Lifting solutions of quasilinear convection-dominated problems

The steady state of the quasilinear convection-diffusion-reaction equation ut −∇(D(u)∇u) + b(u)∇u+ c(u) = 0 (1) is studied. Depending on the ratio between convection and diffusion coefficients, equation (1) ranges from parabolic to almost hyperbolic. From a numerical point of view two main difficulties can arise related with the existence of layers and/or the non smoothness of the coefficients. In this talk we present a new numerical method for solving the steady state equation associated with (1). This method is based on the idea of solving an associated modified problem whose solution corresponds to a lifting of u. The method introduced here avoids an a priori knowledge of the layer(s) location and allows an efficient handling of the lack of the smoothness of the coefficients. Numerical results are included.