@Article{IJNAM-10-4,
author = {Clavero, C. and L., J., Gracia and Shishkin, I., G. and Shishkina, P., L.},
title = {Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-Hand Side},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2013},
volume = {10},
number = {4},
pages = {795--814},
abstract = {<p style="text-align: justify;">The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from
the lateral boundary inside the domain in the case when the convective flux degenerates inside
the domain and the right-hand side has the first kind discontinuity on the degeneration line. The
high-order derivative in the equation is multiplied by $\varepsilon^2$, where $\varepsilon$ is the perturbation parameter, $\varepsilon\in (0,1]$. For small values of $\varepsilon$, an interior layer appears in a neighbourhood of the set where the
right-hand side has the discontinuity. A finite difference scheme based on the standard monotone
approximation of the differential equation in the case of uniform grids converges only under the
condition $N^{-1} = o(\varepsilon)$, $N^{-1}_0 = o(1)$, where $N +1$ and $N_0+1$ are the numbers of nodes in the space
and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform
grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges $\varepsilon$-uniformly at the rate $\mathcal{O}(N^{-1}lnN+N^{-1}_0)$. Numerical experiments confirm the theoretical results.</p>},
issn = {2617-8710},
doi = {https://doi.org/2013-IJNAM-596},
url = {https://global-sci.com/article/83433/grid-approximation-of-a-singularly-perturbed-parabolic-equation-with-degenerating-convective-term-and-discontinuous-right-hand-side}
}