Numerical solution of quasilinear attractive turning point problems (Q1205885)

A quasilinear Dirichlet problem with a small parameter \(-\varepsilon u''- xb(x,u)u'+c(x,u)=0\), \(x\in[-1,1]\) is considered. In order to construct a numerical method uniformly convergent in \(\varepsilon\), certain properties of the solution are investigated. The boundedness of the unique solution \(u_ \varepsilon\) is proved and the distance of the solution and its derivatives from the solution of the corresponding reduced problem \((\varepsilon=0)\) is estimated. The authors propose a difference scheme generated by non-uniform mesh points determined by a function dependent on \(\varepsilon\). The uniqueness of the discrete solution and the convergence in the discrete \(L_ 1\)-norm uniform with respect to \(\varepsilon\) is established. Numerical results testing the method are included. The paper is an extension of the result obtained previously for the semilinear case by the first author [RAIRO, Modélisation Math. Anal. Numér. 24, No. 6, 765-783 (1990; Zbl 0716.65075)].