HIGH-ORDER GALERKIN APPROXIMATIONS FOR PARAMETRIC SECOND-ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

Abstract

Let D ⊂ ℝd, d = 2, 3, be a bounded domain with piecewise smooth boundary, Y = ℓ∞(ℕ) and U = B1(Y), the open unit ball of Y. We consider a parametric family (Py)y∈U of uniformly strongly elliptic, second-order partial differential operators Py on D. Under suitable assumptions on the coefficients, we establish a regularity result for the solution u of the parametric boundary value problem Py 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 [Formula: see text] of Kondrat'ev type. We prove that the (Py)y ∈ U admit a shift theorem that is uniform in the parameter y ∈ U. Specifically, if the coefficients of P satisfy [Formula: see text], y = (yk)k≥1 ∈ U and if the sequences [Formula: see text] are p-summable in k, for 0 < p< 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials Lν(y) such that the corresponding sequence [Formula: see text], where [Formula: see text]. We also show optimal algebraic orders of convergence for the Galerkin approximations uℓ 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 [Formula: see text], 0 < p < 1. We show that, for each m ∈ ℕ, there exists a sequence {Sℓ}ℓ≥0 of nested, finite-dimensional spaces Sℓ ⊂ L2(U;V) such that the Galerkin projections uℓ ∈ Sℓ of u satisfy ‖u - uℓ‖L2(U;V) ≤ C dim (Sℓ)- min {s, t} ‖f‖Hm-1(D), dim (Sℓ) → ∞. The sequence Sℓ is constructed using a sequence Vμ⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace Sℓ is defined by a finite subset [Formula: see text] of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λℓ in the Legendre chaos expansion of u which are approximated by vν ∈ Vμ(ℓ, ν), for each ν ∈ Λℓ, with a suitable choice of μ(ℓ, ν).