We present a method to compute the magnetic moment of a bulk, finite-size, three-dimensional, anisotropic superconductor. Our numerically implemented perturbative procedure is based on a solution of the nonlinear Maxwell-London electrodynamic equations, where we include the nonlinear relation between current and gauge invariant velocity. The method exploits the small ratio of the finite penetration depths to the sample size. We show how to treat the open boundary conditions over an infinite domain and the continuity requirement at the interface. We demonstrate how our method substantially reduces the computational work required, and discuss its implementation to an oblate spheroid. The numerical solution is obtained from a finite-difference method. We briefly discuss the relevance of this work to similar problems in other fields.