Thesis

Positivity-preserving discretisations on general meshes

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Awarding institution
  • University of Strathclyde
Date of award
  • 2026
Thesis identifier
  • T17593
Person Identifier (Local)
  • 202291444
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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 modelswhere positivity of the solution is a key physical requirement. Examples include nonlinear reaction-diffusionsystems 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] tovarious settings. First, we develop a method for the steady-state convection-diffusion equation and establishits 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 itseffectiveness 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 ongoingdevelopment of stable and accurate finite element techniques for PDEs, ensuring solutions remain physicallymeaningful within computational simulations.
Advisor / supervisor
  • Barrenechea, Gabriel R.
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