I am aware that the classical way for initial-boundary-value problems is a time-stepping through a sequence of spatial boundary-value problems. Still in terms of global error it may be interesting to formulate such a problem by a space-time FE and apparently it needs only a change in the application of the boundary conditions (not at begin and end, but only at begin, but both initial position and initial velocity).
For instance, let’s think of the hyperbolic wave equation in 1D
u_{tt}-u_{xx}=0 on a space time domain [0,T]\times[0,L].
Then I would apply the boundary conditions at x=0 and x=L as usual, similarly I would set the initial position at t=0, the question is how to disable the BC at t=T (maybe by a dot product of gradient and normal) and most important impose u_t(x, t=0) instead, either by specifiying a time derivative at t=0 or by another position at time t=\Delta t matching the initial velocity?
Could this be done by a manual assembly/manipulation of the system matrix?