  We consider the numerical approximation of the Landau-Lifshitz-Gilbert
equation, which describes the dynamics of the magnetization in ferromagnetic
materials. In addition to the classical micromagnetic contributions, the energy
comprises the Dzyaloshinskii-Moriya interaction, which is the most important
ingredient for the enucleation and the stabilization of chiral magnetic
skyrmions. We propose and analyze three tangent plane integrators, for which we
prove (unconditional) convergence of the finite element solutions towards a
weak solution of the problem. The analysis is constructive and also establishes
existence of weak solutions. Numerical experiments demonstrate the
applicability of the methods for the simulation of practically relevant problem
sizes.
