Interior numerical approximation of boundary value problems with a distributional data

Ivo Babuška, Victor Nistor

Research output: Contribution to journalArticle

9 Citations (Scopus)

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

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Numerical Approximation
Boundary value problems
Generalized Finite Element Method
Sobolev spaces
Interior
Boundary Value Problem
Finite element method
Harmonic functions
Sobolev Spaces
Set theory
Polynomials
Approximation Space
Local Approximation
Approximation Property
Approximation
Harmonic Functions
Bounded Domain
Polynomial
Subset
Energy

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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

In: Numerical Methods for Partial Differential Equations, Vol. 22, No. 1, 01.01.2006, p. 79-113.

Research output: Contribution to journalArticle

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