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The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem

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Abstract

We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire (Arch Rational Mech Anal 113(3): 209–259,1990. doi:10.1007/BF00375065, Arch Rational Mech Anal 113(3): 261–298, 1990. doi:10.1007/BF00375066) with a lognormal random permeability. Specifically, the permeability 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 the estimation of the statistical mean field with the same asymptotical 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.

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Notes

  1. CSC - IT Center for Science, www.csc.fi.

  2. Matlab R2010a, compiler version 4.11, gcc version 4.4.3.

  3. OpenMPI version 1.4.3.

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Acknowledgments

This research was supported in part by the Swiss National Science Foundation grant No. SNF 200021-120290/1 and by the European Research Council under grant ERC AdG 247277 to CS.

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Author notes

  1. Claude J. Gittelson

    Present address: Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

Authors and Affiliations

  1. Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, 8092 , Zurich, Switzerland

    Claude J. Gittelson & Christoph Schwab

  2. Wärtsilä Finland Oy, Järvikatu 2-4, 65101 , Vaasa, Finland

    Juho Könnö

  3. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076, Aalto, Finland

    Rolf Stenberg

Authors
  1. Claude J. Gittelson
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  2. Juho Könnö
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  3. Christoph Schwab
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  4. Rolf Stenberg
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Correspondence to Christoph Schwab.

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Gittelson, C.J., Könnö, J., Schwab, C. et al. The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem. Numer. Math. 125, 347–386 (2013). https://doi.org/10.1007/s00211-013-0537-5

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  • DOI: https://doi.org/10.1007/s00211-013-0537-5

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