On the Solvability of a Nonlinear Discrete Problem Corresponding to a Higher-order Nite Volume Approximation in 2d Mirko Rokyta

When studying non-linear (higher order MUSCL type) nite volume approximation of a linear convection diiusion problem, one is confronted with a question whether the corresponding nonlinear discrete problem is solvable. In this contribution we note that the corresponding discrete problem L h u h = f h in two dimensions in general need not to possess a solution. However, we show that to every right-hand side f h there is e f h , which lies in the range of L h and is \reasonably close" to f h. Moreover, we show that in one space dimension this does not happen, i.e., that the analogous discrete one-dimensional problem is always solvable.