A finite element scheme for an initial value problem

TL;DR

This paper develops a finite element scheme based on a convolutional extremum principle for initial value problems, demonstrating its effectiveness through tests on harmonic oscillator models for free and forced vibrations.

Contribution

It introduces a novel finite element formulation using a convolutional Hamilton principle tailored for dynamical initial value problems.

Findings

The scheme accurately models free and forced vibrations.

A recurrent algorithm for IVP solutions is derived from the local analysis.

Computational tests validate the scheme's effectiveness.

Abstract

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.