  A new Hamilton principle of convolutional type, completely compatible with
the initial conditions of an IVP, has been proposed in a recent publication
arXiv:1912.08490v1 [math-ph]. In the present paper the possible use of this
principle for the formulation of a FE scheme adjusted to dynamical problems is
investigated. To this end, a FE scheme based on a convolutional extremum
principle for the harmonic oscillator (used as an exemplary initial value
problem) is developed and presented in detail. Besides, from the local finite
element analysis a recurrent (one-step) algorithm arises which provides an
approximate solution to the IVP, as well.T he succeeded schemes are
computationally tested for both free and forced vibration problems.
