The unit circle trap in boundary elements redux

This paper amplifies and subtends the BEM 27 paper: The unit circle trap in boundary elements. The paper deals with some traditional boundary element formulations that suffer from apparent instability problems when a unit circle is involved in the discretization. Such problems have been discussed informally and handled empirically by researchers, but apparently were never formalized with a numerical convergence study in the literature until done so by the authors in the proceedings of BEM 27. The present paper details how the problem arises, identifies sources for its occurrence, cites past allusions to the problem that have come to light since the presentation of the BEM 27 conference paper, and suggests remedies to avoid it. A theoretical outline is provided and extensive numerical case studies are presented. Particular emphasis is placed on the Poisson equation and its synthesis.