Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere

TL;DR

This paper compares different numerical methods for computing harmonic maps into the sphere, focusing on discretization techniques and solvers, and evaluates their effectiveness especially near singularities.

Contribution

It introduces a comparative analysis of two finite element discretizations and two solver methods for harmonic maps, highlighting their performance and convergence properties.

Findings

Nonconforming discretization handles singularities better.

Trust-region method converges faster than gradient flow.

Both methods are energy-decreasing and globally convergent.

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 stationary points of the discretized Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior for smooth problems, the nonconforming discretization handles singularities better. On the solver side, the second-order trust-region method converges after few steps, whereas the number of gradient-flow steps increases proportionally to the inverse grid element diameter.