Journal of Scientific Computing

, Volume 77, Issue 3, pp 1468–1489 | Cite as

Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions

  • Sheng Chen
  • Jie Shen


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.


Weakly singular solution Spectral-Galerkin method Enriched space Jacobi polynomials Error estimate 

Mathematics Subject Classification

65N35 41A10 41A30 41A99 


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China
  2. 2.Beijing Computational Science Research CenterBeijingPeople’s Republic of China
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA
  4. 4.School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific ComputingXiamen UniversityXiamenPeople’s Republic of China

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