Uniform Shift Estimates for Transmission Problems and Optimal Rates of Convergence for the Parametric Finite Element Method

Abstract

Let Ω ⊂ ℝ^d, $d \geqslant 1$, 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 = (P _y ) _y ∈ U of uniformly strongly elliptic, second order partial differential operators P _y on Ω. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution u: Ω×U → ℝ of the parametric, elliptic boundary value/transmission problem P _y u _y  = f _y , 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 of Babuška-Kondrat’ev type in Ω, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs (P _y ) _y ∈ U admit a shift theorem that is uniform in the parameter y ∈ U. In turn, this then leads to h ^m-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).