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Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions

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Abstract

Usual spectral methods are very effective for problems with smooth solutions, but their accuracy will be severely limited if solution of the underlying problems exhibits singular behavior. We develop in this paper enriched spectral-Galerkin methods (ESG) to deal with a class of problems for which the form of leading singular solutions can be determined. Several strategies are developed to overcome the ill conditioning due to the addition of singular functions as basis functions, and to efficiently solve the approximate solution in the enriched space. We validate ESG by solving a variety of elliptic problems with weakly singular solutions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, People’s Republic of China

    Sheng Chen

  2. Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Sheng Chen

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-1957, USA

    Jie Shen

  4. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Jie Shen

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  1. Sheng Chen
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  2. Jie Shen
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Correspondence to Jie Shen.

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Dedicated to Bernardo Cockburn on the occasion of his 60th birthday.

Sheng Chen is partially supported by the Foundation of Jiangsu Normal University (Grant No. 17XLR013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. BK20181002), the National Natural Science Foundation for the Youth of China (Grant No. 11801235) and the Postdoctoral Science Foundation of China (Grant No. BX20180032). Jie Shen is partially supported by NSF Grants DMS-1620262, DMS-1720442 and AFOSR Grant FA9550-16-1-0102.

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Chen, S., Shen, J. Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions. J Sci Comput 77, 1468–1489 (2018). https://doi.org/10.1007/s10915-018-0862-z

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  • DOI: https://doi.org/10.1007/s10915-018-0862-z

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