A semi-implicit scheme based on Arrow-Hurwicz method for saddle point problems

We search saddle points for a large class of convex-concave Lagrangian. A generalized explicit iterative scheme based on Arrow-Hurwicz method converges to a saddle point of the problem. We also propose in this work, a convergent semi-implicit scheme in order to accelerate the convergence of the iterative process. Numerical experiments are provided for a nontrivial numerical problem modeling an optimal shape problem of thin torsion rods. This semi-implicit scheme is figured out in practice robustly efficient in comparison with the explicit one.