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Zbl 1103.65084
Kadalbajoo, Mohan K.; Patidar, Kailash C.
$\varepsilon$-uniformly convergent fitted mesh finite difference methods for general singular perturbation problems. (English)
[J] Appl. Math. Comput. 179, No. 1, 248-266 (2006). ISSN 0096-3003

The authors consider a two point boundary value problem of a scalar linear ordinary differential equation (ODE) of second order. In this equation, a parameter $\varepsilon$ implies a singular perturbation problem for small values of the parameter. A Shishkin mesh is used to discretise the domain of dependence following a strategy introduced {\it G. I. Shishkin} [Zh. Vychisl. Mat. Fiz. 28, No.~11, 1649--1662 (1988; Zbl 0662.65086)]. Thereby, piecewise equidistant grids are applied, where smaller step sizes arise in the boundary layers of the exact ODE solution. A finite difference scheme tailored to the ODE is constructed on an arbitrary fitted mesh to obtain a linear system for the numerical approximations. The authors prove that the finite difference method using the Shishkin mesh is uniformly convergent for all $0 < \vert \varepsilon\vert \le 1$. Thereby, the convergence rate $\mathcal{O}(\log^2(n) / n^2)$ is achieved, where $n$ denotes the total number of subintervals. In contrast, straightforward techniques exhibit just a rate of $\mathcal{O}(\log(n) / n)$. Numerical simulations of five examples, where the exact solution is known, verify the predicted convergence properties.
[Roland Pulch (Wuppertal)]

MSC 2000:: *65L10 Boundary value problems for ODE (numerical methods)
65L12 Finite difference methods for ODE
65L20 Stability of numerical methods for ODE
34B05 Linear boundary value problems of ODE
34E15 Asymptotic singular perturbations, general theory (ODE)
65L50 Mesh generation and refinement (ODE)

Keywords: Shishkin mesh; numerical examples; convergence; two point boundary value problem

Citations: Zbl 0662.65086

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