Conservative Upwind Finite Element Approximation and Its Applications

We present a finite element approximation which is effective for diffusion problems with dominated non-linear convection terms. This approximation is conservative and of upwind type. We analyse it in connection with L² -theory, L¹ -contraction and monotonicity. As an application we consider a non-linear elliptic problem and give error analysis. In the process the existence of the exact solution is also proved.