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Interior numerical approximation of boundary value problems with a distributional data
Author(s) -
Babuška Ivo,
Nistor Victor
Publication year - 2006
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20086
Subject(s) - mathematics , sobolev space , bounded function , domain (mathematical analysis) , mathematical analysis , space (punctuation) , function space , boundary value problem , function (biology) , pure mathematics , philosophy , linguistics , evolutionary biology , biology
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 = , for a smooth, bounded domain , we obtain that the GFEM‐approximation u S ∈ S of u satisfies ‖ u − u S ‖ H 1( A )≤ Ch γ ‖ u ‖ H 1− k (), 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 (), we need also the duals of the Sobolev spaces H m (), m ∈ ℤ + . © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006