation-with-an-exponential-boundary-layer-in-the-solution}
}