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hp-FEM for second moments of elliptic PDEs with stochastic data Part 1: Analytic regularity

by B. Pentenrieder and Ch. Schwab

(Report number 2010-08)

Abstract
For a linear second order elliptic partial differential operator A:V!V', we consider the boundary value problems Au=f with stationary Gaussian random data f over the dual V' of the separable Hilbert space V in which the solution u is sought. The operator A is assumed to be deterministic and bijective. The unique solution u= A-1f is a Gaussian random field over V. It is characterized by its mean field Eu and its covariance Cu 2 V V. For a class of piecewise analytic covariance kernels Cf2V'V' for Gaussian data f, we prove analytic regularity of the covariance Cu of the Gaussian solution u in families of countably normed spaces. To this end, we investigate shift theorems for the (non-hypoelliptic) deterministic tensor PDEs (AA)Cu=Cf proposed in (14) for the covariance Cu. The non-hypoelliptic nature of AA implies that sing supp(Cu) is in general strictly larger than sing supp(Cf). Based on our regularity results, we outline an hp-Finite Element strategy from (7,8) to approximate Cu stemming from covariances of stationary Gaussian data f. In the second part (8) of this work, we prove that this discretization gives exponential rates of convergence of the FE approximations, in terms of the number of degrees of freedom.

Keywords:

@Techreport{PS10_72,
  author = {B. Pentenrieder and Ch. Schwab},
  title = {hp-FEM for second moments of elliptic PDEs with stochastic data Part 1: Analytic regularity},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-08.pdf },
  year = {2010}
}

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