We present a computational framework for the efficient numerical solution of the Landau- Lifshitz-Gilbert equation by exploiting projection-based finite elements [1, 2] and opti- mization on manifolds [3]. As a sample application, domain formation in cylindrical nanodots [4] is considered. The proposed framework leverages tools reported in our previ- ous work on geometrically non-linear Reissner-Mindlin shells [5] and results in an efficient, robust and objective formulation. The micromagnetic model is based on the vector potential and the magnetization director as independent fields. While the vector potential is embedded in the d-dimensional Eu- clidean space, the magnetization director resides on a d−1-dimensional manifold, that is, on the unit sphere embedded into the d-dimensional Euclidean space. The latter property comes along with a number of challenges associated with, for example, objectivity require- ments and the geometrically meaningful integration in numerical schemes. Notably, the salient features of the magnetization prohibit the use of trigonometric functions for its spatial discretization. In the present approach, we overcome the mentioned challenges by the use of projection- based finite elements and the optimization on manifolds, which specifically eliminates the need for explicit parameterization of the unit sphere. As a result, our formulation does not require any artificial constraints (often integrated by means of penalty formulations, renormalization strategies or Lagrange-multiplier methods [6, 7, 8]), but ensures a geo- metrically consistent representation of magnetization by the very definition of the search space. This definition, in turn, yields a further key advantage of our approach — the reduced number of degrees of freedom associated with the magnetization. In contrast to classical finite element implementations that require d degrees of freedom per node in d-dimensional space, our formulation copes with d−1 degrees of freedom per node. Taken altogether, our model facilitates computationally efficient yet accurate numerical simulations in micromagnetics and therewith offers promising avenues towards the realization of micromagnetic precision in large-scale practical applications. To showcase the efficacy of our formulation, we validate it with semi-analytical results of domain formation in cylindrical nanodots and apply it to a variety of more complex scenarios.