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An Algorithm for Computing Fekete Points in the Triangle

On the line and its tensor products, Fekete points are known to be the Gauss--Lobatto quadrature points. But unlike high-order quadrature, Fekete points generalize to non-tensor-product domains such as the triangle. Thus Fekete points might serve as an alternative to the Gauss--Lobatto points for certain applications. In this work we present a new algorithm to compute Fekete points and give results up to degree 19 for the triangle. For degree d > 10 these points have the smallest Lebesgue constant currently known. The computations validate a conjecture of Bos [ J. Approx. Theory, 64 (1991), pp. 271--280] that Fekete points along the boundary of the triangle are the one-dimensional Gauss--Lobatto points.