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

Fingerprint

Optimal Rate of Convergence
Transmission Problem
Finite Element Method
Polynomials
Finite element method
Sobolev spaces
Banach spaces
Set theory
Regularity
Chaos theory
Estimate
Lagrange's polynomial
Chaos Expansion
Uncertainty Quantification
Polynomial Chaos
Elliptic PDE
Approximation Space
Partial Differential Operators
Weighted Sobolev Spaces
Constant function

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

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
Li, Hengguang ; Nistor, Victor ; Qiao, Yu. / Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method. Numerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers. 2013. pp. 12-23 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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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. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8236 LNCS, pp. 12-23, 5th International Conference on Numerical Analysis and Applications, NAA 2012, Lozenetz, Bulgaria, 6/15/13. https://doi.org/10.1007/978-3-642-41515-9_2

Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method. / Li, Hengguang; Nistor, Victor; Qiao, Yu.

Numerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers. 2013. p. 12-23 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8236 LNCS).

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

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N2 - 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).

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Li H, Nistor V, Qiao Y. 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. 2013. p. 12-23. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-41515-9_2