A Diagonal Mass Matrix Triangular Spectral Element Method based on Cubature Points

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 methods. On quadrilateral elements Legendre-Gauss-Lobatto points are both good interpolation and integration points but on triangles analogous points have not yet been found. In this paper we use a promising set of points for the triangle which were only available for polynomial degree N ≤ 5. However, we generalize the procedure used to derive these points to obtain degree N ≤ 7 points which we refer to as cubature points because the points are selected 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 on any type of domain. The fact that these cubature points integrate up to order 2N along the element boundaries and yield a diagonal mass matrix may 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. In this paper we show how to implement this DMM TSE for a variety of applications including 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.