Mathematics Colloquia and Seminars

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We consider Chorin's projection method combined with a finite element spatial discretization for the time-dependent incompressible Navier-Stokes equations. The projection method advances the solution in two steps: A prediction step which computes an intermediate velocity field that is generally not divergence-free, and a projection step which enforces (approximate) incompressibility by projecting this velocity onto the (approximately) divergence-free subspace. We establish convergence, up to a subsequence, of the numerical approximations generated by the projection method with finite element spatial discretization to a Leray-Hopf solution of the incompressible Navier-Stokes equations, without any additional regularity assumptions beyond square-integrable initial data and square-integrable forcing. A discrete energy inequality yields a priori estimates, which we combine with a new compactness result to prove precompactness of the approximations in $L^2([0,T]\times\Omega)$, where $[0,T]$ is the time interval and $\Omega$ is the spatial domain. Passing to the limit as the discretization parameters vanish, we obtain a weak solution of the Navier–Stokes equations. A central difficulty is that different a priori bounds are available for the intermediate and projected velocity fields; our compactness argument carefully integrates these estimates to complete the convergence proof. If time permits, I will also discuss how the proof can be adapted to prove convergence of a second-order in time method.