Boundary Integral Solutions of Laplace's Equation

Laplace's equation frequently arises in modeling physical problems, especially in electromagnetism, in thermal flow, and in fluid flow. In two dimensions, Laplace's equation is + _ dy 2797 2 dx 2 Fig. 1--Region used in an analysis of an electrostatic lens. and in three dimensions, d 2 4> do; 2 + d2 dy 2 + d2 _ dz2 To complete the specification of a particular problem, a region on which to solve Laplace's equation must be specified, plus boundary conditions on the boundary of the region. As compensation for the simplicity of the partial differential equation, the region over which Laplace's equation is to be solved is often complicated. Figure 1 shows the region used by the author in an unpublished analysis of an electrostatic lens. The solution is singular at the re-entrant corners. (By singular, we mean that $ has a finite limit as the corner is approached, but that some derivatives of The singularity is a consequence of the region itself, not of any particular boundary conditions. In fact, the solution is singular unless very special boundary conditions are prescribed. Even with a rectangular region, the solution can be singular at isolated points. Figure 2 is an example, a thin-film capacitor with metal top and bottom contacts. To obtain its capacitance, Laplace's equation must be solved inside the rectangle. The boundary conditions are $ = 1 on the top contact, $ = 0 on the bottom contact, and zero normal derivative, d$/dn = 0, on the remainder of the boundary.