Thesis

Positivity-preserving discretisations on general meshes

Downloadable Content

Download PDF
Creator
Rights statement
Awarding institution
  • University of Strathclyde
Date of award
  • 2026
Thesis identifier
  • T17593
Person Identifier (Local)
  • 202291444
Qualification Level
Qualification Name
Department, School or Faculty
Abstract
  • This thesis focuses on the development and analysis of bound-preserving finite element methods for solving partial differential equations (PDEs), particularly convection-diffusion and reaction-diffusion problems. Bound-preserving methods are crucial for ensuring numerical stability and accuracy, especially in models where positivity of the solution is a key physical requirement. Examples include nonlinear reaction-diffusion systems modeling chemical concentrations, phase-field equations with global extrema constraints, and turbulence models (i.e. see [53,112]). Violations of these bounds can lead to unphysical solutions and instabilities, particularly in coupled systems where errors may propagate and amplify [66, 81]. To address these challenges, we extend the bound-preserving finite element method introduced in [12] to various settings. First, we develop a method for the steady-state convection-diffusion equation and establish its well-posedness and error estimates. Next, we extend this approach to time-dependent reaction-convectiondiffusion equations, proving stability and error bounds for the implicit Euler time-stepping scheme. Finally, we adapt the method for polytopic meshes within the discontinuous Galerkin framework, demonstrating its effectiveness regardless of the geometry of the mesh. The thesis presents mathematical analysis, including well-posedness proofs and error estimates, alongside numerical experiments that validate the proposed methods. These results contribute to the ongoing development of stable and accurate finite element techniques for PDEs, ensuring solutions remain physically meaningful within computational simulations.
Advisor / supervisor
  • Barrenechea, Gabriel R.
Resource Type
DOI
Funder

Relations

Items

Thumbnail Title Date Uploaded Visibility Actions
file details PDF of thesis T17593 2026-02-17 Public Download