Benchmarking numerical algorithms for harmonic maps into the sphere
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Sören Bartels
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.
Funding source: Deutsche Forschungsgemeinschaft (DFG)
Award Identifier / Grant number: 3013
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: The authors gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft (Funder DOI: https://dx.doi.org/10.13039/501100001659) in the Research Unit 3013 Vector- and Tensor-Valued Surface PDEs within the sub-projects TP3: Heterogeneous thin structures with prestrain and TP4: Bending plates of nematic liquid crystal elastomers.
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Data availability: Not applicable.
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