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Time integration and the Trefftz Method Part I - First-order and parabolic problems

The finite element method is applied in the time domain to establish formulations for the integration of first-order and parabolic (transient) problems. The modal decomposition concept is applied using two distinct approaches. The first is based on modal decomposition in the space domain to recover the well-established method for uncoupling the parabolic system of equations. To overcome the limitations of this approach in the implementation of large-scale, non-linear problems, the second approach that is reported consists in inducing uncoupling through modal decomposition in the time domain without using the periodic approximations that characterise analyses in the frequency domain. The methods of modal decomposition are related with the implementation of the Trefftz concept in both time and space.