Approximation theory methods for solving elliptic eigenvalue problems

Georg J. Still

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Abstract

Eigenvalue problems with elliptic operators L on a domain G C R2 are considered. By applying results from complex approximation theory we obtain results on the approximation properties of special classes of solutions of Lu = 0 on G . These solutions are used as trial functions in a method for solving the eigenvalue problem which is based on a-posteriori error bounds. Singular trial functions are applied to smooth the problem at corner points of G . In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. We discuss two problems from plasma physics (relaxed plasma, MHD-equation).
Original languageEnglish
Pages (from-to)468-478
Number of pages11
JournalZeitschrift für angewandte Mathematik und Mechanik
Volume83
Issue number7
DOIs
Publication statusPublished - 2003

Keywords

  • METIS-214127
  • Elliptic eigenvalue problems
  • Complex approximation
  • A-posteriori error bounds
  • Defect-minimization method
  • Degree of approximation

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