Analysis and PDE theory are widely used to establish theoretical guarantees for numerical PDE methods. Can numerical PDEs, in turn, be applied to rigorously analyze PDEs?

In this talk, we will discuss recent applications of numerical methods to finite-time blowup in the incompressible Euler equations and to the stability analysis of nonlinear PDEs. Numerical methods play a crucial role in constructing approximate solutions that provide critical small parameters for the stability analysis of the PDEs. By developing an a posteriori error estimate for the numerical error and treating it as a small perturbation in the stability analysis, we integrate the numerical solution and stability analysis into a rigorous PDE proof.