Abstract
For a Dirichlet problem for an one-dimensional singularly perturbed parabolic convection-diffusion equation, a difference scheme of the solution decomposition method is constructed. This method involves a special decomposition based on the asymptotic construction technique in which the regular and singular components of the grid solution are solutions of grid subproblems solved on uniform grids, moreover, the coefficients of the grid equations do not depend on the singular component of the solution unlike the fitted operator method. The constructed scheme converges in the maximum norm \(\varepsilon \)-uniformly (i.e., independent of a perturbation parameter \(\varepsilon \), \(\varepsilon \in (0,1]\)) at the rate \(\mathcal{O}\left ({N}^{-1}\ln N + N_{0}^{-1}\right )\) the same as a scheme of the condensing grid method on a piecewise-uniform grid (here N and N 0 define the numbers of the nodes in the spatial and time meshes, respectively).
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Notes
- 1.
Notation L (j. k) (\(\overline{G}_{(j.k)},M_{(j.k)}\)) means that this operator (domain, constant) is introduced in the formula (j. k).
- 2.
We denote by M (by m) sufficiently large (small) positive constants that do not depend on the value of the parameter \(\varepsilon \). In the case of grid problems, these constants are also independent of the stencils of the difference schemes.
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Acknowledgements
This research was supported by the Russian Foundation for Basic Research under grant No. 10-01-00726.
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Shishkina, L., Shishkin, G. (2013). Difference Scheme of the Solution Decomposition Method for a Singularly Perturbed Parabolic Convection-Diffusion Equation. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_33
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