The authors consider the numerical solution of the Laplace equation: (i) in the interior of a simply connected region \(D \subset \mathbb{R}^3\) with Dirichlet boundary condition and (ii) in the exterior of \(D\) with Neumann boundary condition. The boundary \(S\) of \(D\) is either smooth or piecewise smooth. The boundary integral equation of the second kind is used to solve the problem. The surface \(S\) is triangulated. The collocation method with piecewise quadratic isoparametric interpolation is considered. Numerical integration methods in evaluating the collocation integrals are discussed. Error estimates are studied. Several numerical examples are given and the results are compared with the theoretical predictions.