Pressure-robust and quasioptimal Discontinuous Galerkin discretisations of the $p$-Stokes problem

Abstract

In the present paper, we propose Local Discontinuous Galerkin (LDG) approximations for a nonlinear system of $p$-Stokes type, having $(p,\delta )$-structure. On the basis of the primal formulation, we prove well-posedness and stability (a priori estimates) of the methods under truly minimal regularity assumptions. We show that the first method possesses a pressure-robust and quasi-optimal error estimate, and discuss its consequences. Moreover, we propose a second method, for which we show a pressure-robust error estimate and prove convergence and convergence rates, which are optimal for linear ansatz functions for all $p\in (1,\infty )$ and $\delta \ge 0$.