The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter $\varepsilon$ taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost $\varepsilon$-uniformly, i.e., with an error that weakly depends on the parameter $\varepsilon$: $|u(x,t)-z(x,t)|\le M[N_1^{-1}\ln^2N_1+N_0^{-1}\ln N_0+\varepsilon^{-1}N_1^{-K}\ln^{K-1}N_1]$, $(x,t)\in\bar G_h$, where $N_1+1$ и $N_0+1$ are the numbers of grid points in $x$ and $t$, respectively; $K$ is the number of refinement iterations (with respect to $x$) in the adapted grid; and $M=M(K)$. Outside the $\sigma$-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges $\varepsilon$-uniformly at a rate $O(N_1^{-1}\ln^2N_1+N_0^{-1}\ln N_0)$, причем $\sigma\le MN_1^{-K+1}\ln^{K-1}N_1$ for $K\ge2$.