Boundary layers for cellular flows at high Péclet numbers

We analyze the behavior of solutions of steady advection-diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ≫ 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(ε ^1/2), ε = Pe ^-1, around the flow separatrices. We construct an ε-dependent approximate "water pipe problem" purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ε-independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ε-independent problem on this graph. Finally, we show that the Dirichlet-to-Neumann map of the water pipe problem approximates the Dirichlet-to-Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers.