  Over the past decade, Finite Element Method (FEM) has served as a
foundational numerical framework for approximating the terms of Time Series
Expansion (TSE) as solutions to transient Partial Differential Equation (PDE).
However, the application of high-order Finite Element (FE) to certain classes
of PDEs, such as diffusion equations and the Navier-Stokes (NS) equations,
often leads to numerical instabilities. These instabilities limit the number of
valid terms in the series, though the efficiency of time series integration
even when resummation techniques like the Borel-Pad\'e-Laplace (BPL)
integrators are employed. In this study, we introduce a novel variational
formulation for computing the terms of a TSE associated with a given PDE using
higher-order FEs. Our approach involves the incorporation of artificial
diffusion terms on the left-hand side of the equations corresponding to each
power in the series, serving as a stabilization technique. We demonstrate that
this method can be interpreted as a minimization of an energy functional,
wherein the total variations of the unknowns are considered. Furthermore, we
establish that the coefficients of the artificial diffusion for each term in
the series obey a recurrence relation, which can be determined by minimizing
the condition number of the associated linear system. We highlight the link
between the proposed technique and the Discrete Maximum Principle (DMP) of the
heat equation. We show, via numerical experiments, how the proposed technique
allows having additional valid terms of the series that will be substantial in
enlarging the stability domain of the BPL integrators.
