Article

Numerische Mathematik

, Volume 109, Issue 3, pp 385-414

First online:

Sparse second moment analysis for elliptic problems in stochastic domains

  • Helmut HarbrechtAffiliated withInstitut für Numerische Simulation, Universität Bonn Email author 
  • , Reinhold SchneiderAffiliated withInstitut für Mathematik, Technische Universität Berlin
  • , Christoph SchwabAffiliated withSeminar für Angewandte Mathematik, Eidgenössische Technische Hochschule Zürich

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Abstract

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially \({\mathcal{O}(N)}\) work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain.

Mathematics Subject Classification (2000)

35J20 35R60 65N38