Approximation of Systems of Singularly Perturbed Elliptic Reaction-Diffusion Equations with Two Parameters

TLDR: In this article, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered, where the higher order derivatives of the ith equation are multiplied by the perturbation parameter ǫi2 (i = 1, 2).

Abstract: In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ɛi2 (i = 1, 2). The parameters ɛi take arbitrary values in the half-open interval (0, 1]. When the vector parameter ɛ = (ɛ1, ɛ2) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ɛ1 and (or) ɛ2 tend to zero, a double boundary layer with the characteristic width ɛ1 and ɛ2 appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge ɛ-uniformly at the rate of O(N−2ln2N) are constructed, where N = min Ns, Ns + 1 is the number of mesh points on the axis xs.