A Rapid Semi-Analytic Method for Solving Laplace's Equation

A relatively new semi-analytic method for solving Laplace's equation has been studied in detail and compared with various long-established techniques, which are also briefly reviewed. The method involves dividing the boundary of a region into N segments, each of which has an unknown surface charge. The surface charge on each segment is assumed to be constant or to be represented by a low-degree polynomial. In the latter case, charge and even derivative continuity at the segment boundaries can be imposed. The potential produced by each segment is expressed analytically and satisfaction of the boundary conditions determines the unknown charges. The latter calculation requires the solution of N linear simultaneous equations. The resulting charge pattern enables one to evaluate the potential and/or electric field at any point with no further difference approximations. This procedure compares favorably both in computation speed and accuracy with other numerical and semi-analytic techniques, viz., net, Monte Carlo, and collocation methods.