The authors analyse steady Bingham fluid flow in cylindrical pipe under a uniform pressure drop per unit length, and present a novel iterative method for numerical solutions of such problems. This primal formulation leads to a variational inequality of the second kind, a variational problem involving a non-differentiable functional. The authors provide an equivalent primal-dual formulation which is the basis of the computational method which possesses convergence properties of the well-known Uzawa’s method.
The main goal is to analyse a time-dependent model whose steady-state solution is a solution of the original problem, and to develop a novel iterative method for its numerical solution. The method presented is based on the primal-dual formulation and on an implicit scheme of backward Euler type, derived from the parabolic variational inequality problem describing a time-dependent model discussed. The authors present several types of algorithms and prove that the schemes are stable and convergent. They discuss a dynamical Tikhonov regularization of the fixed point relation, which plays an important role in improving the conditioning and well-posedness properties of the dual problem at each time step. Then the unique solution of regularized problem and the convergence of the schemes (based on the Opial’s lemma) are established. The iterative solution of the primal-dual problem at each time step is discussed, and a convergence theorem is proved. The finite element approximation is analysed, and it is shown that for numerical computations a fast Poisson solvers can be used. The numerical experiments are presented.