WIT Press

The Unit Circle Trap In Boundary Elements


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G. S. Gipson & B. W. Yeigh


Some of the traditional boundary element formulations suffer from instability problems when a unit circle is involved in the discretization. This is a problem that has been discussed frequently and handled empirically, but apparently never formalized as a topic in the literature. This paper addresses how the problem arises, identifies why the problem occurs, and demonstrates remedies to avoid it. A theoretical outline will be provided and numerical case studies are presented. Particular emphasis is placed on the Poisson equation and its synthesis. Keywords: unit circle, Poisson equation, convergence, boundary elements, numerical instability. 1 Introduction This work focuses on a complaint that has been frequently discussed casually among boundary element researchers but apparently has never been addressed formally in the literature. The issue has to do with the unit circle which is typical in normalized formulations of circular geometry. The purpose of this paper is to first show one application in the Poisson equation where the problem arises and to illustrate why it happens. The cause is identified and a solution is proposed. This presentation is unusual in that it is best done using constant elements since, as will be demonstrated, use of the more sophisticated, higher-order elements exacerbate the problem. This lends credence to referring to the problem as a \“trap.” 2 Background The boundary element method is predicated upon defining a boundary value problem in terms of equations involving surface integrals. Ideally, one would


unit circle, Poisson equation, convergence, boundary elements, numerical instability