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

Let D subset of R-d, d = 2, 3, be a bounded domain with piecewise smooth boundary, Y = l(infinity) (N) and U = B-1(Y), the open unit ball of Y. We consider a parametric family (P-y)(y is an element of 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 P(y)u(x, y) = f(x, y), x is an element of D, y is an element of U, with mixed Dirichlet-Neumann boundary conditions on partial derivative D-d and, respectively, on partial derivative D-n. Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces K-a+1(m+1)(D) of Kondrat'ev type. We prove that the (Py) y is an element of U admit a shift theorem that is uniform in the parameter y is an element of U. Specifically, if the coefficients of P satisfy a(ij) (x, y) = Sigma(k) yk psi(ij)(k) , y = (yk)(k >= 1) is an element of U and if the sequences parallel to psi(ij)(k)parallel to W-m,W-8 infinity (D) are p-summable in k, for 0 < p < 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials L.(y) such that the corresponding sequence u = (u(nu)) is an element of l(p) (F; K-a+1(m+1) (D)), where F = N (N) 0. 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 nu) is an element of 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 subset of L-2 (U; V) such that the Galerkin projections u(l) is an element of S-l of u satisfy parallel to u - u(l)parallel to(L2) ((U; V)) <= C dim(S-l)(-min{s, t}) parallel to f parallel to(Hm-1(D)), dim(S-l) -> infinity. The sequence S-l is constructed using a sequence V-mu subset of V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace S-l is defined by a finite subset L-l subset of F of "active polynomial chaos" coefficients u(nu) is an element of V, nu is an element of Lambda(l) in the Legendre chaos expansion of u which are approximated by v(nu) is an element of V-mu(l,V-nu), for each nu is an element of Lambda(l), with a suitable choice of mu(l, nu).