Analysis of a singular limit of boundary conditions for convection–diffusion equations

In this work we justify the way to set the boundary conditions by certain numerical methods to solve convection–diffusion problems, in particular convection–diffusion problems that appear in turbulence models. To do it, we analyze the limit of convection–diffusion equations when the total flux is imposed in the inflow boundary, and Newmann boundary conditions are imposed in the remaining of the boundary. We prove that the solution converges in L² to the solution of the pure convection problem, with Dirichlet boundary conditions in the inflow boundary, in both the steady and evolution problems. In addition, the convective derivatives also converge in L², and the convective traces in the inflow and outflow boundaries converge in spaces of L² kind.