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Volume 7, Issue 3
Pointwise Approximation of Corner Singularities for Singularly Perturbed Elliptic Problems with Characteristic Layers

V. B. Andreev

Int. J. Numer. Anal. Mod., 7 (2010), pp. 416-427.

Published online: 2010-07

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  • Abstract

A Dirichlet problem for a singularly perturbed steady-state convection-diffusion equation with constant coefficients on the unit square is considered. In the equation under consideration the convection term is represented by only a single derivative with respect to one coordinate axis. This problem is discretized by the classical five-point upwind difference scheme on a rectangular piecewise uniform mesh that is refined in the neighborhood of the regular and the characteristic boundary layers. It is proved that, for sufficiently smooth right-hand side of the equation and the restrictions of the continuous boundary function to the sides of the square, without additional compatibility conditions at the corners, the error of the discrete solution is $O(N^{-1}\ln^2 N)$ uniformly with respect to the small parameter, in the discrete maximum norm, where $N$ is the number of mesh points in each coordinate direction.

  • Keywords

Parabolic boundary layers, elliptic equation, piecewise uniform mesh, corner singularities.

  • AMS Subject Headings

65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-7-416, author = {V. B. Andreev}, title = {Pointwise Approximation of Corner Singularities for Singularly Perturbed Elliptic Problems with Characteristic Layers}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2010}, volume = {7}, number = {3}, pages = {416--427}, abstract = {

A Dirichlet problem for a singularly perturbed steady-state convection-diffusion equation with constant coefficients on the unit square is considered. In the equation under consideration the convection term is represented by only a single derivative with respect to one coordinate axis. This problem is discretized by the classical five-point upwind difference scheme on a rectangular piecewise uniform mesh that is refined in the neighborhood of the regular and the characteristic boundary layers. It is proved that, for sufficiently smooth right-hand side of the equation and the restrictions of the continuous boundary function to the sides of the square, without additional compatibility conditions at the corners, the error of the discrete solution is $O(N^{-1}\ln^2 N)$ uniformly with respect to the small parameter, in the discrete maximum norm, where $N$ is the number of mesh points in each coordinate direction.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/728.html} }
TY - JOUR T1 - Pointwise Approximation of Corner Singularities for Singularly Perturbed Elliptic Problems with Characteristic Layers AU - V. B. Andreev JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 416 EP - 427 PY - 2010 DA - 2010/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/728.html KW - Parabolic boundary layers, elliptic equation, piecewise uniform mesh, corner singularities. AB -

A Dirichlet problem for a singularly perturbed steady-state convection-diffusion equation with constant coefficients on the unit square is considered. In the equation under consideration the convection term is represented by only a single derivative with respect to one coordinate axis. This problem is discretized by the classical five-point upwind difference scheme on a rectangular piecewise uniform mesh that is refined in the neighborhood of the regular and the characteristic boundary layers. It is proved that, for sufficiently smooth right-hand side of the equation and the restrictions of the continuous boundary function to the sides of the square, without additional compatibility conditions at the corners, the error of the discrete solution is $O(N^{-1}\ln^2 N)$ uniformly with respect to the small parameter, in the discrete maximum norm, where $N$ is the number of mesh points in each coordinate direction.

V. B. Andreev. (2010). Pointwise Approximation of Corner Singularities for Singularly Perturbed Elliptic Problems with Characteristic Layers. International Journal of Numerical Analysis and Modeling. 7 (3). 416-427. doi:
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