Least-squares finite elements and constrained evolution systems.

Although semigroup theory provides a thorough analytic framework for solving constrained evolution systems, the treatment of constraints in the approximation of such systems can greatly affect the accuracy of the approximate solution. Herein, a method based on least-squares finite elements is developed which maintains the simplicity and efficiency of typical approaches for unconstrained problems, while guaranteeing accuracy. The method is developed in a general fashion which can be applied to a wide range of linear and non-linear constrained evolution systems and can employ numerous techniques from the optimization literature. This new method is illustrated on some examples including the wave equation and Maxwell's equations.