A Lobatto interpolation grid in the tetrahedron

A sequence of increasingly refined interpolation grids inside the tetrahedron is proposed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The number of interpolation nodes, N, corresponds to the number of terms in the complete mth-order polynomial expansion with respect to the three tetrahedral barycentric coordinates. The proposed grid is constructed by deploying Lobatto interpolation nodes over the faces of the tetrahedron, and then computing interior nodes using a simple formula that involves the zeros of the Lobatto polynomials. Numerical computations show that the Lebesgue constant and interpolation accuracy of the proposed grid compare favourably with those of alternative grids constructed by solving optimization problems. The condition number of the mass matrix is significantly lower than that of the uniform grid and comparable to that of optimal grids proposed by previous authors.