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Title of Journal: Numer Math

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Abbravation: Numerische Mathematik

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Springer Berlin Heidelberg

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DOI

10.1007/bf00319454

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0945-3245

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The multilevel Monte Carlo finite element method

Authors: Claude J Gittelson Juho Könnö Christoph Schwab Rolf Stenberg
Publish Date: 2013/03/21
Volume: 125, Issue: 2, Pages: 347-386
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Abstract

We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire Arch Rational Mech Anal 1133 209–2591990 doi 101007/BF00375065 Arch Rational Mech Anal 1133 261–298 1990 doi 101007/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 dtimes 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 multilevel Monte Carlo MLMC discretization of the stochastic Brinkman problem and prove that the MLMCMFEM 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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