Abstract

The classical Minty lemma relies on monotonicity and some notion of continuity of the operator involved. For discontinuous relations a suitable extension has been proved in Bulicek et al. (2012, 2016), which was applied to show existence of weak solutions to fluid equations for non-Newtonian fluids with discontinuous constitutive relation, cf. Bulicek et al. (2012). I will present a splitting and regularising strategy to show convergence (up to subsequences) of finite element approximations using this Minty type convergence lemma. Furthermore, I shall introduce approximations for discontinuous relations satisfying a variant of the Minty type convergence result. This is useful to establish convergence of numerical approximations without splitting the limits. If time allows I will introduce a class of non-monotone relations and sketch issues around the identification of the relation in the existence proof. Parts of this talk are based on joint work with Endre Süli.