Stabilized time-series expansions for high-order finite element solutions of partial differential equations

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    Abstract

    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–Padé–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.

    Original languageBritish English
    Article numbere12708
    JournalStudies in Applied Mathematics
    Volume153
    Issue number2
    DOIs
    StatePublished - Aug 2024

    Keywords

    • divergent series resummation
    • finite element method
    • stabilization technique
    • time integration
    • time-series expansion

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