It is well known that, when a boundary value problem contains a boundary singularity, the accuracy and rate of convergence of approximations produced by standard numerical methods are worse than those for the smooth cases. As a result great effort has been expended by many researchers over the last half century in devising numerical schemes which overcome these deficiencies, see e.g. [24], [35], [16]. In the main these numerical schemes have been produced using knowledge of the forms of the singularities, and the production of theoretical error estimates for the numerical approximations demands knowledge of the regularity of the (weak) solutions of the problems. As a result the evaluation of the forms of the singularities and their effective treatment both constructively and numerically collectively now constitute a major field of research, see e.g. the books of Grisvard, Wendland, Whiteman [16], Grisvard [14], [15], Dauge [9], Leguillon and Sanchez-Palencia [20], Kufner and Sändig [19] and the habilitation or doctoral theses of Abdel-Messieh [1], Beagles [3], Becker [6], Dobrowolski [11], von Petersdorff [25], Rank [27] and Volk [30]. The solving of eigenvalue problems is a continually occurring feature of the methods for determining the forms of the singularities.


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© Springer Basel AG 1991

Authors and Affiliations

  • J. R. Whiteman
    • 1
  1. 1.BICOM, Institute of Computational Mathematics, Department of Mathematics and StatisticsBrunei UniversityUxbridgeEngland

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