A diagonal-mass-matrix triangular-spectral-element method based on cubature points

Abstract

The cornerstone of nodal spectral-element methods is the co-location of the interpolation and integration points, yielding a diagonal mass matrix that is efficient for time-integration. On quadrilateral elements, Legendre–Gauss–Lobatto points are both good interpolation and integration points; on triangles, analogous points have not yet been found. In this paper, a promising set of points for the triangle that were only available for polynomial degree N ≤ 5 are used. However, the derivation of these points is generalized to obtain degree N ≤ 7 points, which are referred to as cubature points because their selection is based on their integration accuracy. The diagonal-mass-matrix (DMM) triangular-spectral-element (TSE) method based on these points can be used for any set of equations and any type of domain. Because these cubature points integrate up to order 2N along the element boundaries and yield a diagonal mass matrix, they allow the triangular spectral elements to compete with quadrilateral spectral elements in terms of both accuracy and efficiency, while offering more geometric flexibility in the choice of grids. It is shown how to implement this DMM TSE for a variety of applications involving elliptic and hyperbolic equations on different domains. The DMM TSE method yields comparable accuracy to the exact integration (non-DMM) TSE method, while being far more efficient for time-dependent problems.