Fekete Collocation Points for Triangular Spectral Elements

We propose a new spectral element method based on Fekete points. We use the Fekete criterion to compute points which are almost optimal for approximation, diierentiation and quadrature. These grids have the same number of points as the dimension of the associated polynomial space, thus allowing us to use a cardinal function basis which leads to a diagonal mass matrix. For quadrilaterals, Gauss-Lobatto points are the unique Fekete points, making the method equivalent to the standard spectral element method. But unlike the Gauss-Lobatto points, Fekete points generalize to other domains such as the triangle. Furthermore, numerical and theoretical evidence suggests that the element boundary points are the Gauss-Lobatto points, making Fekete point triangular elements and quadrilateral elements naturally conform. Thus triangles and quadrilaterals can be combined in the same grid while retaining a diagonal mass matrix. We present an algorithm to compute Fekete points along with results for the triangle up to degree 19. For degree d > 10 these points have the smallest Lebesgue constant currently known.