Positive definite matrices and Sylvester's criterion
American Mathematical Monthly
Matrices with positive definite Hermitian part: inequalities and linear systems
SIAM Journal on Matrix Analysis and Applications
Convergence analysis of some algorithms for solving nonsmooth equations
Mathematics of Operations Research
Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces
Nonlinear Analysis: Theory, Methods & Applications
Newton and Quasi-Newton Methods for a Class of Nonsmooth Equations and Related Problems
SIAM Journal on Optimization
The Primal-Dual Active Set Strategy as a Semismooth Newton Method
SIAM Journal on Optimization
An Absolutely Stable Pressure-Poisson Stabilized Finite Element Method for the Stokes Equations
SIAM Journal on Numerical Analysis
Semismooth Newton and Augmented Lagrangian Methods for a Simplified Friction Problem
SIAM Journal on Optimization
SIAM Journal on Scientific Computing
Path-following Methods for a Class of Constrained Minimization Problems in Function Space
SIAM Journal on Optimization
An Iterative Method for the Stokes-Type Problem with Variable Viscosity
SIAM Journal on Scientific Computing
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This 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 P"1)-Q"0 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.