Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer
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 ε before th...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2019-12-01
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| Series: | Учёные записки Казанского университета. Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/uz-eng-phm-2019-4-2.html |
| _version_ | 1797970769738727424 |
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| author | I.A. Blatov N.A. Zadorin |
| author_facet | I.A. Blatov N.A. Zadorin |
| author_sort | I.A. Blatov |
| collection | DOAJ |
| description | 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 ε 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 ε, 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 ε, 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. |
| first_indexed | 2024-04-11T03:21:09Z |
| format | Article |
| id | doaj.art-2173c96630c24fc1ad6f421630953575 |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2024-04-11T03:21:09Z |
| publishDate | 2019-12-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета. Серия Физико-математические науки |
| spelling | doaj.art-2173c96630c24fc1ad6f4216309535752023-01-02T08:42:36ZengKazan Federal UniversityУчёные записки Казанского университета. Серия Физико-математические науки2541-77462500-21982019-12-01161449750810.26907/2541-7746.2019.4.497-508Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layerI.A. Blatov0N.A. Zadorin1Povolzhskiy State University of Telecommunications and Informatics, Samara, 443010 RussiaSobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 RussiaInterpolation 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 ε 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 ε, 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 ε, 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.https://kpfu.ru/uz-eng-phm-2019-4-2.htmlfunction of one variableboundary layerbakhvalov meshpiecewise linear interpolationnumerical differentiationε -uniform error estimation |
| spellingShingle | I.A. Blatov N.A. Zadorin Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer Учёные записки Казанского университета. Серия Физико-математические науки function of one variable boundary layer bakhvalov mesh piecewise linear interpolation numerical differentiation ε -uniform error estimation |
| title | Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer |
| title_full | Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer |
| title_fullStr | Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer |
| title_full_unstemmed | Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer |
| title_short | Interpolation on the Bakhvalov mesh in the presence of an exponential boundary layer |
| title_sort | interpolation on the bakhvalov mesh in the presence of an exponential boundary layer |
| topic | function of one variable boundary layer bakhvalov mesh piecewise linear interpolation numerical differentiation ε -uniform error estimation |
| url | https://kpfu.ru/uz-eng-phm-2019-4-2.html |
| work_keys_str_mv | AT iablatov interpolationonthebakhvalovmeshinthepresenceofanexponentialboundarylayer AT nazadorin interpolationonthebakhvalovmeshinthepresenceofanexponentialboundarylayer |