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Expansion of random boundary excitations for elliptic PDEs

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  • Sabelfeld Karl

    (Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, and Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Lavrentieva str. 6, 630090 Novosibirsk, Germany/Russia. Email: sabelfel@wias-berlin.de)

Abstract

In this paper we deal with elliptic boundary value problems with random boundary conditions. Solutions to these problems are inhomogeneous random fields which can be represented as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen–Loève series expansion which is based on the eigen-decomposition of the covariance operator. It can be applied to simulate both homogeneous and inhomogeneous random fields. We study the correlation structure of solutions to some classical elliptic equations in respond to random excitations of functions prescribed on the boundary. We analyze the stochastic solutions for Dirichlet and Neumann boundary conditions to Laplace equation, biharmonic equation, and to the Lamé system of elasticity equations. Explicit formulae for the correlation tensors of the generalized solutions are obtained when the boundary function is a white noise, or a homogeneous random field on a circle, a sphere, and a half-space. These exact results may serve as an excellent benchmark for developing numerical methods, e.g., Monte Carlo simulations, stochastic volume and boundary element methods.

Suggested Citation

  • Sabelfeld Karl, 2008. "Expansion of random boundary excitations for elliptic PDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 13(5-6), pages 405-453, January.
    • Handle: RePEc:bpj:mcmeap:v:13:y:2008:i:5-6:p:405-453:n:5
    DOI: 10.1515/mcma.2007.022
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    References listed on IDEAS

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    1. Kurbanmuradov O. & Sabelfeld K., 2006. "Stochastic Spectral and Fourier-Wavelet Methods for Vector Gaussian Random Fields," Monte Carlo Methods and Applications, De Gruyter, vol. 12(5), pages 395-445, November.
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