Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method

Hengguang Li, Victor Nistor, Yu Qiao

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Let Ω ⊂ ℝd, d ≥ 1, be a bounded domain with piecewise smooth boundary ∂Ω and let U be an open subset of a Banach space Y. Motivated by questions in "Uncertainty Quantification," we consider a parametric family P = (Py)y∈U of uniformly strongly elliptic, second order partial differential operators Py on Ω. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution u: Ω x U → ℝ of the parametric, elliptic boundary value/transmission problem Py u y = fy, y ∈ U, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for d = 2. Our regularity and well-posedness results are formulated in a scale of broken weighted Sobolev spaces K̂ a+1m+1 (Ω) of Babuška-Kondrat'ev type in Ω, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs (Py)y∈U admit a shift theorem that is uniform in the parameter y ∈ U. In turn, this then leads to hm-quasi-optimal rates of convergence (i. e., algebraic orders of convergence) for the Galerkin approximations of the solution u, where the approximation spaces are defined using the "polynomial chaos expansion" of u with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab (2010).

Original languageEnglish (US)
Title of host publicationNumerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers
Pages12-23
Number of pages12
DOIs
StatePublished - Nov 7 2013
Event5th International Conference on Numerical Analysis and Applications, NAA 2012 - Lozenetz, Bulgaria
Duration: Jun 15 2013Jun 20 2013

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8236 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other5th International Conference on Numerical Analysis and Applications, NAA 2012
CountryBulgaria
CityLozenetz
Period6/15/136/20/13

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Li, H., Nistor, V., & Qiao, Y. (2013). Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method. In Numerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers (pp. 12-23). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8236 LNCS). https://doi.org/10.1007/978-3-642-41515-9_2