3:00 pm Friday, November 19, 2010
Math/ICES Center of Numerical Analysis Seminar : Corrector Theory for MsFEM and HMM in Random Media by
Wenjia Jing (Columbia) in ACE 6.304
Corrector theory, i.e., theory about the deviation from homogenization, for some linear equations in heterogeneous random media has been established in the stationary ergodic setting. Here, we develop methods to assess whether certain numerical schemes, which successfully approximate the homogenized solution, are able to capture the right corrector indicated by the theory, as the discretization size goes to zero. We analyze, in particular, the Multi-scale Finite Element Method (MsFEM) and the Heterogeneous Multi-scale Method (HMM) applied to a one-dimensional second order ODE with elliptic random coefficient. The corrector for this equation is characterized in [Bourgeat and Piatnitski 1999] for short range medium, and in [Bal, Garnier, Motsch, and Perrier 2008] for long range medium, as Gaussian processes. Our analysis on the numerical schemes shows that, the MsFEM corrector converges to the right one but the HMM corrector converges to an amplified version, the amplification factor depending on a parameter in the scheme and on the parameter describing how much the random medium is correlated. We then propose modifications of HMM that eliminate this amplification effect. Our proofs are based on detailed analysis of the structure of the stiffness matrix and statistics of its entries, combined with central limit theorems and tools to prove weak convergence in the space of continuous paths. This is a joint work with Guillaume Bal.
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