WIAS Preprint No. 3036, (2023)

Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients


Authors

  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Hegemann, Nando

2020 Mathematics Subject Classification

  • 65N12 65N15 65N50 65Y20 68Q25

Keywords

  • Uncertainty quantification, adaptivity, convergence, parametric PDEs, residual error estimator, lognormal diffusion

DOI

10.20347/WIAS.PREPRINT.3036

Abstract

Solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin and stochastic collocations methods. This work investigates a residual based adaptive algorithm used to approximate the solution of the stationary diffusion equation with lognormal coefficients. It is known that the refinement procedure is reliable, but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical results by providing a quasi-error reduction results for the adaptive solution of the lognormal stationary diffusion problem. A computational example supports the theoretical statement.

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