Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: $\varepsilon$-uniformly convergent schemes

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 6, pp. 1014-1033

Résumé

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter $\varepsilon$, the size of the uniform mesh in $x$, the desired accuracy of the grid solution, and the prescribed number of iterations $K$ used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on $\varepsilon$. The scheme converges almost $\varepsilon$-uniformly; namely, it converges under the condition $N^{-1}=o(\varepsilon^\nu)$, where $\nu=\nu(K)$ can be chosen arbitrarily small when $K$ is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final $K$ th iteration, the difference scheme converges $\varepsilon$-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in $x$ on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known $\varepsilon$-uniformly convergent schemes on piecewise uniform grids.