After recalling the notion of harmonic maps into spheres, we discuss two variational formulations of the corresponding Euler--Lagrange equations. The second variational formulation leads easily to a linearization of the nonlinear equation. Subsequently, we focus on the gradient flow approach and recall known results for the linearly implicit Euler method, namely, energy decay (stability) and constraint violation properties.Our contribution concerns the application of the linearly implicit two-step BDF method to the gradient flow problem. More precisely, we devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The considered problem serves as a model for partial differential equations with holonomic constraint.For the performance of the method, illustrated via the computation of stationary harmonic maps and bending isometries, we refer to the manuscript on which the talk is based.The talk is based on the manuscript: G.A., Sören Bartels, Christian Palus: Quadratic constraint consistency in the projection-free approximation for harmonic maps and bending isometries. Submitted for publication. arXiv:2310.00381, http://arxiv.org/abs/2310.0038.
