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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.
##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations
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