Grid approximation of a singularly perturbed quasilinear parabolic convection-diffusion equation on a priori adapted meshes

G. I. Shishkin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 149 (2007) no. 4, pp. 146-172

Résumé

An initial-boundary value problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation. For such a problem, a solution of a classical difference scheme on uniform grid converges at the rate $\mathcal O((\varepsilon+N^{-1})^{-1}N^{-1}+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of nodes in the meshes in $x$ and $t$ respectively; the scheme converges only under the condition $N^{-1}\ll\varepsilon$. In the present paper, nonlinear and linearized finite difference schemes are constructed on a priori sequentially adapted grids, and their convergence is studied. The construction of the schemes is carried out on the basis of a majorant to the singular component of the discrete solution on uniform grids that allows us to find a priori subdomains where the computed solution requires a further improvement. Such subdomain is defined by the perturbation parameter $\varepsilon$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the grid solution and the prescribed number $K$ of iterations to refine the solution. The advantage of this approach consists in the uniform meshes used. The error of the discrete solution depends weakly on the parameter $\varepsilon$. The schemes that are constructed in the iterative process converge almost $\varepsilon$-uniformly, namely, under the condition $N^{-1}\ll\varepsilon^{\nu}$, where the value $\nu=\nu(K)$ can be chosen arbitrarily small for sufficiently large $K$.