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The multi-level Monte Carlo Finite Element Method for a stochastic Brinkman problem

by C. J. Gittelson and J. Könnö and Ch. Schwab and R. Stenberg

(Report number 2011-31)

Abstract
We present the formulation and the numerical analysis of the Brinkman problem derived rigorously in [2, 3] with a random permeability tensor. The random permeability tensor is assumed to be a lognormal random field taking values in the symmetric matrices of size $d\times d$, where $d$ denotes the spatial dimension of the physical domain $D$. We prove that the solutions admit bounded moments of any finite order with respect to the random input's Gaussian measure. We present a Mixed Finite Element discretization in the physical domain $D$ which is uniformly stable with respect to the realization of the lognormal permeability field. Based on the error analysis of this Mixed Finite Element Method (MFEM), we develop a Multi-Level Monte Carlo (MLMC) discretization of the stochastic Brinkman problem and prove that the MLMC-MFEM allows to estimate the statistical mean field with asymptotically the same accuracy versus work as the MFEM for a single instance of the stochastic Brinkman problem. The robustness of the MFEM implies in particular that the present analysis also covers the Darcy diffusion limit. Numerical experiments confirm the theoretical results.

Keywords:

@Techreport{GKSS11_94,
  author = {C. J. Gittelson and J. K\"onn\"o and Ch. Schwab and R. Stenberg},
  title = {The multi-level Monte Carlo Finite Element Method for a stochastic Brinkman problem },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-31},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-31.pdf },
  year = {2011}
}

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