On certain configurations of points in Rn which are unisolvent for polynomial interpolation
Journal of Approximation Theory
The spectral element method for the shallow water equations on the sphere
Journal of Computational Physics
From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex
SIAM Journal on Numerical Analysis
Monomial cubature rules since “Stroud”: a compilation—part 2
Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
A generalized diagonal mass matrix spectral element method for non-quadrilateral elements
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Numerical Initial Value Problems in Ordinary Differential Equations
Numerical Initial Value Problems in Ordinary Differential Equations
Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation
SIAM Journal on Numerical Analysis
An Algorithm for Computing Fekete Points in the Triangle
SIAM Journal on Numerical Analysis
An encyclopaedia of cubature formulas
Journal of Complexity
Improved Lebesgue constants on the triangle
Journal of Computational Physics
Nodal configurations and voronoi tessellations for triangular spectral elements
Nodal configurations and voronoi tessellations for triangular spectral elements
A Cardinal Function Algorithm for Computing Multivariate Quadrature Points
SIAM Journal on Numerical Analysis
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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.