Treatment of corners in BIE analysis of potential problems

At corners or edges in the boundary of the domain of a potential problem the local normal gradient of potential is double‐valued. When Dirichlet boundary conditions are specified there are thus two unknowns at a single nodal point, and the sets of equations resulting from the usual BIE discretization are rendered indeterminate. We discuss here earlier approaches to the resolving of this problem, and describe a further approach which appears to offer some advantages. Both normal gradients can be approximated directly from local potential boundary conditions, showing the problem indeed to be formally overdetermined. This ability is discarded, in favour of yielding a robust and well‐conditioned relationship between the two gradients. This, in conjunction with the BIE analysis, permits solutions of considerable accuracy to be found, including the gradients at such corner nodes. Illustrative calculations are presented for rectilinear and curvilinear domains. These show that, even with as few elements as there are corners, and thus one and a half times as many unknowns as there are nodal points, good approximations to the gradients can be obtained. The need for progressively finer discretization as a corner is approached is thus much reduced.