### Abstract

Let D ⊂ ℝ^{d}, 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 (P_{y})_{y∈U} of uniformly strongly elliptic, second-order partial differential operators P_{y} 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 ∈ 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 K_{a+1}^{m+1}(D) of Kondratév type. We prove that the (P_{y})_{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) = ∑_{k}y_{k}ψ_{k} ^{ij}, y = (y_{k})_{k≥1} ∈ U and if the sequences ||ψ_{k}^{ij}||W^{m}, ∞(D) are p-summable in k, for 0 < p< 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials L_{ν}(y) such that the corresponding sequence u = (u_{ν}) ∈ ℓ^{p}(ℱ; K_{a+1}^{m+1}(D)), where ℱ = ℕ_{0} ^{(N)}. 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_{ν}) ∈ 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} ⊂ L^{2}(U;V) such that the Galerkin projections u_{ℓ} ∈ S_{ℓ} of u satisfy ||u - u_{ℓ}||_{L2(U;V)} ≤ C dim(S _{ℓ})^{-min{s, t}} ||f||_{Hm-1(D)}, dim(S _{ℓ}) → ∞. The sequence S_{ℓ} is constructed using a sequence V_{μ}⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace S _{ℓ} is defined by a finite subset Λ_{ℓ} ⊂ ℱ of "active polynomial chaos" coefficients u_{ν} ∈ V, ν ∈ Λ_{ℓ} in the Legendre chaos expansion of u which are approximated by v_{ν} ∈ V _{μ(ℓ, ν)}, for each ν ∈ Λ_{ℓ}, with a suitable choice of μ(ℓ, ν).

Original language | English (US) |
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Pages (from-to) | 1729-1760 |

Number of pages | 32 |

Journal | Mathematical Models and Methods in Applied Sciences |

Volume | 23 |

Issue number | 9 |

DOIs | |

State | Published - Aug 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Modeling and Simulation
- Applied Mathematics

### Cite this

*Mathematical Models and Methods in Applied Sciences*,

*23*(9), 1729-1760. https://doi.org/10.1142/S0218202513500218

}

*Mathematical Models and Methods in Applied Sciences*, vol. 23, no. 9, pp. 1729-1760. https://doi.org/10.1142/S0218202513500218

**High-order Galerkin approximations for parametric second-order elliptic partial differential equations.** / Nistor, Victor; Schwab, Christoph.

Research output: Contribution to journal › Article

TY - JOUR

T1 - High-order Galerkin approximations for parametric second-order elliptic partial differential equations

AU - Nistor, Victor

AU - Schwab, Christoph

PY - 2013/8/1

Y1 - 2013/8/1

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).

UR - http://www.scopus.com/inward/record.url?scp=84877876545&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877876545&partnerID=8YFLogxK

U2 - 10.1142/S0218202513500218

DO - 10.1142/S0218202513500218

M3 - Article

AN - SCOPUS:84877876545

VL - 23

SP - 1729

EP - 1760

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 9

ER -