Cubature versus Fekete-Gauss nodes for spectral element methods on simplicial meshes

International audienceIn a recent JCP paper (by Y. Liu et al., vol. 336, p. 458, 2017) a higher order triangular spectral element method (T SEM) is proposed to address seismic wave field modeling. The main interest of this T SEM is that the mass matrix is diagonal, so that an explicit time marching becomes very cheap. This property results from the fact that, similarly to the usual SEM, the basis functions are Lagrange polynomials based on a set of points that shows both nice interpolation and quadrature properties. In the quadrangle, i.e. for the QSEM, the set of points is simply obtained by tensorial product of Gauss-Lobatto-Legendre (GLL) points. In the triangle, finding such an appropriate set of points is however not trivial. Thus, the recent JCP work follows anterior works that started in 2000's and now provides cubature nodes and weights up to N = 9, where N is the total degree of the polynomial approximation. Here we wish to evaluate the accuracy of this cubature nodes T SEM with respect to the Fekete-Gauss one, that makes use of two sets of points, namely the Fekete points and the Gauss points of the triangle for interpolation and quadrature, respectively. Because the Fekete-Gauss T SEM is in the spirit of any nodal hp-finite element methods, one may expect that the conclusions of this Note will remain relevant if using other sets of carefully defined interpolation points