A posteriori error analysis of mixed finite element methods for a regularized μ(I)-rheology model of granular materials

Abstract

We consider a Banach spaces-based mixed variational formulation recently proposed for the stationary $\mu \left(I\right)$-rheology model of granular materials, and develop the first reliable and efficient residual-based a posteriori error estimator for its associated mixed finite element scheme in both 2D and 3D, considering PEERS and AFW-based discretizations. For the reliability analysis, and due to the nonlinear nature of the problem, we employ the first-order Gâteaux derivative of the global operator involved in the problem, combined with appropriate small-data assumptions, a stable Helmholtz decomposition in nonstandard Banach spaces, and local approximation properties of the Raviart–Thomas and Clément interpolants. In turn, inverse inequalities, the localization technique based on bubble functions in local $L^p$-spaces, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical examples confirming the theoretical properties of the estimator and illustrating the performance of the associated adaptive algorithms are reported. In particular, the case of fluid flow through a 2D cavity with two circular obstacles is considered.