Preconditioned iteration for saddle-point systems with bound constraints arising in contact problems

We introduce a new algorithm, which we shall call Mixed Iteration, or by the acronym Mix-It, for the solution of saddle point problems arising from minimization problems under equality and/or inequality constraints. Such constrained problems are found in many applications and we shall be specially interested in contact problems in solid mechanics. The algorithm is however general and can be applied to other situations. We present a real saddle point algorithm in the sense that the primal variables and Lagrange multipliers unknowns are solved simultaneously. The effectiveness of the proposed method for large-scale simulations is improved by a block preconditioner and an adequate treatment of inequality constraints. The block preconditioner involves the solution of two subsystems associated respectively with the primal and the dual variables. We present some numerical results to illustrate the performances of the proposed method.