Computer difference scheme for a singularly perturbed elliptic convection-diffusion equation in the presence of perturbations

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 5, pp. 814-831

Résumé

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection-diffusion equation with a perturbation parameter $\varepsilon$, $\varepsilon\in(0, 1]$, multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge $\varepsilon$-uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if $N_1^{-1}N_2^{-1}\ll\varepsilon$, where $N_1$ and $N_2$ are the numbers of grid intervals in $x$ and $y$, respectively, the scheme is not -uniformly well-conditioned or $\varepsilon$-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on $\varepsilon$, $N_1$, $N_2$, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as $N_1$, $N_2\to\infty$, $\varepsilon\in(0, 1]$. The difference schemes constructed in the presence of the indicated perturbations that converges as $N_1$, $N_2\to\infty$ for fixed $\varepsilon$, $\varepsilon\in(0, 1]$, is called a computer difference scheme. Schemes converging $\varepsilon$-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.