Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions

Ivo Babuška, Victor Nistor, Nicolae Tarfulea

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29 Scopus citations

Abstract

We introduce a method for approximating essential boundary conditions-conditions of Dirichlet type-within the generalized finite element method (GFEM) framework. Our results apply to general elliptic boundary value problems of the form - ∑i, j = 1n (aij uxi)xj + ∑i = 1n bi uxi + cu = f in Ω, u = 0 on ∂ Ω, where Ω is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, { Sμ }μ ≥ 1, which are nonconforming (that is Sμ ⊄ H01 (Ω)). We assume that ∥ v ∥L2 (∂ Ω) ≤ Chμm ∥ v ∥H1 (Ω), for all v ∈ Sμ, and there exists uI ∈ Sμ such that ∥ u - uIH1 (Ω) ≤ Chμj ∥ u ∥Hj + 1 (Ω), 0 ≤ j ≤ m, where u ∈ Hm + 1 (Ω) is the exact solution, m is the expected order of approximation, and hμ is the typical size of the elements defining Sμ. Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence uμ ∈ Sμ of u. Next, we extend our analysis to the inhomogeneous boundary value problem - ∑i, j = 1n (aij uxi)xj + ∑i = 1n bi uxi + cu = f in Ω, u = g on ∂ Ω. Finally, we outline the construction of a sequence of GFEM spaces Sμ ⊂ over(S, ̃)μ, μ = 1, 2, ..., that satisfies our assumptions.

Original languageEnglish (US)
Pages (from-to)175-183
Number of pages9
JournalJournal of Computational and Applied Mathematics
Volume218
Issue number1
DOIs
StatePublished - Aug 15 2008

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

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