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Sparse finite elements for elliptic problems with stochastic loading

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We formulate elliptic boundary value problems with stochastic loading in a bounded domain D⊂ℝd. We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in D×D for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic finite element (FE) spaces in D yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in D×D allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation.

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Authors and Affiliations

  1. Seminar for Applied Mathematics, ETH-Zentrum, 8092, Zürich, Switzerland

    Christoph Schwab & Radu-Alexandru Todor

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  1. Christoph Schwab
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  2. Radu-Alexandru Todor
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Correspondence to Christoph Schwab.

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Mathematics Subject Classification (2000):  65N30

Research performed under IHP network Breaking Complexity of the EC, contract number HPRN-CT-2002-00286, and supported in part by the Swiss Federal Office for Science and Education under grant number BBW 02.0418.

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Schwab, C., Todor, RA. Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95, 707–734 (2003). https://doi.org/10.1007/s00211-003-0455-z

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  • DOI: https://doi.org/10.1007/s00211-003-0455-z

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