Ap­p­lied Ma­the­ma­tics Col­lo­qui­um: Ge­or­gi­os Akri­vis (Io­an­ni­na, Gree­ce), D1.320

The linearly implicit two-step BDF method for harmonic maps into spheres

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. We devise a projection-free iterative scheme for the approximation of harmonic maps that is unconditionally energy stable and provides a second-order accuracy of the constraint violation under a mild, sharp 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.
The talk is based on joint work with Sören Bartels and Christian Palus (Albert-Ludwigs-Universität Freiburg).