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

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Abstract

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.

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Michael Daniel Samson & Li-Lian Wang

  2. Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Huiyuan Li

Authors
  1. Michael Daniel Samson
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  2. Huiyuan Li
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  3. Li-Lian Wang
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Correspondence to Li-Lian Wang.

Additional information

The research of Michael Daniel Samson and Li-Lian Wang is partially supported by Singapore AcRF Tier 1 Grant RG58/08.

The work of Huiyuan Li was supported by National Natural Science Foundation of China (NSFC) Grants 10601056 and 10971212.

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Samson, M.D., Li, H. & Wang, LL. A new triangular spectral element method I: implementation and analysis on a triangle. Numer Algor 64, 519–547 (2013). https://doi.org/10.1007/s11075-012-9677-4

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  • DOI: https://doi.org/10.1007/s11075-012-9677-4

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