Solution of Potential Problems Using an Overdetermined Complex Boundary Integral Method

The advantages of solving potential problems using an overdetermined boundary integral element method are examined. Representing a 2-dimensional potential solution by an analytic complex function forms two algebraic systems from the real and imaginary parts of the discretized form of the Cauchy theorem. Depending on which boundary condition is prescribed, the real or the imaginary algebraic system is diagonally dominant. Computations show that the errors of the strong system (diagonally dominant) often have almost the same value as those of weak system (diagonally non-dominant) but with the opposite sign. The overdetermined system, composed of the combination of the real and imaginary parts, tends to average these errors, especially for circular contours. An error analysis and convergence studies for several geometries and boundary conditions are performed. A methodology for handling computational difficulties with contour corners is outlined. A further modification is proposed and tested that shows exponential convergence for circular contours.