SC Seminar: Alexander Linke

Dr. Alexander Linke , Chair for Scientific Computing (SciComp), University of Kaiserslautern-Landau (RPTU)

Title: On the Discretization of the Incompressible Navier–Stokes Equations, an Elephant in the Room, and a Conceptual Update for Discretizing Constrained PDEs


The dynamics of the incompressible Navier-Stokes equations are closely related to equivalence classes of forces and an associated semi-norm, in the kernel of which all gradient fields lie. The corresponding numerical treatment of gradient fields in the momentum balance, which only change the pressure but not the velocity of an incompressible flow, has been repeatedly addressed in research since the 1980s, and a number of very different algorithmic approaches have been proposed to avoid this numerical source of error. For many years, however, the research question remained the proverbial ‘elephant in the room’, whose relevance for simulation practice was assessed very differently in the research community. The talk provides an overview of the historical development of the research question and discusses its practical relevance based on completely different physical regimes such as hydrostatics and high Reynolds number vortex flows. In particular, it discusses how recently developed pressure-robust methods using H(div)-conforming finite elements could solve the numerical challenge in a fundamental way and thus contribute to the solution of other previously unsolved problems. Furthermore, a conceptual update for the discretization of PDEs with constraints is proposed, which replaces the historical Stokes model problem of classical mixed finite element theory by a set of model problems from which the relevance of H(div)-conforming algorithms for the discretization of incompressible flows immediately emerges. Finally, considerations on the relevance of the obtained results for the numerical treatment of related problems, such as the compressible Navier–Stokes equations, are discussed.

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