Publication:
Approximation of Fractional Harmonic Maps

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Authors
Antil, Harbir
Zichao, Wendy Di
Khatri, Ratna
Subjects
fractional derivatives
harmonic maps
nonlocality
compactness
finite element method
algorithms
convergence analysis
spectral method
spin chains
defects
Advisors
Date of Issue
2021
Date
Publisher
ArXiv
Language
Abstract
This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields. We devise and analyze numerical methods for the approximation of various partial differential equations related to fractional harmonic maps. The compactness results imply the convergence of numerical approximations. Numerical examples on spin chain dynamics and point defects are presented to demonstrate the effectiveness of the proposed methods.
Type
Preprint
Description
The article of record as published may be located at https://doi.org/10.48550/arXiv.2104.10049
Funded by Naval Postgraduate School
Department
Organization
Naval Postgraduate School
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NPS Report Number
Sponsors
HA is partially supported by NSF grants DMS-1818772 and DMS-1913004, the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036, and the Department of the Navy, Naval Postgraduate School under Award NO: N00244-20-1-0005. SB acknowledges support by the DFG via the Research Unit FOR 3013 Vector- and tensor-valued surface PDEs. AS is supported by NSF Career DMS-2044898 and Simons foundation grant no 579261.
Funder
HA is partially supported by NSF grants DMS-1818772 and DMS-1913004, the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036, and the Department of the Navy, Naval Postgraduate School under Award NO: N00244-20-1-0005. SB acknowledges support by the DFG via the Research Unit FOR 3013 Vector- and tensor-valued surface PDEs. AS is supported by NSF Career DMS-2044898 and Simons foundation grant no 579261.
Format
30 p.
Citation
Antil, Harbir, Sören Bartels, and Armin Schikorra. "Approximation of fractional harmonic maps." arXiv preprint arXiv:2104.10049 (2021).
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