Fully discrete finite element approximation of unsteady flows of
implicitly constituted incompressible fluids

Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalised Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\Omega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully-discrete approximation scheme, using a spatial mixed finite element approximation on general shape-regular simplicial meshes combined with backward Euler time-stepping. We consider the case when the velocity field belongs to the space of solenoidal functions contained in $L^\infty(0,T;L^2(\Omega)^d)\cap L^q(0,T; W^{1,q}_0(\Omega)^d)$ with $q\in (2d/(d+2), \infty)$, which is the maximal range of $q$ with respect to existence of weak solutions. In order to facilitate passage to the limit with the discretisation parameters for the sub-range $q \in (2d/(d+2),(3d+2)/(d+2))$ we introduce a regularisation of the momentum equation by means of a penalty term and first show convergence of a subsequence of approximate solutions to a weak solution of the regularised problem; we then pass to the limit with the regularisation parameter. This is achieved by the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness techniques. For $q \geq (3d+2)/(d+2)$ convergence of a subsequence of approximate solutions to a weak solution can be shown directly, without the regularisation term.