A pressure-robust Discrete de Rham scheme for the Navier-Stokes equations

In this work we design and analyse a Discrete de Rham (DDR) method for theincompressible Navier-Stokes equations. Our focus is, more specifically, on theSDDR variant, where a reduction in the number of unknowns is obtained usingserendipity techniques. The main features of the DDR approach are the supportof general meshes and arbitrary approximation orders. The method we develop isbased on the curl-curl formulation of the momentum equation and, throughcompatibility with the Helmholtz-Hodge decomposition, delivers pressure-robusterror estimates for the velocity. It also enables non-standard boundaryconditions, such as imposing the value of the pressure on the boundary.In-depth numerical validation on a complete panel of tests including generalpolyhedral meshes is provided. The paper also contains an appendix where boundson DDR potential reconstructions and differential operators are proved in themore general framework of Polytopal Exterior Calculus.