The content discusses the issue of boundary resonance error in numerical homogenization of multiscale elliptic equations. Key points:
Numerical homogenization typically requires solving a microscale problem on a domain whose size η does not match the true periodicity ε of the heterogeneous media, leading to a boundary or "cell resonance" error that is proportional to ε/η.
This error can dominate discretization error and pollute the entire homogenization scheme, so its reduction is important.
The authors present an alternative procedure to reduce the resonance error, based on the observation that the error itself is an oscillatory function of the domain size η.
For 1D problems, the authors derive an explicit expression for the resonance error, showing it can be written as 1/x P(x) + R(x), where x = η/ε, P is periodic, and R is a corrector term.
For 2D "tubular" domains, the authors show the resonance error can also be written in this form, with R given by the average of a locally ε-periodic function.
More generally, the authors propose a numerical strategy that solves the microscale problem for a sequence of domain sizes η, and averages the results against kernels satisfying certain moment conditions to accelerate the convergence of the resonance error to zero.
Numerical examples in 1D and 2D illustrate the effectiveness of the approach.
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by Sean P. Carn... at arxiv.org 04-03-2024
https://arxiv.org/pdf/2308.07563.pdfDeeper Inquiries