Allaire G., Amar M. Boundary layer tails in periodic homogeneization. Technical report ercim.cnr.ian//1998-1087, 1998. |

Abstract (English) |
This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. Indeed, boundary layers are often more important for improving the rate of convergence than the usual periodic correctors. In truth we are not interested in computing exactly the full set of boundary layers (neither are we interested in obtaining a complete asymptotic expansion, valid at any order). Rather, we seek the non-oscillating tails of such boundary layers away from the boundary, and we determine if their knowledge improves, or not, the convergence rate of the homogenization process. It turns out that these boundary layers tails can be incorporated into the homogenized equation by adding dispersive terms and a Fourier boundary condition. We therefore derived optimal interior estimates using these simple boundary layer tails. Another feature of our work is that we focus on error estimates in the $L^2$-norm rather than in the $H^1$-norm as usual. The reason for this is that, in many applications, it is preferable to have a good approximation of the unknown itself rather than of its gradient. Boundary layers are often negligible for interior error estimates in the $H^1$-norm at first order, but not in the $L^2$-norm at second-order (recall that the $L^2$ error estimate at first-order is trivial). Part of the novelty of our work comes from this focus on higher order interior error estimates in the $L^2$-norm. | |

Subject | Boundary layers |

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