Strongly stable generalized finite element method: Application to interface problems

Ivo Babuška, Uday Banerjee, Kenan Kergrene

Research output: Contribution to journalArticlepeer-review

56 Scopus citations


In this paper, we present the theoretical justification for the Stable Generalized Finite Element Method (SGFEM) when applied to smooth interface problems. We prove that a Generalized Finite Element Method (GFEM) is stable, i.e., its conditioning is(a) not worse than that of the Finite Element Method, and (b) robust with respect to the mesh, if the enrichment space of the GFEM satisfies two axioms. We provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied. We show that the enrichment space of the GFEM, used to address the interface problems in 2D, satisfies the sufficient conditions and thus the two axioms yielding an SGFEM. The idea of a strongly stable GFEM associated with one of the two axioms has been introduced; strong stability of the GFEM is important for designing efficient iterative methods to solve the underlying linear system. A proof of the optimal convergence of the GFEM for the interface problem has also been derived in this paper. The work in this paper is the continuation of Kergrene et al. (2016), where the stability of the GFEM, when applied to interface problems, was established through numerical experiments. The numerical results in Kergrene et al. (2016) indicated that the GFEM is indeed strongly stable.

Original languageEnglish (US)
Pages (from-to)58-92
Number of pages35
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Dec 1 2017


  • Conditioning
  • GFEM
  • Interface
  • Robustness
  • XFEM

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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