Boyd considers the use of prolate spheroidal wave functions: prolate elements for computing the grid points, quadrature weights, and derivatives for spectral element methods. This is done by first computing the prolate nodal basis, and the appropriate quadrature and weights that replace the Legendre-Lobatto grid points, quadrature weights, and cardinal function derivative matrices. The logic of the spectral element method is not modified. The resulting method is the so-called prolate element method. The author developed the method as a library, resulting in software that can be applied to various classes of partial differential equations, linear and nonlinear.

The author successfully introduces the transformation from the modal to the nodal basis, and also reports some results on its condition number. Another area of improvement is the initialization for the computation of weights and grid points that replace the continuation algorithm used previously. Boyd is still testing the software on the shallow water problem.