## 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) |
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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 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics