High-order Galerkin approximations for parametric second-order elliptic partial differential equations

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) = ∑_ky_kψ_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 = ℕ₀ ^(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 ⊂ L²(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).