DFG-Project: Numerical solution of quadratic operator eigenvalue problems from continuum mechanics

Subprojects AP 72/1-1, 1-2, 1-3; Funding period: July 2002 - June 2006
  • Prof. Dr. Thomas Apel



  • Dr.rer.nat. Cornelia Pester



  • Sven Beuchler
  • Dominique Leguillon
  • Arnd Meyer
  • Serge Nicaise
  • Anna-Margarethe Sändig
  • Sergey I. Solov'ev
  • David Watkins
  • Zohar Yosibash


Student assistants in larger subprojects: 

  • Mojca Miklavec
  • Jan Rosam



  • Study of singular parts of solutions of elliptic boundary value problems in domains with concave (re-entrant) corners
  • Application in continuum mechanics in the analysis of stress concentrations in the vicinity of corners and crack tips and thus in the prediction of crack initiation and crack propagation
  • Consideration of bodies made of anisotropic materials or composites possible
  • Determination of singular functions leads to eigenvalue problems for (mostly quadratic) operator pencils with complex eigenvalues and eigenfunctions
  • Special structure of the spectrum (Hamiltonian eigenvalue symmetry)
  • Of interest are the eigenvalues with smallest positive real part
  • Finite element discretization and suitable linearization of the problem leads to a standard eigenvalue problem for a Hamiltonian matrix
  • Further development of a priori and a posteriori finite element failure analysis
    Dissertation (C. Pester):
    • Derivation of a residual-based a posteriori error estimator for the solutions of eigenvalue problems for arbitrary operator pencils
    • Model examples: Laplace equation, linear elasticity problem.
      Derivation of the associated eigenvalue problems when considering corner singularities
    • Construction of an adaptive finite element method
    • Special features:
      • Complex solutions can occur
      • Consideration of vector functions
      • Definition domain is generally a part of the sphere surface
      Error estimator theory is also applicable for arbitrary two-dimensional manifolds and independent of the origin of the eigenvalue problem
  • Implementation of fast and stable algorithms for solving the algebraic eigenvalue problem by exploiting the structure of the problem
  • Effectiveness improvements: Comparison of different methods, comparison of different packages for solving large weakly populated systems of equations
  • Verification of all investigations by appropriate numerical tests




  • Th. Apel, V. Mehrmann, and D. Watkins Numerical solution of large scale structured polynomial or rational eigenvalue problems. In: F. Cucker, R. DeVore, P. Olver, and E. Süli, editors, Foundations of Computational Mathematics, Minneapolis 2002, volume 312 of Lecture Note Series, Cambridge, 2004. London Mathematical Society, Cambridge University Press. Download
  • Th. Apel, A.-M. Sändig, S.I. Solov'ev Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes. Math. Model. Numer. Anal., 36:1043-1070, 2002. Download
  • Th. Apel, A. Meyer, C. Pester Corrigendum: Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes. Download
  • Th. Apel, C. Pester Clement-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation. IMA J. Numer. Anal. 25, No.2, 310-336, 2005. Download
  • Th. Apel, C. Pester Quadratic eigenvalue problems in the analysis of cracks in brittle materials. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS). Jyväskylä, 2004. Download
  • C. Pester Hamiltonian eigenvalue symmetry for quadratic operator eigenvalue problems. J. Integral Equations Appl., 17:71-89, 2005. Download
  • A. Meyer, C. Pester The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problems. Math. Methods Appl. Sci. 30:751-777, 2007. Download
  • C. Pester A residual a posteriori error estimator for the eigenvalue problem for the Laplace-Beltrami operator. Preprint SFB393/05-01, Preprint-Reihe des SFB393 der Technischen Universität Chemnitz, 2005. Download
  • C. Pester CoCoS -- Computation of corner singularities (Dokumentation zum Programmpaket CoCoS). Preprint SFB393/05-03, Preprint-Reihe des SFB393 der Technischen Universität Chemnitz, 2005. Download
  • C. Pester A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3d vertex singularities. Dissertation. Technische Universität Chemnitz, 2006. Download

 Student Research Projects: 

  • Chr. Gay Solving of Poisson equations with singularities. Rapport de stage de fin d'études, TU Chemnitz/ENSTA Paris, 2002.
  • C. Pester Fehlerschätzer für lineare Eigenwertprobleme. Diplomarbeit. TU Chemnitz, 2002
  • J. Rosam Berechnung der Rissgeometrie bei spröden elastischen Körpern. Diplomarbeit, TU Chemnitz, 2004.
  • S. Trebesius A singular function method for elliptic boundary value problems in three dimensional domains. Diplomarbeit, TU Chemnitz, 2004.