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), 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 separatirces, 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. c © 2000 Wiley Periodicals, Inc.