Finite Element Approximation of Steady Flows of Colloidal Solutions

We consider the mathematical analysis and numerical approximation of a system
of nonlinear partial differential equations that arises in models that have
relevance to steady isochoric flows of colloidal suspensions. The symmetric
velocity gradient is assumed to be a monotone nonlinear function of the
deviatoric part of the Cauchy stress tensor. We prove the existence of a unique
weak solution to the problem, and under the additional assumption that the
nonlinearity involved in the constitutive relation is Lipschitz continuous we
also prove uniqueness of the weak solution. We then construct mixed finite
element approximations of the system using both conforming and nonconforming
finite element spaces. For both of these we prove the convergence of the method
to the unique weak solution of the problem, and in the case of the conforming
method we provide a bound on the error between the analytical solution and its
finite element approximation in terms of the best approximation error from the
finite element spaces. We propose first a Lions-Mercier type iterative method
and next a classical fixed-point algorithm to solve the finite-dimensional
problems resulting from the finite element discretisation of the system of
nonlinear partial differential equations under consideration and present
numerical experiments that illustrate the practical performance of the proposed
numerical method.