Numerical Algorithms

, Volume 64, Issue 3, pp 519–547 | Cite as

A new triangular spectral element method I: implementation and analysis on a triangle

  • Michael Daniel Samson
  • Huiyuan Li
  • Li-Lian Wang
Original Paper


This paper serves as our first effort to develop a new triangular spectral element method (TSEM) on unstructured meshes, using the rectangle–triangle mapping proposed in the conference note (Li et al. 2011). Here, we provide some new insights into the originality and distinctive features of the mapping, and show that this transform only induces a logarithmic singularity, which allows us to devise a fast, stable and accurate numerical algorithm for its removal. Consequently, any triangular element can be treated as efficiently as a quadrilateral element, which affords a great flexibility in handling complex computational domains. Benefited from the fact that the image of the mapping includes the polynomial space as a subset, we are able to obtain optimal L 2- and H 1-estimates of approximation by the proposed basis functions on triangle. The implementation details and some numerical examples are provided to validate the efficiency and accuracy of the proposed method. All these will pave the way for developing an unstructured TSEM based on, e.g., the hybridizable discontinuous Galerkin formulation.


Rectangle–triangle mapping Consistency condition Triangular spectral elements Spectral accuracy 

Mathematics Subject Classifications (2010)

65N35 65N22 65F05 35J05 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Michael Daniel Samson
    • 1
  • Huiyuan Li
    • 2
  • Li-Lian Wang
    • 1
  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  2. 2.Institute of SoftwareChinese Academy of SciencesBeijingChina

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