Homogenization approximation for PDEs with non-separated scales
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Fri, 11/06/2010 12:30 |
Lei Zhang (Hausdorff Center for Mathematics) |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
Numerical homogenization/upscaling for problems with multiple scales have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation.
In this talk, we present some rigorous results on homogenization of divergence form scalar and vectorial elliptic equations with  rough coefficients which allow for a continuum of scales. The first approach is based on a new type of compensation phenomena for scalar elliptic equations using the so-called “harmonic coordinates”. The second approach, the so-called “flux norm approach” can be applied to finite dimensional homogenization approximations of both scalar and vectorial problems with non-separated scales. It can be shown that in the flux norm, the error associated with approximating the set of solutions of the PDEs with rough coefficients, in a properly defined finite-dimensional basis, is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard finite element space. We will also talk about the ongoing work on the localization of the basis in the flux norm approach. |
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