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$\epsilon$-uniformly convergent fitted mesh finite difference methods for general singular perturbation problems. (English) Zbl 1103.65084

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 $\epsilon$ 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 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 $0<|\epsilon |\le 1$. Thereby, the convergence rate $𝒪\left({log}^{2}\left(n\right)/{n}^{2}\right)$ is achieved, where $n$ denotes the total number of subintervals. In contrast, straightforward techniques exhibit just a rate of $𝒪\left(log\left(n\right)/n\right)$. Numerical simulations of five examples, where the exact solution is known, verify the predicted convergence properties.

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