Using the technique of majorant functions in approximation of a singular perturbed parabolic convection–diffusion equation on adaptive grids

Abstract

A grid approximation of the Dirichlet problem is considered on a segment for a parabolic convection-diffusion equation; the high derivative of the equation contains a parameter ε taking arbitrary values from the half-interval (0,1]. A difference scheme on a posteriori adaptive grids is constructed for the boundary value problem. The classic approximation of the equation on uniform grids in the main domain and also in domains refined to improve the accuracy of the grid solution are used. Such subdomains are determined based on majorant functions for singular components of the solutions to the boundary value problem and the difference scheme. Special schemes on a posteriori piecewise-uniform grids are constructed, these schemes permit to obtain approximations convergent in the whole grid domain 'almost e-uniformly', namely, with an error weakly depending on the value of the parameter ε; the scheme converges ε-uniformly with the first order of accuracy (up to logarithmic cofactors) outside a sufficiently small neighborhood of the 'outlet' part of the boundary through which the characteristics of the limit equation leave the domain.