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A nodally bound-preserving finite element method for time-dependent convection-diffusion equations

Abdolreza Amiri, Gabriel R. Barrenechea, Tristan Pryer

TL;DR

This paper tackles the challenge of numerically solving time-dependent convection-diffusion equations while preserving physical bounds of the solution. It introduces a nodally bound-preserving finite element method that, at each time step, projects the solution onto an admissible set defined by nodal bounds and stabilises with a continuous interior penalty term; this framework is analysed for well-posedness and provides stability and optimal-order error estimates for the implicit Euler time discretisation, with strong numerical evidence of bound preservation for Crank-Nicolson. The key contributions include the construction of the admissible set, the projection-based discretisation, and the rigorous stability and error theory, complemented by comprehensive numerical experiments showing robustness in convection-dominated regimes and compatibility with non-Delaunay meshes. The method avoids CFL-type restrictions and offers a practically effective way to enforce global bounds without sacrificing high-order accuracy or requiring mesh refinements for stability. Overall, the approach advances reliable, physically consistent simulations of time-dependent transport phenomena in settings where conventional Galerkin discretisations produce spurious oscillations.

Abstract

This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential equation. The method is built by defining, at each time step, a convex set of admissible finite element functions (that is, the ones that satisfy the global bounds at their degrees of freedom) and seeks for a discrete solution in this admissible set. A family of $θ$-schemes is used as time integrators, and well-posedness of the discrete schemes is proven for the whole family, but stability and optimal-order error estimates are proven for the implicit Euler scheme. Nevertheless, our numerical experiments show that the method also provides stable and optimally-convergent solutions when the Crank-Nicolson method is used.

A nodally bound-preserving finite element method for time-dependent convection-diffusion equations

TL;DR

This paper tackles the challenge of numerically solving time-dependent convection-diffusion equations while preserving physical bounds of the solution. It introduces a nodally bound-preserving finite element method that, at each time step, projects the solution onto an admissible set defined by nodal bounds and stabilises with a continuous interior penalty term; this framework is analysed for well-posedness and provides stability and optimal-order error estimates for the implicit Euler time discretisation, with strong numerical evidence of bound preservation for Crank-Nicolson. The key contributions include the construction of the admissible set, the projection-based discretisation, and the rigorous stability and error theory, complemented by comprehensive numerical experiments showing robustness in convection-dominated regimes and compatibility with non-Delaunay meshes. The method avoids CFL-type restrictions and offers a practically effective way to enforce global bounds without sacrificing high-order accuracy or requiring mesh refinements for stability. Overall, the approach advances reliable, physically consistent simulations of time-dependent transport phenomena in settings where conventional Galerkin discretisations produce spurious oscillations.

Abstract

This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential equation. The method is built by defining, at each time step, a convex set of admissible finite element functions (that is, the ones that satisfy the global bounds at their degrees of freedom) and seeks for a discrete solution in this admissible set. A family of -schemes is used as time integrators, and well-posedness of the discrete schemes is proven for the whole family, but stability and optimal-order error estimates are proven for the implicit Euler scheme. Nevertheless, our numerical experiments show that the method also provides stable and optimally-convergent solutions when the Crank-Nicolson method is used.

Paper Structure

This paper contains 11 sections, 7 theorems, 87 equations, 10 figures.

Table of Contents

  1. Introduction
  2. General setting and the model problem
  3. The model problem
  4. Space discretisation
  5. The admissible set
  6. The finite element method
  7. Well-posedness
  8. Stability and Error analysis
  9. Numerical experiments
  10. Conclusions and outlook
  11. Acknowledgements

Key Result

Lemma 2.2

Let the operator $(\cdot)^{+}$ be defined in posoperator. There exists a constant $C>0$, independent of $h$, such that for all $t\in [0,T]$ the following holds for all $w_{h},v_{h}\in V_{\mathcal{P}}$.

Figures (10)

  • Figure 1: Three coarse level indicative meshes used in the experiments all with $N=5$.
  • Figure 2: Comparison of the error of the approximated solution by the BP-Euler method and BP-CN method with the exact solution in $\parallel \cdot\parallel_{0,\Omega}$-norm (using mesh \ref{['e11']}).
  • Figure 3: Using norm (\ref{['norm22']}) for the comparison of the error of the approximated solution by the BP-Euler method with the exact solution (using mesh \ref{['e11']}).
  • Figure 4: The average number of the Richardson iterations of 1000 time steps ($T=1$) needed to reach convergence using $\mathbb{P}_{1}$ and $\mathbb{Q}_{1}$ elements and BP-Euler and BP-CN methods and the meshes \ref{['e11']} and \ref{['Q11']}.
  • Figure 5: Initial data $u^{0}$ for rotating body problem.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Lemma 4.1
  • Lemma 4.2
  • ...and 5 more