In some circumstances, the mixed finite-element method, in which both the pressure and the velocities are approximated, offers significant advantages over the classical method, in which only the pressure is approximated. Mixed finite-element methods produce better velocity fields than those derived from classical methods leading to more robust and reliable pathline calculations. However, the system of discrete equations produced by the mixed finite-element method is much larger than that produced by the classical method. Furthermore, the coefficient matrix is no longer positive definite and this makes it difficult to apply standard iterative techniques to solve the equations. New techniques for solving the discrete mixed finite-element equations have recently been developed in collaboration with Robert Scheichl and Ivan Graham at the University of Bath.

The basic idea is to apply a transformation to the discrete equations that decouples the velocity and pressure calculations. The velocity is computed first. The coefficient matrix for this calculation is much smaller than that for the original equations and is also symmetric positive definite, which means that standard iterative techniques can be applied. Once the velocity field is known, the pressure can be computed by solving a triangular system of equations, which is also relatively cheap.

Robert Scheichl was one of the three winners of the prestigious 2000 SIAM Student Paper Prize for a paper that he wrote on this work for the 2000 SIAM Annual Meeting.