High order Galerkin appoximations for parametric second order elliptic partial differential equations

LetD ⊂ Rd, d = 2, 3, be a bounded domain with piecewise smooth boundary ∂D and let U be an open subset of a Banach space Y . We consider a parametric family Py of uniformly strongly elliptic, parametric second order partial differential operators Py on D in divergence form, where the parameter y ranges in the parameter domain U so that, for a given set of data fy , the solution u and the coefficients of the parametric boundary value problem Pyu = fy are functions of (x, y) ∈ D × U . Under suitable regularity assumptions on these coefficients and on the source term f , we establish a regularity result for the solution u : D×U → R of the parametric, elliptic boundary value problem Pyu(x, y) = fy(x) = f(x, y), x ∈ D, y ∈ U , with mixed DirichletNeumann boundary conditions. Let ∂D = ∂dD ∪ ∂nD denote decomposition of the boundary into a part on which we assign Dirichlet boundary conditions and the part on which we assign Neumann boundary conditions. We assume that ∂dD is a finite union of closed polygonal subsets of the boundary such that no adjacent faces have Neumann boundary conditions (i. e., there are no Neumann-Neumann corners or edges). Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces K a+1 (D) of Kondrat’ev type in D. We prove that the parametric, elliptic PDEs (Py)y∈U admit a shift theorem which is uniform in the parameter sequence y ∈ U . Specifically, if coefficients a pq(x, y) depend on the parameter sequence y = (yk)k≥1 in an affine fashion, ie. a pq = a pq0 + ∑ k≥1 ykψ ij pqk, and if the sequences ‖ψ pqk‖Wm,∞(D) are p-summable for some 0 < p < 1, then the parametric solution uy admit an expansion into tensorized Legendre polynomials Lν(y) such that the corresponding coefficient sequence u = (uν) ∈ #p(F ;K a+1 (D)). Here, we denote by F ⊂ N0 the set of sequences {kn}n∈N with kn ∈ N0 with only finitely many non-zero terms, and by Y = #∞(N) and U = B1(Y ), the open unit ball of Y . We identify the parametric solution u with its coefficient vector u = (uν)ν∈F , uν ∈ V , in the “polynomial chaos” expansion with respect to tensorized Legendre polynomials on U . We also show quasioptimal algebraic orders of convergence for Finite Element approximations of the parametric solutions u(y) from suitable Finite Element spaces in two and three dimensions. Let t = m/d and s = 1/p − 1/2 for some p ∈ (0, 1] such that u = (uν) ∈ #p(F ;K a+1 (D)). We then show that, for each m ∈ N, exists a sequence {S"}"≥0 of nested, finite dimensional spaces S" ⊂ L2(U, μ;V ) such that M" = dim(S") → ∞ and such that the Galerkin projections u" ∈ S" of the solution u onto S" satisfy ‖u− u"‖L2(U,μ;V ) ≤ C dim(S")‖f‖Hm−1(D) . The sequence S" is constructed using a nested sequence Vμ ⊂ V of Finite Element space in D with graded mesh refinements toward the singular boundary points of the domain D as in [7, 9, 27]. Our sequence Vμ is independent of y. Each subspace S" is then defined by a finite subset Λ" ⊂ F of “active polynomial chaos” coefficients uν ∈ V , ν ∈ Λ" in the Legendre chaos expansion of u which, in turn, are approximated by vν ∈ Vμ(",ν) for each ν ∈ Λ", with a suitable choice of μ(#, ν). 0 V. NISTOR AND CH. SCHWAB