Sobolev gradients are an efficient method of calculating solutions to a wide variety of systems of partial differential equations. Successful applications have been made to problems in transonic flow, Ginsburg-Landau equations for superconductivity, elasticity, minimal surfaces and oil-water separation problems.

I will discuss why the poor numerical performance of ordinary gradients and the good performance of Sobolev gradients give an instance of the first law of numerical analysis: "Analytical Difficulties and Numerical Difficulties Always Come in Pairs."

I will show how the method can also be successfully applied to the hyperbolic fully non-linear Monge-Ampere equation:

Det(D^2 z) = \kappa (1 + {z_x}^2 + {z_y}^2)^2

on the unit square, where \kappa = -1, represents the Gaussian curvature of the surface described by z.