Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier–Stokes equations

Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier–Stokes equations. The spatial discretization based on inf–sup stable pairs of finite element spaces is stabilized using a one-level local projection stabilization method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for both the semidiscrete case and the fully discrete case. Numerical results support the theoretical predictions.

Equal-Order Stabilized Finite Element Approximation of the p-Stokes Equations on Anisotropic Cartesian Meshes

The p-Stokes equations with power-law exponent {p\in(1,2)} describes non-Newtonian, shear-thinning, incompressible flow. In many industrial applications and natural settings, shear-thinning flow takes place in very thin domains. To account for such anisotropic domains in simulations, we here study an equal-order bi-linear anisotropic finite element discretization of the p-Stokes equations, and extend a non-linear Local Projection Stabilization to anisotropic meshes. We prove an a priori estimate and illustrate the results with two numerical examples, one confirming the rate of convergence predicted by the a-priori analysis, and one showing the advantages of an anisotropic stabilization compared to an isotropic one.