### Abstract

We study the approximation properties of a harmonic function u ∈H ^{1-k}(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = script O sign, for a smooth, bounded domain script O sign, we obtain that the GFEM-approximation u _{s} ∈ S of u satisfies ∥u - u _{s}∥H ^{1}(A) ≤ Ch ^{γ}∥u∥ _{H1-k},(script O sign), where h is the typical size of the "elements" defining the GFEM-space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super-approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161-168; Math Comput 28 (1974), 937-958). In addition to the usual "energy" Sobolev spaces H ^{1}(script O sign), we need also the duals of the Sobolev spaces H ^{m}(script O sign), m ∈ ℤ _{+}.

Original language | English (US) |
---|---|

Pages (from-to) | 79-113 |

Number of pages | 35 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*Numerical Methods for Partial Differential Equations*,

*22*(1), 79-113. https://doi.org/10.1002/num.20086

}

*Numerical Methods for Partial Differential Equations*, vol. 22, no. 1, pp. 79-113. https://doi.org/10.1002/num.20086

**Interior numerical approximation of boundary value problems with a distributional data.** / Babuška, Ivo; Nistor, Victor.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Interior numerical approximation of boundary value problems with a distributional data

AU - Babuška, Ivo

AU - Nistor, Victor

PY - 2006/1/1

Y1 - 2006/1/1

N2 - We study the approximation properties of a harmonic function u ∈H 1-k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = script O sign, for a smooth, bounded domain script O sign, we obtain that the GFEM-approximation u s ∈ S of u satisfies ∥u - u s∥H 1(A) ≤ Ch γ∥u∥ H1-k,(script O sign), where h is the typical size of the "elements" defining the GFEM-space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super-approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161-168; Math Comput 28 (1974), 937-958). In addition to the usual "energy" Sobolev spaces H 1(script O sign), we need also the duals of the Sobolev spaces H m(script O sign), m ∈ ℤ +.

AB - We study the approximation properties of a harmonic function u ∈H 1-k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = script O sign, for a smooth, bounded domain script O sign, we obtain that the GFEM-approximation u s ∈ S of u satisfies ∥u - u s∥H 1(A) ≤ Ch γ∥u∥ H1-k,(script O sign), where h is the typical size of the "elements" defining the GFEM-space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super-approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161-168; Math Comput 28 (1974), 937-958). In addition to the usual "energy" Sobolev spaces H 1(script O sign), we need also the duals of the Sobolev spaces H m(script O sign), m ∈ ℤ +.

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U2 - 10.1002/num.20086

DO - 10.1002/num.20086

M3 - Article

AN - SCOPUS:33645293766

VL - 22

SP - 79

EP - 113

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 1

ER -