Geodesic
The Lagrange interpolation and the Newton--Cotes formulas for functions with a~boundary layer component on piecewise-uniform meshes
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 3, pp. 289-303

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The interpolation problem of a one-variable function, which can be considered as a solution of a boundary value problem for an equation with a small parameter $\varepsilon$ with a higher derivative is investigated. The application of the Lagrange interpolation for such a function on a uniform grid can result in serious errors. In the case of the Shishkin mesh, $\varepsilon$-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The $\varepsilon$-uniform error estimates of the Newton–Cotes formulas on such meshes are obtained. Numerical experiments have been carried out. The results obtained confirm the theoretical estimates. 
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     author = {A. I. Zadorin},
     title = {The {Lagrange} interpolation and the {Newton--Cotes} formulas for functions with a~boundary layer component on piecewise-uniform meshes},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {289--303},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2015_18_3_a3/}
}
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A. I. Zadorin. The Lagrange interpolation and the Newton--Cotes formulas for functions with a~boundary layer component on piecewise-uniform meshes. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 3, pp. 289-303. http://geodesic.mathdoc.fr/item/SJVM_2015_18_3_a3/