## Abstract

Let D ⊂ ℝ^{d}, d = 2, 3, be a bounded domain with piecewise smooth boundary, Y = ℓ^{∞}(ℕ) and U = B _{1}(Y), the open unit ball of Y. We consider a parametric family (P_{y})_{y∈U} of uniformly strongly elliptic, second-order partial differential operators P_{y} on D. Under suitable assumptions on the coefficients, we establish a regularity result for the solution u of the parametric boundary value problem P_{y} u(x, y) = f(x, y), x ∈ D, y ∈ U, with mixed Dirichlet-Neumann boundary conditions on ∂_{d} D and, respectively, on ∂_{n} D. Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces K_{a+1}^{m+1}(D) of Kondratév type. We prove that the (P_{y})_{y ∈ U} admit a shift theorem that is uniform in the parameter y ∈ U. Specifically, if the coefficients of P satisfy a ^{ij}(x, y) = ∑_{k}y_{k}ψ_{k} ^{ij}, y = (y_{k})_{k≥1} ∈ U and if the sequences ||ψ_{k}^{ij}||W^{m}, ∞(D) are p-summable in k, for 0 < p< 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials L_{ν}(y) such that the corresponding sequence u = (u_{ν}) ∈ ℓ^{p}(ℱ; K_{a+1}^{m+1}(D)), where ℱ = ℕ_{0} ^{(N)}. We also show optimal algebraic orders of convergence for the Galerkin approximations u_{ℓ} of the solution u using suitable Finite Element spaces in two and three dimensions. Namely, let t = m/d and s = 1/p-1/2, where u = (u_{ν}) ∈ ℓ^{p}(ℱ; K _{a+1}^{m+1}(D)), 0 < p < 1. We show that, for each m ∈ ℕ, there exists a sequence {S_{ℓ}}_{ℓ≥0} of nested, finite-dimensional spaces S_{ℓ} ⊂ L^{2}(U;V) such that the Galerkin projections u_{ℓ} ∈ S_{ℓ} of u satisfy ||u - u_{ℓ}||_{L2(U;V)} ≤ C dim(S _{ℓ})^{-min{s, t}} ||f||_{Hm-1(D)}, dim(S _{ℓ}) → ∞. The sequence S_{ℓ} is constructed using a sequence V_{μ}⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace S _{ℓ} is defined by a finite subset Λ_{ℓ} ⊂ ℱ of "active polynomial chaos" coefficients u_{ν} ∈ V, ν ∈ Λ_{ℓ} in the Legendre chaos expansion of u which are approximated by v_{ν} ∈ V _{μ(ℓ, ν)}, for each ν ∈ Λ_{ℓ}, with a suitable choice of μ(ℓ, ν).

Original language | English (US) |
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Pages (from-to) | 1729-1760 |

Number of pages | 32 |

Journal | Mathematical Models and Methods in Applied Sciences |

Volume | 23 |

Issue number | 9 |

DOIs | |

State | Published - Aug 1 2013 |

## All Science Journal Classification (ASJC) codes

- Modeling and Simulation
- Applied Mathematics