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A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes

by M. Eigel and C.J. Gittelson and Ch. Schwab and E. Zander

(Report number 2014-01)

Abstract
We analyze a-posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countably-parametric, elliptic boundary value problems. A residual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges, and to this end we establish a contraction property satisfied by its iterates. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results.

Keywords: partial differential equations with random coefficients, parabolic differential equations, uncertainty quantification, stochastic finite element methods, operator equations, adaptive methods

@Techreport{EGSZ14_551,
  author = {M. Eigel and C.J. Gittelson and Ch. Schwab and E. Zander},
  title = {A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-01},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-01.pdf },
  year = {2014}
}

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