A constrained string method is developed to solve the saddle-point problem with constraints. Based on the intrinsic description of the string method, Lagrange multipliers are employed for treating the constraints. Various mathematical properties are established such as the conservation of the constraints and the energy dissipation law. We also investigate time discretization schemes for the constrained string method and discuss possible alternative ways to enforce the constraints. Some numerical examples are presented for illustration.
"A constrained string method and its numerical analysis." Commun. Math. Sci. 7 (4) 1039 - 1051, December 2009.