Finite volume schemes for non-coercive elliptic problems with Neumann boundary conditions

We consider a convective-diffusive elliptic problem with Neumann boundary conditions: the presence of the convective term entails the non-coercivity of the continuous equation and, because of the boundary conditions, the equation has a kernel. We discretize this equation with finite volume techniques and in a general framework which allows to consider several treatments of the convective term: either via a centered scheme, an upwind scheme (widely used in fluid mechanics problems) or a Scharfetter-Gummel scheme (common to semiconductor literature). We prove that these schemes satisfy the same properties as the continuous problem (one-dimensional kernel spanned by a positive function for instance) and that their kernel and solution converge to the kernel and solution of the PDE. We also present several numerical implementations, studying the effects of the choice of one scheme or the other in the approximation of the solution or the kernel.