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Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs

by M. Hansen and Ch. Schwab

(Report number 2011-29)

Abstract
We investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a-priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor-, Legendre- and Chebyshev polynomials on the infinite-dimensional parameter domain. We deduce rates of convergence for N term truncated approximations of expansions of the parametric solution. We also deduce spatial regularity of the solution, and establish convergence rates of N -term discretizations of the parametric solutions with respect to these polynomials in parameter space and with respect to a multilevel hierarchy of Finite Element spaces in the spatial domain of the PDE.

Keywords: Semilinear Elliptic Partial Differential Equations, Infinite Dimensional Spaces, N-term approximation, Analyticity in Infinite Dimensional Spaces, Tensor Product Taylor-, Legendre- and Chebyshev polynomial Approximation

@Techreport{HS11_96,
  author = {M. Hansen and Ch. Schwab},
  title = {Analytic regularity and nonlinear approximation of a class of parametric semilinear elliptic PDEs },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-29},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-29.pdf },
  year = {2011}
}

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