Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere

Abstract

We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite elements. While the first method enforces the unit-length constraint only at the Lagrange nodes, the other one adds a pointwise projection to fulfill the constraint everywhere. For the solution of the resulting algebraic problems we compare a nonconforming gradient flow with a Riemannian trust-region method. Both are energy-decreasing and can be shown to converge globally to a stationary point of the Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior, the second-order trust-region method vastly outperforms the solver based on gradient flow.