An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch

We describe a pseudo-spectral method for rapidly calculating an analytic approximation to a 4-sided PDE surface patch. The method generates an approximate solution consisting of three parts: an eigenfunction solution and a polynomial solution, both of which satisfy the generating partial differential equation exactly, and a third function, or 'remainder' term that ensures that the boundary conditions are satisfied exactly. Being analytic, the approximation allows an arbitrary degree of surface refinement thereby facilitating physical analysis.