Computation of harmonic functions on higher genus surfaces
<|reference_start|>Computation of harmonic functions on higher genus surfaces: We introduce a method to compute efficiently and with arbitrary precision a basis of harmonic functions with prescribed singularities on a general compact surface of genus two and more. This basis is obtained as a composition of theta functions and the Abel-Jacobi map, which is approximated at spectral speed by complex polynomials. We then implement this method to compute harmonic extensions on genus $2$ surfaces with boundary, that are described by their Fenchel-Nielsen coordinates and a smooth parametrization of the boundary. Finally, we prove the spectral convergence of the method for the harmonic extension.<|reference_end|>
@article{nahon2024computation,
    title={Computation of harmonic functions on higher genus surfaces},
    author={Micka"el Nahon, 'Edouard Oudet},
    journal={arXiv preprint arXiv:2410.06763},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.06763},
    primaryClass={math.NA cs.NA math.AP}
}