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Reducing the Boundary Resonance Error in Numerical Homogenization


Core Concepts
The cell resonance error, which can dominate discretization error in numerical homogenization, can be asymptotically reduced by taking a weighted average of the homogenized coefficients computed at different domain sizes.
Abstract
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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Deeper Inquiries

How would the proposed method perform in the case of non-periodic or random heterogeneous media, where the homogenized coefficients cannot be expressed in closed form

In the case of non-periodic or random heterogeneous media, where the homogenized coefficients cannot be expressed in closed form, the proposed method may still be applicable. The method focuses on reducing the boundary resonance error by taking a weighted average at several domain sizes. This approach is based on the observation that the error itself is an oscillatory function of the domain size. By averaging the estimates of the homogenized coefficients over different domain sizes, the method aims to improve the accuracy of the approximation. For non-periodic or random heterogeneous media, the method could still be effective in reducing the boundary resonance error. Even though closed-form expressions for the homogenized coefficients may not be available, the numerical averaging technique can help in obtaining more accurate estimates. The key lies in choosing appropriate averaging kernels with specific properties that accelerate the convergence of the error to zero. By applying these kernels to the numerical solutions obtained for different domain sizes, the method can potentially provide improved approximations for the homogenized coefficients in non-periodic or random media.

Can the theoretical analysis of the resonance error be extended to higher dimensional domains beyond the "tubular" case considered here

The theoretical analysis of the resonance error can be extended to higher dimensional domains beyond the "tubular" case considered in the context. While the specific analysis in the context focused on a two-dimensional tubular domain, the underlying principles and techniques can be generalized to higher dimensions. The key lies in understanding the behavior of the cell resonance error as a function of the domain size and developing strategies to reduce this error systematically. In higher dimensional domains, the resonance error may exhibit more complex behavior due to the increased dimensionality. However, by applying similar decomposition and averaging techniques, it should be possible to characterize the error and devise methods to mitigate its impact. Extending the analysis to higher dimensions would involve adapting the theoretical framework to account for the additional complexities introduced by higher-dimensional geometries and the corresponding solutions to the microscale problems.

What are the potential connections between the proposed method and existing reduced basis techniques in numerical homogenization, and how could they be leveraged to further improve computational efficiency

There are potential connections between the proposed method for reducing the boundary resonance error and existing reduced basis techniques in numerical homogenization. Reduced basis (RB) techniques are commonly used to reduce the computational cost of solving parametrized PDEs by constructing low-dimensional approximations of the solution space. In the context of numerical homogenization, RB techniques can be leveraged to efficiently compute the homogenized coefficients for a wide range of parameter values. The proposed method could benefit from RB techniques by incorporating the reduced basis approximation into the process of solving the microscale problems at different domain sizes. By precomputing and storing solutions for a reduced set of parameters, the method can accelerate the computation of the homogenized coefficients for varying domain sizes. This integration of RB techniques can lead to significant computational savings, especially when dealing with high-dimensional parameter spaces or complex heterogeneous media. By combining the proposed method for reducing the boundary resonance error with RB techniques, it is possible to enhance the overall efficiency and scalability of numerical homogenization algorithms, making them more suitable for practical applications involving non-periodic or random heterogeneous media.
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