Interior numerical approximation of boundary value problems with a distributional data

Ivo Babuška, Victor Nistor

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9 Scopus citations

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 languageEnglish (US)
Pages (from-to)79-113
Number of pages35
JournalNumerical Methods for Partial Differential Equations
Volume22
Issue number1
DOIs
StatePublished - Jan 1 2006

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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