Grid approximation of quasi-linear singularly perturbed elliptic and parabolic equations with mixed boundary conditions

G. I. Shishkin

Matematičeskoe modelirovanie, Tome 7 (1995) no. 10, pp. 111-126.

Résumé

Boundary value problems for quasi-linear elliptic and parabolic equations with mixed boundary condition are considered on a restangular domain. Highest derivatives of these equations are multiplied by a parameter which can get any value in the interval $(0,1]$. When the parameter is equal to zero the elliptic equations are reduced to the zero order equations and the parabolic equations are reduced to the first order equations which don't contain spatial derivatives. For such problems error of classical approximations of the boundary value problem on a uniform grid increases infinitely, when the parameter tends to zero. Using of special condensing grids we construct the special difference schemes, which converge uniformly with respect to the parameter.