Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer

I. A. Blatov ; N. A. Zadorin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 161 (2019) no. 4, pp. 497-508

Résumé

Interpolation of the function of one variable with large gradients in the region of the exponential boundary layer was studied. The interpolated function corresponds to solution of a boundary value problem for an ordinary differential equation of the second order with the small parameter $\varepsilon$ before the highest derivative. Applying classical polynomial interpolation formulas on a uniform mesh to this function can lead to unacceptable errors. In the paper, the error of the piecewise linear interpolation formula on the Bakhvalov mesh condensing in the region of the boundary layer was estimated. The Bakhvalov mesh is used in a number of works when constructing difference schemes for singularly perturbed problems; therefore, estimating the error of interpolation formulas on this mesh is of interest. An error estimate of the order of $O(1/N^2)$ was obtained uniformly with respect to the parameter $\varepsilon, $ where $N$ is the number of mesh nodes. The problem of computing the derivative of the function with large gradients given in the nodes of the Bakhvalov mesh was investigated. The classical difference formula with two nodes was considered obtained by differentiating the linear interpolant studied above. An estimate of the relative error of the order of $O(1/N),$ uniform in the parameter $\varepsilon,$ was obtained. The results of the numerical experiments consistent with the obtained error estimates were presented. Numerical comparison of the errors obtained during the interpolation and numerical differentiation on the Bakhvalov mesh with errors on the Shishkin mesh and on the uniform mesh was carried out.

Détail

Keywords: function of one variable, boundary layer, Bakhvalov mesh, piecewise linear interpolation, numerical differentiation, $\varepsilon$-uniform error estimation.

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