Prof. Jan Giesselmann (TU Darmstadt)

Can a numerical approximation establish that an exact solution of a nonlinear PDE exists on a given time interval? For equations whose solutions may develop singularities, this requires more than observing that a simulation remains regular. We will discuss how conditional a posteriori error estimates can be combined with continuation criteria to turn numerical approximations into rigorous existence results.

The main example is the three-dimensional incompressible Navier–Stokes system with periodic boundary conditions. A conditional stability estimate, together with a blow-up criterion in the critical space (L^\infty(0,T;L^3)), yields an a posteriori criterion for the existence of strong solutions. The criterion involves negative Sobolev norms of the residual and can be evaluated through computable bounds for standard mixed finite element approximations. We will explain the analytical mechanism and how reconstruction connects the discrete approximation to the continuous stability theory.

A second example, the parabolic–elliptic Keller–Segel system, illustrates how the same principle can verify the existence of weak solutions. In this setting, we also explain how round-off errors and errors from inexact algebraic solves can be incorporated into the a posteriori bounds. We will discuss the scope and limitations of these criteria, including the distinction between failure of verification and formation of a singularity. The talk is based on joint work with Aaron Brunk and Tabea Tscherpel (Navier–Stokes), and with Marc Hoffmann (Keller–Segel).