The Lagrange interpolation and the Newton-Cotes formulas for functions with a boundary layer component on piecewise-uniform meshes
Alexander Ivanovich Zadorin
Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Pevtsova st., 13, Omsk, Russia, 644099
Keywords: one-variable function, boundary layer, high gradients, Shishkin mesh, Lagrange interpolation, Newton, Cotes formula, error estimate
Abstract
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 ε 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, ε-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The ε-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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