A Nonlinear, Subgridscale Model for Incompressible viscous Flow Problems

We consider a nonlinear subgridscale model of the Navier--Stokes equations resulting in a Ladyzhenskaya-type system. The difference is that the power "$p$" and scaling coefficient $\mu(h) \doteq O(h^{\sigma})$ do not arise from macroscopic fluid properties and can be picked to ensure both $L^{\infty}$-stability and yet be of the order of the basic discretization error in smooth regions. With a properly scaled $p$-Laplacian-type artificial viscosity one can construct a higher-order method which is just as stable as first-order upwind methods.