Higher-order accurate method for a quasilinear singularly perturbed elliptic convection-diffusion equation

G. I. Shishkin

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 1, pp. 81-108

Résumé

We consider the Dirichlet problem on a rectangle for a quasilinear singularly perturbed elliptic convection-diffusion equation in the case when the domain has no characteristic parts of its boundary; the higher derivatives of the equation contain a parameter е that takes arbitrary values in the half-interval (0,1]. For a linear problem of this type, the $\varepsilon$-uniform rate of convergence for well-known schemes has not higher than the first order (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge $\varepsilon$-uniformly at the rate $O(N^{-2}\ln^2N)$, where $N$ specifies the number of mesh points in each variable. The piecewise uniform meshes, condensing in the boundary layer, are used. When the values of the parameter are small as compared to the effective mesh size, we apply the domain decomposition method, which is motivated by “asymptotic constructions”. We use monotone approximations of “auxiliary” subproblems that describe the main terms of asymptotic representations of the solutions inside and outside the vicinity of the regular and the angular boundary layers. The above subproblems are solved sequentially on subdomains using uniform meshes. If the values of the parameter are not sufficiently small (as compared to the effective mesh size), then classical finite difference schemes are employed, where the first derivatives are approximated by central difference derivatives. Note that the computation of solutions of the constructed difference scheme, based on the method of “asymptotic constructions”, is essentially simplified for sufficiently small values of the parameter $\varepsilon$.