A New Triangular Spectral Element Method I: Implementation and Analysis on a Triangle

TL;DR

This paper introduces a novel triangular spectral element method using a rectangle-triangle mapping that enables efficient, accurate, and flexible numerical solutions on unstructured meshes, with proven optimal approximation estimates.

Contribution

It develops a new TSEM with a unique mapping that handles singularities effectively, allowing for efficient implementation and optimal approximation on triangles.

Findings

The mapping induces only a logarithmic singularity.

The method achieves optimal $L^2$- and $H^1$-estimates.

Numerical examples confirm efficiency and accuracy.

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 [21]. 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.