A Mixed Finite Element Approach for A Variational-Hemivariational Inequality of Incompressible Bingham Fluids

In this paper, we consider an incompressible Bingham fluid with a nonsmooth and nonconvex slip boundary condition and establish its variational formulation, i.e., the Bingham type variational-hemivariational inequality. The existence and uniqueness of solutions are investigated based on the minimization argument. To solve this problem, we construct an equivalent mixed hemivariational inequality based on the minimax principles and employ an Uzawa type iteration algorithm to approximate it. Then we give a convergence analysis for such an algorithm. Further, we adopt a P2-P1 finite element to discretize the mixed hemivariational inequality and obtain the error estimates. Finally, the numerical test results are presented, which are in accordance with the previous error estimates.