Performance of numerically computed quadrature points

In this work we focus on the diagonal-mass-matrix spectral element method (DMM) in non-tensor product domains such as triangles and tetrahedra. For many problems, the DMM method is more efficient than methods which require the inversion of a wide bandwidth mass matrix. For elements which are tensor product domains, the method is formulated using a nodal basis derived from Gauss–Lobatto points which simultaneously provide well-conditioned interpolants and high quality quadrature. In non-tensor product domains such as triangles and tetrahedra such points are not known analytically. DMM therefore relies on numerically computed interpolation and quadrature points. Here we compare three point sets for use in DMM methods on triangles: Fekete points and two new point sets that have improved quadrature properties but less optimal interpolation properties. We show convergence results for the linear advection and Poisson problems, and the growth of the eigenspectrum for the first and second derivative operators. We conclude that these points sets, when used for DMM on triangles, suffer from numerical stability problems which can be overcome by employing common filtering techniques such as the erfc-log filter [J.P. Boyd, The erfc-log filter and the asymptotics of the Euler and Vandeven sequence accelerations, in: Proc. of 3rd Int. Conf. on Spectral and High Order Methods, 1996, pp. 267–276]. We show the growth of the maximum eigenvalues for both the first and second derivative operator is no worse than for that of tensor product domains. We also show that for the solution of the Poisson equation in weak form, the new points, which have improved quadrature properties, are superior to Fekete points.