Grid approximation of boundary value problems for singularly perturbed quasi-linear elliptic equations with interior layer

The first boundary value problem for two dimension quasi-linear elliptic equation of second order is considered. Highest derivatives of the equation are multiplied by a parameter which can get any value on interval $(0,1]$. When the parameter is equal to zero the reduced equation is a quasi-linear first order equation. An interior layer appears when the parameter tends to zero. The considered problem is as model for problems which appear when the non-linear shock waves are investigated. With using of special condensing grids we construct the special difference schemes, which converge uniformly with respect the parameter.