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

• Published in: Journal of computational and applied mathematics Vol. 470; p. 116691
• Main Authors: Amiri, Abdolreza, Barrenechea, Gabriel R., Pryer, Tristan
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.12.2025
• Subjects: Positivity preservation; Stabilised finite-element approximation; Time-dependent convection–diffusion equation; Variational inequality; Time-dependent convection–diffusion equation; Variational inequality; Stabilised finite-element approximation; Positivity preservation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2025.116691

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. •Positivity-preserving method for time-dependent convection–diffusion problems.•Physical bounds enforced using convex admissible sets at each time step.•Stability and error estimate proven for the implicit Euler method.•Crank–Nicolson and Euler schemes yield stable, convergent numerical results.