TY - GEN

T1 - Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric finite element method

AU - Li, Hengguang

AU - Nistor, Victor

AU - Qiao, Yu

PY - 2013

Y1 - 2013

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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84886812579&partnerID=8YFLogxK

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U2 - 10.1007/978-3-642-41515-9_2

DO - 10.1007/978-3-642-41515-9_2

M3 - Conference contribution

AN - SCOPUS:84886812579

SN - 9783642415142

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 12

EP - 23

BT - Numerical Analysis and Its Applications - 5th International Conference, NAA 2012, Revised Selected Papers

T2 - 5th International Conference on Numerical Analysis and Applications, NAA 2012

Y2 - 15 June 2013 through 20 June 2013

ER -