Geodesic
Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 13 (2007) no. 2, pp. 218-233 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter $\varepsilon^2$, $\varepsilon\in (0,1]$. For small values of the parameter $\varepsilon$, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width $\varepsilon$), respectively, which have bounded smoothness for fixed values of the parameter $\varepsilon$. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge $\varepsilon$-uniformly with the second order of accuracy in $x$ and the first order of accuracy in $t$, up to logarithmic factors. 
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G. I. Shishkin. Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 13 (2007) no. 2, pp. 218-233. http://geodesic.mathdoc.fr/item/TIMM_2007_13_2_a17/

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