Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling

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

“…In a recent JCP paper [9], 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.…”

“…In the triangle, finding such an appropriate set of points is however not trivial. Thus, the work of [9] follows anterior works that started in 2000's [2,6,11] 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, see e.g.…”

“…Of course, N should be chosen as small as possible to avoid a useless increase of the inner nodes number: In [2] and posterior works, N is chosen such that it exists a cubature rule exact for polynomials of degree N + N − 2. The determination of N , together with the cubature points and weights gives rise to a difficult optimization problem, see [9] for details. It turns out that N − N increases monotonically with N : N = N + 1, for 1 < N < 5, N = N + 2 for N = 5 and N = N + 3 for 5 < N < 10.…”

“…On the contrary, the edge cubature points are not of Gauss type. In [9], where the wave equation is considered, one only makes use of Dirichlet conditions via eventually a PML (perfectly matched layer) to address the absorbing wall case. This is why it is of interest to consider again the previous elliptic problem but with now the Robin condition ∂ n u + αu = g on the boundary.…”

“…We are grateful to Dr Youshan Liu, Chinese Academy of Sciences, Beijing, for transmitting us the cubature points and weights used in [9].…”