Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods

AbstractThis paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P1)–Q0 elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method

Finite element error analysis for state-constrained optimal control of the Stokes equations

An optimal control problem for 2d and 3d Stokes equations is investigated with pointwise inequality constraints on the state and the control. The paper is concerened with the full discretization of the control problem allowing for different types of discretization of both the control and the state. For instance, piecewise linear and continuous approximations of the control are included in the present theory. Under certain assumptions on the $L^\infty$-error of the finite element discretization of the state, error estimates for the control are derived which can be seen to be optimal since their order of convergence coincides with the one of the interpolation error. The assumptions of the $L^\infty$-finite-element-error can be verified for different numerical settings. The theoretical results are confirmed by numerical examples

Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints

Revised version of the preprint first published 06. December 2005In this paper we consider the distributed optimal control of the Navier-Stokes equations in presence of pointwise mixed control-state constraints. After deriving a first order necessary condition, the regularity of the mixed constraint multiplier is investigated. Second-order sufficient optimality conditions are studied as well. In the last part of the paper, a semi-smooth Newton method is applied for the numerical solution of the control problem. The convergence of the method is proved and numerical experiments are carried out