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Some uniformly convergent schemes on Shishkin mesh. (English) Zbl 1029.65506
Given the selfadjoint problem \[ \begin{gathered} Ly\equiv-\varepsilon y''+p(x)y=f(x),\quad x\in I=(0,1)\\ y(0)=0,\quad y(1))=0 \end{gathered} \] where \(0<\varepsilon\ll 1\) is a small parameter, and \(p,f\in C^2(I)\), \(p(x)\geq\beta^2>0\), a difference scheme on the non-uniform mesh was derived by the use of cubic spline. The proposed scheme is analyzed and proved to be uniformly convergent with the order \(O(n^{-2}\ln^2n)\). The uniform convergence of the non-uniform mesh of the Shiskin type is achieved without the exponential fitting. A numerical example is included to demonstrate the described scheme.
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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