## Monotome iterations for spectral approximation of nonlinear layer problem.(English)Zbl 1029.65504

Given a singularly perturbed problem $\begin{gathered} Ly\equiv-\varepsilon^2 y''(x)+b(x,y)=0,\quad x\in[0,1]\\ y(0)=0, \quad y(1))=0 \end{gathered}$ where $$0<\varepsilon\ll 1$$ is a small parameter, a continuous method for approximation of the solution, based on the use of spectral approximation, is presented. The original problem is transformed so that standard spectral technique can be applied and the approximate domain decomposition is determined by introducing numerical layer length. The monotone iterations are applied for construction of the sequence of spectral approximations and the error estimate is provided by the use of asymptotic behavior of the exact solution at the endpoints of the layer subintervals.

### MSC:

 65L10 Numerical solution of boundary value problems involving ordinary differential equations
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