On the stability and convergence of discontinuous Galerkin schemes for incompressible flows

Abstract

Abstract The property that the velocity $\textbf{u}$ belongs to $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretization methods are uniformly stable in the $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$-norm. In this work, we establish that this is indeed the case for discontinuous Galerkin (DG) discretizations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretization is equivalent to the RadauIIA Implicit Runge–Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo–Nirenberg-type inequalities on DG spaces, which might be of independent interest.