Singularities and treatments of elliptic boundary value problems

This paper provides a survey for treatments for singularity problems of elliptic equations. We take the Laplace equation on polygons as an example, and choose Motz's problem as a benchmark of singularity problems. The conformal transformation method of Whiteman and Papamichael [1] and Rosser and Papamichael [2] is the most accurate method for seeking the leading coefficients of the solution expansions of Motz's problem. In this paper, we provide the first 100 leading coefficients D"0-D"9"9, where d"0 and D"9"9 have 199 and 89 significant decimal digits, respectively, which serve as the test coefficients for other numerical methods. Other treatments may be classified into three categories: local refinements, singular function method of Fix [3], Wigley [4], and Blum and Dobrowolski [5], and the combined methods. The local refinements have a wide range of applications due to less requirements; the combined methods are most efficient because the singular functions may be chosen to fit best the singularity of the solutions. Also, the natural boundary element method may handle well the unbounded domain problems. Moreover, the combined methods are introduced briefly with the coupling techniques, when other numerical methods are interpreted as a kind of finite element methods by using different admissible functions and different integration rules. Different coupling techniques are, indeed, intended to remedy the nonconformality of the admissible functions along the common (interior) boundary of different numerical methods used. Besides, about 300 publications related to singularities and their treatments are collected in this paper.

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