A nodally bound-preserving finite element method for time-dependent convection–diffusion equations
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 conv...
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| Published in | Journal of computational and applied mathematics Vol. 470; p. 116691 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
15.12.2025
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| Online Access | Get full text |
| ISSN | 0377-0427 1879-1778 |
| DOI | 10.1016/j.cam.2025.116691 |
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| 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.
•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. |
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| AbstractList | 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. |
| ArticleNumber | 116691 |
| Author | Barrenechea, Gabriel R. Amiri, Abdolreza Pryer, Tristan |
| Author_xml | – sequence: 1 givenname: Abdolreza surname: Amiri fullname: Amiri, Abdolreza email: abdolreza.amiri@strath.ac.uk organization: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, G1 1XH, Glasgow, Scotland, United Kingdom – sequence: 2 givenname: Gabriel R. surname: Barrenechea fullname: Barrenechea, Gabriel R. organization: Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, G1 1XH, Glasgow, Scotland, United Kingdom – sequence: 3 givenname: Tristan surname: Pryer fullname: Pryer, Tristan organization: Department of Mathematical Sciences, University of Bath, Claverton down, Bath, BA2 7AY, UK |
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| Cites_doi | 10.1007/s00211-016-0808-z 10.1142/S0218202524500283 10.1016/j.jcp.2013.01.052 10.1016/j.cma.2004.01.026 10.1016/0045-7825(82)90071-8 10.1007/s00211-018-0955-5 10.1090/S0025-5718-99-01148-5 10.1016/0045-7825(85)90089-1 10.1093/imanum/drad055 10.1137/0727022 10.1137/22M1488934 10.1016/j.cma.2009.11.023 10.1090/S0025-5718-05-01761-8 10.1016/0045-7825(73)90019-4 10.1016/j.cma.2009.02.011 10.1137/0733033 10.1016/j.jcp.2008.12.011 10.1016/j.cma.2003.12.032 10.1017/S0004972700027349 10.1016/0045-7825(89)90111-4 10.1051/m2an:1999145 |
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| Keywords | Time-dependent convection–diffusion equation Variational inequality Stabilised finite-element approximation Positivity preservation |
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| SubjectTerms | Positivity preservation Stabilised finite-element approximation Time-dependent convection–diffusion equation Variational inequality |
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| Title | A nodally bound-preserving finite element method for time-dependent convection–diffusion equations |
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