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

Victor Nistor, Christoph Schwab

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


Let D ⊂ ℝd, 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 (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 Ka+1m+1(D) of Kondratév 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 a ij(x, y) = ∑kykψk ij, y = (yk)k≥1 ∈ U and if the sequences ||ψkij||Wm, ∞(D) 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 u = (uν) ∈ ℓp(ℱ; Ka+1m+1(D)), where ℱ = ℕ0 (N). We also show optimal algebraic orders of convergence for the Galerkin approximations u 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 = (uν) ∈ ℓp(ℱ; K a+1m+1(D)), 0 < p < 1. We show that, for each m ∈ ℕ, there exists a sequence {S}ℓ≥0 of nested, finite-dimensional spaces S ⊂ L2(U;V) such that the Galerkin projections u ∈ S of u satisfy ||u - u||L2(U;V) ≤ C dim(S )-min{s, t} ||f||Hm-1(D), dim(S ) → ∞. The sequence S is constructed using a sequence Vμ⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace S is defined by a finite subset Λ ⊂ ℱ of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λ in the Legendre chaos expansion of u which are approximated by vν ∈ V μ(ℓ, ν), for each ν ∈ Λ, with a suitable choice of μ(ℓ, ν).

Original languageEnglish (US)
Pages (from-to)1729-1760
Number of pages32
JournalMathematical Models and Methods in Applied Sciences
Issue number9
StatePublished - Aug 1 2013

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Applied Mathematics

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