Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection–diffusion equation

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 6, pp. 1010-1041

Résumé

The Dirichlet problem for a singularly perturbed ordinary differential convection–diffusion equation with a small parameter $\varepsilon$ ($\varepsilon\in (0,1]$) multiplying the higher order derivative is considered. For the problem, a difference scheme on locally uniform meshes is constructed that converges in the maximum norm conditionally, i.e., depending on the relation between the parameter $\varepsilon$ and the value $N$ defining the number of nodes in the mesh used; in particular, the scheme converges almost $\varepsilon$-uniformly (i.e., its accuracy depends weakly on $\varepsilon$). The stability of the scheme with respect to perturbations in the data and its conditioning are analyzed. The scheme is constructed using classical monotone approximations of the boundary value problem on a priori adapted grids, which are uniform on subdomains where the solution is improved. The boundaries of these subdomains are determined by a majorant of the singular component of the discrete solution. On locally uniform meshes, the difference scheme converges at a rate of $O(\min[\varepsilon^{-1}N^{-K}\ln N, 1]+N^{-1}\ln N)$, where $K$ is a prescribed number of iterations for refining the discrete solution. The scheme converges almost $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N)$ if $N^{-1}\le \varepsilon^\nu$, where $\nu$ (the defect of $\varepsilon$-uniform convergence) determines the required number $K$ of iterations ($K=K(\nu)\sim \nu^{-1}$) and can be chosen arbitrarily small from the half-open interval $(0, 1]$. The condition number of the difference scheme satisfies the bound $\boldsymbol{\kappa}_P=O(\varepsilon^{-1/K}\ln^{1/K}\varepsilon^{-1}\delta^{-(K+1)/K})$ where $\delta$ is the accuracy of the solution of the scheme in the maximum norm in the absence of perturbations. For sufficiently large $K$, the scheme is almost $\varepsilon$-uniformly strongly stable.