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Sparse finite elements for stochastic elliptic problems - higher order moments

by C. Schwab and R. A. Todor

(Report number 2003-04)

Abstract
We define the higher order moments associated to the stochastic solution of an elliptic BVP in D \subset Rd with stochastic source terms and boundary data. We prove that the k-th moment (or k-point correlation function) of the random solution solves a deterministic problem in Dk \subset Rdk. We discuss well-posedness and regularity in scales of Sobolev spaces with bounded mixed derivatives. We discretize this deterministic k-th moment problem using sparse tensor product FE-spaces and, exploiting a spline wavelet basis, we propose an algorithm of (up to logarithmic terms) the same accuracy and complexity as a multigrid finite element method for the mean field problem in D.

Keywords: stochastic pde, sparse grids, finite elements, wavelets

@Techreport{ST03_316,
  author = {C. Schwab and R. A. Todor},
  title = {Sparse finite elements for stochastic elliptic problems - higher order moments},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2003-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2003/2003-04.pdf },
  year = {2003}
}

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