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Nodal Auxiliary Space Preconditioning for the Surface de Rham Complex

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Abstract

This work develops optimal preconditioners for the discrete H(curl) and H(div) problems on two-dimensional surfaces by nodal auxiliary space preconditioning (Hiptmair and Xu in SIAM J Numer Anal 45:2483–2509, 2007). In particular, on unstructured triangulated surfaces, we develop fast and user-friendly preconditioners for the edge and face element discretizations of curl–curl and grad–div problems based on inverting several discrete surface Laplacians. The proposed preconditioners lead to efficient iterative methods for computing harmonic tangential vector fields on discrete surfaces. Numerical experiments on two- and three-dimensional hypersurfaces are presented to test the performance of those surface preconditioners.

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Acknowledgements

We would like to thank Professors Jinchao Xu and Ludmil Zikatanov for stimulating discussions about iterative methods of singular linear systems.

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  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, 310058, China

    Yuwen Li

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Communicated by Doug Arnold.

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Li, Y. Nodal Auxiliary Space Preconditioning for the Surface de Rham Complex. Found Comput Math 24, 1019–1048 (2024). https://doi.org/10.1007/s10208-023-09611-0

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