Approximate and low regularity dirichlet boundary conditions in the generalized finite element method

Ivo Babuška, Victor Nistor, Nicolae Tarfulea

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4 Scopus citations

Abstract

We propose a method for treating Dirichlet boundary conditions for the Laplacian in the framework of the Generalized Finite Element Method (GFEM). A particular interest is taken in boundary data with low regularity (possibly a distribution). Our method is based on using approximate Dirichlet boundary conditions and polynomial approximations of the boundary. The sequence of GFEM-spaces consists of nonzero boundary value functions, and hence it does not conform to one of the basic Finite Element Method (FEM) conditions. We obtain quasi-optimal rates of convergence for the sequence of GFEM approximations of the exact solution. We also extend our results to the inhomogeneous Dirichlet boundary value problem, including the case when the boundary data has low regularity (i.e. is a distribution). Finally, we indicate an effective technique for constructing sequences of GFEM-spaces satisfying our assumptions by using polynomial approximations of the boundary.

Original languageEnglish (US)
Pages (from-to)2115-2142
Number of pages28
JournalMathematical Models and Methods in Applied Sciences
Volume17
Issue number12
DOIs
StatePublished - Dec 2007

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Applied Mathematics

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