The Landau-Lifshitz-Gilbert equation describes the dynamic behavior of a ferromagnetic body under the influence of a magnetic field. Considering the numerical treatment it makes for an interesting equation due to its strong nonlinearity and non-convex side constraint. In this thesis a finite element scheme for the solution of the equation is developed. A well-known scheme is extended to include the total magnetic field, extending it by the contributions of the anisotropy energy, magnetostatic energy and the Zeeman energy. The term of the exchange energy is being treated implicitly, while the other terms enter the scheme explicitly. This gives rise to the numerical treatment of the stray field using established solution methods. The presented scheme is shown to be conditionally weakly convergent (up to a subsequence) to a weak solution of the equation. Two iterations are presented, which can be used for solving the equation, followed by their MATLAB implementation and numerical treatment of standard examples.