Numerically solving PDEs typically requires compressing infinite information into a finite system of algebraic equations. Pragmatically, we usually follow a recipe: “Assume solutions of form X; substitute into PDE Y; discard terms by rule Z.” In contrast, Lanczos’s pioneering “tau method” prescribes modifying the PDE to form an exact finite system. Crucially, any recipe-based method can be viewed as adding a small equation correction, enabling us to compare multiple schemes independently of the solver. 

This talk also addresses a paradox: PDEs often admit infinitely many solutions, but finite systems produce only a finite set. When we include a “small” correction, the missing solutions are effectively hidden. I will discuss how tau methods frame this perspective and outline proposals for systematically studying and optimising various residuals.