Interior numerical approximation of boundary value problems with a distributional data

2004/10/06 by Ivo Babuška, Babuska, Ivo, Victor Nistor +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.math/0410184

openalex publication_date 2004/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the approximation properties of a harmonic function u ∈ H\sp1-k(Ω), k > 0, on relatively compact sub-domain A of Ω, using the Generalized Finite Element Method. For smooth, bounded domains Ω, we obtain that the GFEM--approximation u_S satisfies ‖u - u_S‖_H\sp1(A) ≤ C h^γ‖u‖_H\sp1-k(Ω), 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+ 1. The main technical result is an extension of the classical super-approximation results of Nitsche and Schatz \citeNitscheSchatz72 and, especially, \citeNitscheSchatz74. It turns out that, in addition to the usual ``energy'' Sobolev spaces H¹, one must use also the negative order Sobolev spaces H\sp-l, l ≥ 0, which are defined by duality and contain the distributional boundary data.

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