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.

翻译：我们认为,数学和数字接近是非线性部分差异方程式体系的数学分析和数字近似值,这个体系出现在与稳定的偏差流相联的模型中,它与稳定的相近悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮悬浮的模型中。对准速度梯度假定是Cauchy应力应激悬浮悬浮的偏移部分的单质非线性函数的单向非线性非线性边际函数。我们证明存在这一问题的单向性非线性速度梯度的单向性非线性函数函数。我们首先建议使用狮子-Mercier型迭接浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮浮标,然后采用经典的定点定值算法来解决因非线性元素离质元素分离的有限性定度问题。我们证明了非线性非线性部分性实验法的模拟微变形微等方程式。