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

Gittelson, Claude ; Könnö, Juho ; Schwab, Christoph ; Stenberg, Rolf

In: Numerische Mathematik, 2013, vol. 125, no. 2, p. 347-386

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    Titre
    • The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem
    Auteur
    • Gittelson, Claude. Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland
    • Könnö, Juho. Wärtsilä Finland Oy, Järvikatu 2-4, 65101, Vaasa, Finland
    • Schwab, Christoph. Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland
    • Stenberg, Rolf. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076, Aalto, Finland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Numerische Mathematik, 2013, vol. 125, no. 2, p. 347-386. Springer Berlin Heidelberg
    Autre version électronique
    Classification
    • Informatique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:322495
    Summary
    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