Asymptotic behaviour of three-dimensional singularly perturbed convection–diffusion problems with discontinuous data

We consider three singularly perturbed convection-diffusion problems defined in three-dimensional domains: (i) a parabolic problem $-\epsilon(u_{xx}+u_{yy})+u_t +v_1u_x+v_2u_y=0$ in an octant, (ii) an elliptic problem $-\epsilon(u_{xx}+u_{yy}+u_{zz}) +v_1u_x+v_2u_y+v_3u_z=0$ in an octant and (iii) the same elliptic problem in a half space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter $\epsilon\to 0^+$. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small $\epsilon-$ behaviour of the solution and its derivatives in the boundary layers or the internal layers