On Numerical Solution of Quasilinear Boundary Value Problems with Two Small Parameters

We consider the singular perturbation problem $$-varepsilon^2u"+mu b(x,u)u'+c(x,u)=0,u(0),u(1)$$ given with two small parameters $varepsilon$ and $mu$ , $mu =varepsilon^{1+p},p>0$. The problem is solved numerically by using finite difference schemes on the mesh which is dense in the boundary layers. The convergence uniform in $varepsilon$ is proved in the discrete $L^1$ norm. Some convergence results are given in the maximum norm as well.