Finite Element and Runge-Kutta Methods for Boundary-Value and Optimal Control Problems

Introduction W E consider a class of Ž nite element in time (FET) and some implicit Runge–Kutta (RK) schemes for the solution of boundary-valueproblems.FETs have been recently advocatedas a new way of solving this class of problems,with particular emphasis on optimal control problems (for example, see Ref. 1 and the references therein). The Ž nite element method develops solutions discretizing appropriate weak forms of the equations that describe a given problem. On the other hand, the RK method directly discretizes the governing ordinary differential equations (ODEs). Although the two approacheslook different, in this work we prove that the discreteequationsarisingfrom the classofFETs here considered are linear combinationsof the equationsgeneratedby some implicit RK schemes and that the FET unknowns are linear combinations of the RK unknowns within each time step. Under these circumstances, this means that the two approaches are in reality the same method that yields identical numerical solutions and, hence, enjoys the same numerical properties. The analysis is valid for the p version of the method, that is, for arbitrarily high order. Similar results were derived in Ref. 2 for initial-valueproblems. That work is here extended to cover the case of boundary-valuedifferential-algebraic problems. RK methods are probably the best understood and most widely studied family of integration schemes, which makes them a mature and trusted technology with extensive applications to the class of problems here considered. FETs are less widely known, but their application to certain problems leads to strikingly simple solution procedures,for example, in the case of optimal control. Our hope is that the proof of equivalencehere offered might help close the gulf existing between practitioners using the two methods. We believe that additional insight can be usually obtained by a uniŽ ed view, and we are, therefore, not suggesting to abandon one approach in favor of the other. Furthermore, some developments, for example a posteriori error estimation for adaptive mesh control, might be easier to accomplish in one framework rather than in the other.4