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Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients

Stochastics and Partial Differential Equations: Analysis and Computations volume 8, pages 787–818 (2020)Cite this article

Abstract

We study the iterative algorithm proposed by Armstrong et al. (An iterative method for elliptic problems with rapidly oscillating coefficients, 2018. arXiv preprint arXiv:1803.03551) to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating coefficients and thus usually require very expensive numerical calculations, while this iterative method is comparatively easy to compute. In this article, we strengthen the estimate for the contraction factor achieved by one iteration of the algorithm. We obtain an estimate that holds uniformly over the initial function in the iteration, and which grows only logarithmically with the size of the domain.

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Acknowledgements

I am grateful to Jean-Christophe Mourrat for his suggestion to study this topic, helpful discussions and detailed reading of the article.

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Authors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Chenlin Gu

  2. DMA, Ecole Normale Supérieure, PSL Research University, Paris, France

    Chenlin Gu

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  1. Chenlin Gu
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Correspondence to Chenlin Gu.

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Gu, C. Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients. Stoch PDE: Anal Comp 8, 787–818 (2020). https://doi.org/10.1007/s40072-019-00159-1

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  • DOI: https://doi.org/10.1007/s40072-019-00159-1

Keywords

  • Stochastic homogenization
  • Elliptic equation
  • Numerical algorithm
  • Iterative method