Grid approximation of the domain and solution decomposition method with improved convergence rate for singularly perturbed elliptic equations in domains with characteristic boundaries

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 7, pp. 1196-1212

Résumé

In a rectangle, the Dirichlet problem for singularly perturbed elliptic equations with convection terms is considered in the case when the characteristics of the reduced equations are parallel to the rectangle sides. The higher order derivatives in the equations are multiplied by a perturbation parameter $\tilde\varepsilon=\varepsilon^2$ that can take arbitrary values in the half-open interval $(0,1]$. For such convection–diffusion problems, the order of the $\varepsilon$-uniform convergence (in the maximum norm) of the well-known special schemes on piecewise uniform grids is not higher than unity (with respect to the variable along the flow). In this paper, a scheme on piecewise uniform grids is constructed that converges $\varepsilon$-uniformly at the rate of $O(N^{-2}\ln^2N)$, where $N$ specifies the number of mesh points with respect to each variable. When is not very small (compared to the effective mesh size in the direction of the convective flow) this scheme approximates the equation using central difference derivatives. For small $\tilde\varepsilon$, a domain decomposition method is used; more precisely, the problem is considered separately in a neighborhood of the outflow part of the domain boundary and outside this neighborhood. In the neighborhood of the outflow part of the boundary, central difference derivatives are used. Outside this neighborhood, the solution is decomposed. The regular part of the solution and the parabolic boundary layer are found by solving the corresponding problems, in which the convective term is approximated by the upwind difference derivative. The order of approximation of the convective term is improved due to a correction of the defect.

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