Numerical solution of the Stokes problem under free boundary by the modified projection gradient method

From a variational standpoint, the solution of Stokes equation is reduced to the constrained minimization of the total energy functional over the space of solenoidal fields. The paper presents the finite element method combined with the gradient projection method to obtain the approximate solution of this problem by the unconstrained minimization of the quadratic functional with a reduced number of unknown variables, which has not been previously used for solving such a problem. The numerical solutions of a few test free boundary problems are presented to reveal the advantages of the developed method over the penalty and Lagrangian methods with respect to the accuracy, stability, and computer speed.