Divergence-preserving methods for incompressible flow

  • Date:
  • Time: 14:00 - 15:30
  • Address:
    Sokolovská 83, Praha
  • Room: K3
  • Speaker: Jan Blechta

In this review talk the structure-preserving property of discretizations for incompressible flow that is called pressure robustness will be covered. It is arguably the most important development of the last decade in the field: "[P]ressure-robust space discretisations outperform non-pressure-robust space discretisations for incompressible Navier–Stokes flows, especially at high Reynolds numbers." [Gauger, Linke, Schroeder 2019] "[P]ressure-robust schemes outperform non-pressure-robust schemes for entire classes of transient incompressible flows at high Reynolds numbers." [Lederer, Merdon, Schöberl 2019] Classical discretizations of incompressible flow, such as the Hood–Taylor mixed element or the MINI element, enforce mass conservation in the sense of L2 orthogonality to the pressure space. It turns out that, in the simplest case of the stationary Stokes system, this leads to the velocity discretization error being polluted by the pressure error, which gives the name to the phenomenon of pressure nonrobustness. This can also be viewed as the incompatibility of the discretization with the Helmholtz decomposition. On practical level this leads to various kinds of instabilities. On the other hand the schemes which enforce the mass conservation pointwise are free of these problems. The difficult part is to establish inf-sup stability. In this talk I will cover classical results concerning the stability (or the lack of) for the divergence-free Scott–Vogelius pair, new results giving its stability on special (but useful) classes of meshes, overview its connection to the de Rham complex and the finite element exterior calculus, and mention other types of divergence-preserving discretizations. I will also give few computational examples showing how pressure nonrobustness results in various kind of instabilities. Last but not least, it will be shown how certain simple divergence-preserving schemes can be implemented with ease, thus allowing computational practitioners to ditch the everlasting, pressure non-robust, Hood–Taylor method.