We present a new application of the complex polynomial method variant of the complex variable boundary element method. Instead of fitting the boundary conditions using collocation points, we minimize the error of fit in the l2 norm to minimize the least-squares error. This approach greatly enhances the utility and efficiency of the method, allowing us to apply the method to a variety of engineering problems. Numerical solutions of partial differential equations (PDE) involving the Laplace or Poisson equations are important topics in engineering, physics, and applied mathematics. Some applications include heat transport, Fickian diffusion, groundwater flow, contaminant flow in groundwater, stress–strain including torsion in shafts, and electrostatics. The most popular numerical techniques used to approximate solutions to such boundary value problems of these PDE include real variable boundary element methods and the domain methods of finite-difference, finiteelement methods.