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Stabilized Time Series Expansions for High-Order Finite Element Solutions of Partial Differential Equations

Ahmad Deeb, Denys Dutykh

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.

Stabilized Time Series Expansions for High-Order Finite Element Solutions of Partial Differential Equations

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.
Paper Structure (25 sections, 97 equations, 15 figures, 8 tables, 1 algorithm)

This paper contains 25 sections, 97 equations, 15 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Resummation techniques in case of ODEs
  3. Partial sum
  4. Borel-Padé-Laplace integrator
  5. Continuation
  6. Problem in FEM discretization of parabolic PDEs
  7. FEMF
  8. Case study: heat equation
  9. Improving the stability with mass lumping
  10. Understanding the error amplification
  11. Convergence error in $\mathrm{u}^h_1$
  12. Condition number of mass matrix
  13. Error propagation along $\mathrm{u}_k$
  14. The idea
  15. Optimal choice of $\alpha_k$
  16. ...and 10 more sections

Figures (15)

  • Figure 1: Illustration, in the Borel space, of convergence domain, singularity and a sector for analytical prolongation.
  • Figure 2: Illustration of the continuation process.
  • Figure 3: Absolute errors and their slopes $\mathfrak{s}({h,p})$, in the logarithmic scale, for different order $p$ and different mesh sizes for the specific case
  • Figure 4: The errors in the terms of the series with lumping technique for the first and second order FE, before stabilization technique.
  • Figure 5: Plot of the convergence error ${e}^{h,p}_1$ with respect to $h$ and for different degrees $p$.
  • ...and 10 more figures

Theorems & Definitions (4)

  • remark 1
  • remark 2
  • remark 3
  • remark 4