Free Boundaries and Finite Elements in One Dimension

Two problems in control theory, one with state constraints and the other with control constraints, have been approximated by the finite element method. This discretization has been applied to both the primal and the dual formulation, in order to make a number of observations and comparisons: 1. The rate of convergence as the grid interval h is decreased, for polynomial elements of different degrees. 2. The presence or absence of a boundary layer in the error, concentrated at the "contact points" where the constraints change between binding and nonbinding. 3. The advantages of simpler constraints in the dual formulation, and the disadvantages of replacing strict convexity by ordinary convexity. 4. The numerical efficiency of each possible variation in achieving an approximate solution of reasonable accuracy. We concluded that in our model problems, linear elements and the dual method provide the most efficient combination.