Fekete Points

Global spectral methods give exponential convergence rates and have high accuracy for smooth solutions, but are global and used with simple domains. The spectral element method combines the geometric flexibility of the finite element method with the accuracy and efficiency of the spectral method. The usual implementation on triangles becomes ill-conditioned for degree (maximum degree of the polynomial) greater than about seven. One can use preconditioning to partially improve this. The ill-conditioning is not found in quadrilaterals domains.

Work on this subject with Mark Taylor led to a new way to obtain a globally diagonal mass matrix (without any preconditioning) and an exponentially converging method, for general domains. The first paper, The natural function space for triangular and tetrahedral spectral elements shows that the Koornwinder basis, a set of multi-dimensional orthogonal polynomials, are the eigenfunctions of a Sturm-Liouville problem. This paper was sent to SINUM where it sat for two years before being sent out to review and in that time others published before us. We were invited to submit a paper on simplices to SISC but never did. Our draft of the more general proof for simplices can be found here The Koornwinder polynomials are Lendre polynomials in simplices. These polynomials are the Legendre polynomials in the simplex, and can be generalized to Chebyshevs, or other weights. In this paper on functional spaces for triangles and tetrahedra, we also gave the eigenvalues for an n-dimensional simplex. The second paper, An Algorithm for computing Fekete points in the triangle, shows that using this basis with the optimal points, the Fekete points, one may get well behaved Cardinal Functions, which means we get a globally diagonal mass matrix for explicit methods and an exponentially converging method. In this paper we show how to compute the points, list the points for up to degree 19, and give error estimates for the numerical method. The third paper, A generalized diagonal mass matrix spectral element method for non-quadrilateral elements puts the theory of the first to papers into one general format, showing how you use these two pieces to form a new method.

One of the key differences between triangular and quadrilateral elements is that there exists good quadrature formulas that make the mass matrix diagonal for quadrilaterals more accurate. Another paper showed that Tensor product Gauss-Lobatto points are Fekete points for the cube.

In 2019 I wrote an entry for Wiley StatsRef: Statistics Reference Online titled n-dimensional Quadrature.

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