Finite element and Runge-Kutta methods for boundary-value and optimal control problems

This study addresses a class of finite element in time (FET) and some implicit Runge-Kutta (RK) schemes for the solution of boundary-value problems. It is demonstrated that although the two approaches look different, the discrete equations arising from the class of FETs are linear combinations of the equations generated by some implicit RK schemes and that the FET unknowns are linear combinations of the RK unknowns within each time step.