Dispersion analysis of the spectral element method using a triangular mesh

Abstract

The spectral element method (SEM) is a powerful tool to study wave propagation. Its main advantages are its accuracy and efficiency. Much work has been done to study the accuracy of SEM in quadrilateral elements, but the accuracy of this method using triangular elements is not well understood. In practice triangular elements are preferable to handle irregular geometries, but this introduces additional difficulties when obtaining the interpolation polynomial and quadrature points. In this paper, we show how to circumvent the difficulties using SEM with triangular elements (TSEM), and analyze two different types of nodes (Fekete points and Cohen points). The Fekete points are determined by minimizing the interpolation errors inside the element, while Cohen nodes are obtained by optimizing the accuracy of the quadrature rule. Both nodes have been employed for simulation, but their accuracy has not been studied. Our goal is to analyze the grid dispersion of these two types of nodes by considering the ‘X’ type triangular mesh. The analyses are based on the plane wave assumption by solving an eigenvalue problem. Our results indicate that, considering the same polynomial order, employing Cohen nodes requires more nodes per element but yields more accurate results compared to the Fekete points. Furthermore, the analysis suggests that higher order polynomials will improve the accuracy for both Fekete and Cohen nodes, which is the case for quadrilateral elements.