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Volume 1, Issue 2
A Finite Difference Scheme on $a$ $priori$ Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation

Grigory I. Shishkin

Numer. Math. Theor. Meth. Appl., 1 (2008), pp. 214-234.

Published online: 2008-01

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  • Abstract

A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on $a$ $priori$ (sequentially) adapted meshes and study its convergence. The scheme on $a$ $priori$ adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find $a$ $priori$ a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter $ε$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations $K$ for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter $ε$. The scheme converges almost $ε$-uniformly, precisely, under the condition $N^{-1}=o\left(ε^{\nu}\right)$, where $N$ denotes the number of nodes in the spatial mesh, and the value $\nu=\nu(K)$ can be chosen arbitrarily small for suitable $K$.

  • Keywords

Singular perturbations, convection-diffusion problem, piecewise-uniform mesh, a priori adapted mesh, almost $ε$-uniform convergence.

  • AMS Subject Headings

65M06, 65M12, 65M15, 65M50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-1-214, author = {}, title = {A Finite Difference Scheme on $a$ $priori$ Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2008}, volume = {1}, number = {2}, pages = {214--234}, abstract = {

A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on $a$ $priori$ (sequentially) adapted meshes and study its convergence. The scheme on $a$ $priori$ adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find $a$ $priori$ a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter $ε$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations $K$ for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter $ε$. The scheme converges almost $ε$-uniformly, precisely, under the condition $N^{-1}=o\left(ε^{\nu}\right)$, where $N$ denotes the number of nodes in the spatial mesh, and the value $\nu=\nu(K)$ can be chosen arbitrarily small for suitable $K$.

}, issn = {2079-7338}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nmtma/6049.html} }
TY - JOUR T1 - A Finite Difference Scheme on $a$ $priori$ Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 214 EP - 234 PY - 2008 DA - 2008/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nmtma/6049.html KW - Singular perturbations, convection-diffusion problem, piecewise-uniform mesh, a priori adapted mesh, almost $ε$-uniform convergence. AB -

A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on $a$ $priori$ (sequentially) adapted meshes and study its convergence. The scheme on $a$ $priori$ adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find $a$ $priori$ a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter $ε$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations $K$ for improving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter $ε$. The scheme converges almost $ε$-uniformly, precisely, under the condition $N^{-1}=o\left(ε^{\nu}\right)$, where $N$ denotes the number of nodes in the spatial mesh, and the value $\nu=\nu(K)$ can be chosen arbitrarily small for suitable $K$.

Grigory I. Shishkin. (2020). A Finite Difference Scheme on $a$ $priori$ Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation. Numerical Mathematics: Theory, Methods and Applications. 1 (2). 214-234. doi:
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