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

Victor Nistor, Christoph Schwab

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

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
Volume23
Issue number9
DOIs
StatePublished - Aug 1 2013

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Higher Order Approximation
Galerkin Approximation
Elliptic Partial Differential Equations
Partial differential equations
Chaos theory
Polynomials
Sobolev spaces
Set theory
Coefficient
Boundary value problems
Regularity
Graded Meshes
Finite Element
Parametric Solutions
Chaos Expansion
Polynomial Chaos
Boundary conditions
Partial Differential Operators
Legendre polynomial
Weighted Sobolev Spaces

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Applied Mathematics

Cite this

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title = "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 = 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 Kondrat{\'e}v 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 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 u = (uν) ∈ ℓp(ℱ; K a+1m+1(D)), 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(S ℓ)-min{s, t} ||f||Hm-1(D), dim(S ℓ) → ∞. 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 S ℓ is defined by a finite subset Λℓ ⊂ ℱ of {"}active polynomial chaos{"} coefficients uν ∈ V, ν ∈ Λℓ in the Legendre chaos expansion of u which are approximated by vν ∈ V μ(ℓ, ν), for each ν ∈ Λℓ, with a suitable choice of μ(ℓ, ν).",
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High-order Galerkin approximations for parametric second-order elliptic partial differential equations. / Nistor, Victor; Schwab, Christoph.

In: Mathematical Models and Methods in Applied Sciences, Vol. 23, No. 9, 01.08.2013, p. 1729-1760.

Research output: Contribution to journalArticle

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AU - Nistor, Victor

AU - Schwab, Christoph

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N2 - 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 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 u = (uν) ∈ ℓp(ℱ; K a+1m+1(D)), 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(S ℓ)-min{s, t} ||f||Hm-1(D), dim(S ℓ) → ∞. 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 S ℓ is defined by a finite subset Λℓ ⊂ ℱ of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λℓ in the Legendre chaos expansion of u which are approximated by vν ∈ V μ(ℓ, ν), for each ν ∈ Λℓ, with a suitable choice of μ(ℓ, ν).

AB - 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 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 u = (uν) ∈ ℓp(ℱ; K a+1m+1(D)), 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(S ℓ)-min{s, t} ||f||Hm-1(D), dim(S ℓ) → ∞. 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 S ℓ is defined by a finite subset Λℓ ⊂ ℱ of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λℓ in the Legendre chaos expansion of u which are approximated by vν ∈ V μ(ℓ, ν), for each ν ∈ Λℓ, with a suitable choice of μ(ℓ, ν).

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